Asymptotic behaviour of Christoffel-Darboux kernel via three-term recurrence relation I
aa r X i v : . [ m a t h . SP ] S e p ASYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL VIA THREE-TERMRECURRENCE RELATION I
GRZEGORZ ŚWIDERSKI AND BARTOSZ TROJANAbstract. F or Jacobi parameters belonging to one of the three classes: asymptotically periodic, periodicallymodulated and the blend of these two, we study the asymptotic behavior of the Christoffel functions and thescaling limits of the Christoffel–Darboux kernel. We assume regularity of Jacobi parameters in terms of theStolz class. We emphasize that the first class only gives rise to measures with compact supports. Introduction
Let µ be a probability measure on the real line with infinite support such that for every n ∈ N ,the moments ∫ R x n d µ ( x ) are finite . Let L ( R , µ ) be the Hilbert space of square-integrable functions equipped with the scalar product h f , g i = ∫ R f ( x ) g ( x ) d µ ( x ) . By performing the Gram–Schmidt orthogonalization process on the sequence of monomials ( x n : n ∈ N ) one obtains the sequence of polynomials ( p n : n ∈ N ) satisfying(1.1) h p n , p m i = δ nm where δ nm is the Kronecker delta. Moreover, ( p n : n ∈ N ) satisfies the following recurrence relation(1.2) p ( x ) = , p ( x ) = x − b a , xp n ( x ) = a n p n + ( x ) + b n p n ( x ) + a n − p n − ( x ) , n ≥ where a n = h xp n , p n + i , b n = h xp n , p n i , n ≥ . Notice that for every n , a n > and b n ∈ R . The pair ( a n ) and ( b n ) is called the Jacobi parameters.Let P n be the orthogonal projection in L ( R , µ ) on the space of polynomials of degree at most n . Then P n is given by the Christoffel–Darboux kernel K n , that is P n f ( x ) = ∫ R K n ( x , y ) f ( y ) d µ ( y ) where from (1.1) one can verify that(1.3) K n ( x , y ) = n Õ j = p j ( x ) p j ( y ) . To motivate the study of Christoffel–Darboux kernels see surveys [12] and [18].
Mathematics Subject Classification.
Primary: 42C05, 47B36.
Key words and phrases.
Orthogonal polynomials, asymptotics, Christoffel functions, Scaling limits.
The asymptotic behavior of K n is well understood in the case when the measure µ has compact support. Inthis setup one of the most general results has been proven in [26]. Namely, if I is an open interval containedin supp ( µ ) such that µ is absolutely continuous on I with continuous positive density µ ′ , then(1.4) lim n →∞ n K n (cid:16) x + un , x + v n (cid:17) = ω ′ ( x ) µ ′ ( x ) sin (cid:0) ( u − v ) ω ′ ( x ) (cid:1) ( u − v ) ω ′ ( x ) locally uniformly with respect to x ∈ I and u , v ∈ R , provided that µ is regular (see [19, Definition 3.1.2]).In the formula (1.4), ω ′ denotes the density of the equilibrium measure corresponding to the support of µ ,see (2.4) for details. In the case when supp ( µ ) is a finite union of compact intervals, µ is regular providedthat µ ′ > almost everywhere in the interior of supp ( µ ) . To give some historical perspective, let us alsomention three earlier results. The case supp ( µ ) = [− , ] and u = v = has been examined in [14, Theorem8], and its extension to a general compact supp ( µ ) has been obtained in [25, Theorem 1]. The extension toall u , v ∈ R has been proven in [11].The best understood class of measures with unbounded support is the class of exponential weights (seethe monograph [8]). In [10] and [9, Theorem 7.4] under a number of regularity conditions on the function Q ( x ) = − log µ ′ ( x ) , the following analogue of (1.4) was shown(1.5) lim n →∞ K n ( x , x ) ˜ K n (cid:16) x + u ˜ K n ( x , x ) , x + v ˜ K n ( x , x ) (cid:17) = sin ( u − v ) u − v locally uniformly with respect to x , u , v ∈ R where ˜ K n ( x , y ) = p µ ′ ( x ) µ ′ ( y ) K n ( x , y ) . Unlike (1.4), the formula(1.5) does not give any information if u = v = . It was recently proved in [4] that under some additionalregularity on Q (1.6) lim n →∞ ρ n K n ( x , x ) = π µ ′ ( x ) locally uniformly with respect to x ∈ R where ρ n = n Õ j = a j . By combining (1.5) with (1.6) one obtains(1.7) lim n →∞ ρ n K n (cid:16) x + u ρ n , x + v ρ n (cid:17) = ω ′ ( ) µ ′ ( x ) sin (cid:0) ( u − v ) ω ′ ( ) (cid:1) ( u − v ) ω ′ ( ) where ω ′ ( x ) = (cid:0) π √ − x (cid:1) − is the density of the equilibrium measure for the interval [− , ] .Instead of taking the measure µ as the starting point one can consider polynomials ( p n : n ∈ N ) satisfyingthe three-term recurrence relation (1.2) with a n > and b n ∈ R for any n ∈ N . Then the Favard’s theorem(see, e.g. [17, Theorem 5.10]) states that there is a probability measure µ such that ( p n ) is orthonormal in L ( R , µ ) . The measure µ is unique, if and only if there is exactly one measure with the same moments as µ .It is always the case when the Carleman condition(1.8) ∞ Õ n = a n = ∞ is satisfied (see, e.g. [17, Corollary 6.19]). Moreover, the measure µ has compact support, if and only if theJacobi parameters are bounded.In this article our starting point is the three-term recurrence relation. We study analogues of (1.6) and(1.7) for three different classes of Jacobi parameters: asymptotically periodic, periodically modulated anda blend of these two; for the definitions, see Sections 3.1, 3.2 and 3.3, respectively. The first class onlygives rise to measures with compact supports. The second class introduced in [5] has the Jacobi parametersuniformly unbounded in the sense that lim inf a n = ∞ . The third class has been studied in [1] as an exampleof unbounded Jacobi parameters corresponding to measures with absolutely continuous parts having supportsequal a finite union of closed intervals. SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 3
To simplify the exposition in the introduction, we shall focus on the periodic modulations only. Beforewe formulate our results, let us state some definitions. Let N be a positive integer. We say that sequences ( a n ) , ( b n ) are N -periodically modulated if there are two N -periodic sequences ( α n : n ∈ Z ) and ( β n : n ∈ Z ) of positive and real numbers, respectively, such that(a) lim n →∞ a n = ∞ , (b) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) a n − a n − α n − α n (cid:12)(cid:12)(cid:12)(cid:12) = , (c) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) b n a n − β n α n (cid:12)(cid:12)(cid:12)(cid:12) = . The crucial rôle is played by the N -step transfer matrices defined by X n ( x ) = B n + N − ( x ) B n + N − ( x ) · · · B n ( x ) where B j ( x ) = − a j − a j x − b j a j ! , and X n ( x ) = B n + N − ( x ) B n + N − · · · B n ( x ) where B j ( x ) = − α j − α j x − β j α j ! . The name is justified by the following property (cid:18) p n + N − ( x ) p n + N ( x ) (cid:19) = X n ( x ) (cid:18) p n − ( x ) p n ( x ) (cid:19) . Let r be a positive integer. We say that the sequence ( x n : n ∈ N ) of vectors from a normed space V belongsto D r ( V ) , if it is bounded and for each j ∈ { , . . . , r } , ∞ Õ n = (cid:13)(cid:13) ∆ j x n (cid:13)(cid:13) rj < ∞ where ∆ x n = x n , ∆ j x n = ∆ j − x n + − ∆ j − x n , n ≥ . If X is the real line with an Euclidean norm we shortly write D r = D r ( X ) . Given a compact set K ⊂ C anda vector space R , by D r ( K , R ) we denote the case when X is the space of all continuous mappings from K to R equipped with the supremum norm.Our first result is the following theorem, see Theorem 4.4. Theorem A.
Let N and r be positive integers and i ∈ { , , . . . , N − } . Suppose that K is a compact intervalwith non-empty interior contained in Λ = (cid:26) x ∈ R : lim j →∞ discr X j N + i ( x ) exists and is negative (cid:27) . Assume that lim j →∞ a ( j + ) N + i − a j N + i − = and (cid:0) X j N + i : j ∈ N (cid:1) ∈ D r (cid:0) K , GL ( , R ) (cid:1) . Suppose that X is the limit of (cid:0) X j N + i : j ∈ N (cid:1) . If ∞ Õ j = a j N + i − = ∞ , A discriminant of a × matrix X is discr X = ( tr X ) − X . GRZEGORZ ŚWIDERSKI AND BARTOSZ TROJAN then for x ∈ K (cid:18) ∞ Õ j = a j N + i − (cid:19) − n Õ j = p j N + i ( x ) = |[X( x )] , | π µ ′ ( x ) p − discr X( x ) + E n ( x ) where lim n →∞ sup x ∈ K (cid:12)(cid:12) E n ( x ) (cid:12)(cid:12) = . Let us remark that in Theorem 5.4 we have obtained the quantitative bound on E n in the D setting.Theorem A is an important step in proving the analogues of (1.6) and (1.7). The following theorem (seeTheorem 4.6) provides the analogue of (1.6) for periodic modulations. Similar results are obtained also forthe remaining classes, that is for asymptotically periodic in Theorem 4.7, and for the blend in Theorem 4.10. Theorem B.
Let ( a n ) and ( b n ) be N -periodically modulated Jacobi parameters. Suppose that there is r ≥ such that for every i ∈ { , , . . . , N − } , (cid:18) a k N + i − a k N + i : k ∈ N (cid:19) , (cid:18) b k N + i a k N + i : k ∈ N (cid:19) , (cid:18) a k N + i : k ∈ N (cid:19) ∈ D r , and the Carleman condition (1.8) is satisfied. If | tr X ( )| < , then ρ n K n ( x , x ) = ω ′ ( ) µ ′ ( x ) + E n ( x ) where lim n →∞ | E n ( x )| = locally uniformly with respect to x ∈ R , where ω is the equilibrium measure of (cid:8) x ∈ R : | tr X ( x )| ≤ (cid:9) and ρ n = n Õ j = α j a j . Again, in D setup, we obtained the quantitative bound on E n , see Theorem 5.5 (periodic modulations)and Theorem 5.7 (asymptotically periodic).We emphasize that Theorem B solves [3, Conjecture 1] for a larger class of Jacobi parameters than it wasoriginally stated, see Section 4.3 for details.Lastly, we provide the analogue of (1.7) for periodic modulations (see Theorem 5.13). In view of [24,Corollary 7], the asymptotically periodic case follows from [26]. For the blend, see Theorem 5.15. Theorem C.
Suppose that the hypotheses of Theorem B are satisfied for r = . Then lim n →∞ ρ n K n (cid:16) x + u ρ n , x + v ρ n (cid:17) = ω ′ ( ) µ ′ ( x ) sin (cid:0) ( u − v ) ω ′ ( ) (cid:1) ( u − v ) ω ′ ( ) locally uniformly with respect to x , u , v ∈ R . Let us present some ideas of the proofs. The basic strategy commonly used is to exploit the Christoffel–Darboux formula, that is K n ( x , y ) = a n p n + ( x ) p n ( y ) − p n ( x ) p n + ( y ) x − y , if x , y , p n ( x ) p ′ n + ( x ) − p ′ n ( x ) p n + ( x ) , otherwise.However, it requires the precise asymptotic of the polynomials as well as its derivatives in terms of both n and x . Unfortunately, for the classes of Jacobi parameters we are interested in they are not available. In therecent article [24], we managed to obtain the asymptotic of ( p n ( x ) : n ∈ N ) locally uniformly with respect SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 5 to x . Based on it we develop a method to study K n ( x , y ) . Namely, we use the formula (1.3), which leads tothe need of estimation of the oscillatory sums of a form n Õ k = γ k Í nj = γ j sin (cid:16) n Õ j = θ j ( x n ) + σ ( x n ) (cid:17) sin (cid:16) n Õ j = θ j ( y n ) + σ ( y n ) (cid:17) where x n = x + u ρ n , and y n = x + v ρ n . To deal with the sums we prove two auxiliary results (see Lemma 4.1 and Theorem 5.9) that are valid forsequences not necessarily belonging to D r .The organization of the article is as follows. In Section 2 we present basic definitions used in the article.In Section 3 we collected the definitions and basic properties of the three classes of sequences. Section 4 isdevoted to the general D r setting. In particular, we present there the proofs of Theorem A and B, and weprovide a solution of Ignjatović conjecture. In Section 5 we study the case D where we derive quantitativebound on the error in the asymptotic of the polynomials. We also provide the quantitative versions ofTheorems A and B. Finally, we prove Theorem C. Notation. By N we denote the set of positive integers and N = N ∪ { } . Throughout the whole article,we write A . B if there is an absolute constant c > such that A ≤ cB . Moreover, c stands for a positiveconstant which value may vary from occurrence to occurrence. Acknowledgement.
The first author was partialy supported by the Foundation for Polish Science (FNP).
2. Definitions
Given two sequences a = ( a n : n ∈ N ) and b = ( b n : n ∈ N ) of positive and real numbers, respectively,and k ∈ N , we define k th associated orthonormal polynomials as p [ k ] ( x ) = , p [ k ] ( x ) = x − b k a k , a n + k − p [ k ] n − ( x ) + b n + k p [ k ] n ( x ) + a n + k p [ k ] n + ( x ) = xp [ k ] n ( x ) , ( n ≥ ) , For k = we usually omit the superscript. A sequence ( u n : n ∈ N ) is a generalized eigenvector associatedto x ∈ C , if for all n ≥ , (cid:18) u n u n + (cid:19) = B n ( x ) (cid:18) u n − u n (cid:19) where B n ( x ) = (cid:18) − a n − a n x − b n a n (cid:19) . Let A be the closure in ℓ of the operator acting on sequences having finite support by the matrix(2.1) © « b a . . . a b a . . . a b a . . . a b ... ... ... . . . ª®®®®®®¬ . The operator A is called Jacobi matrix . If the Carleman condition(2.2) ∞ Õ n = a n = ∞ GRZEGORZ ŚWIDERSKI AND BARTOSZ TROJAN is satisfied then the operator A has the unique self-adjoint extension (see e.g. [17, Corollary 6.19]). Let usdenote by E A its spectral resolution of the identity. Then for any Borel subset B ⊂ R , we set µ ( B ) = h E A ( B ) δ , δ i ℓ where δ is the sequence having on the th position and elsewhere. The polynomials ( p n : n ∈ N ) forman orthonormal basis of L ( R , µ ) .In this article the central object is the Christoffel–Darboux kernel defined as(2.3) K n ( x , y ) = n Õ j = p j ( x ) p j ( y ) . We are interested in the class of Jacobi matrices associated to slowly oscillating sequences introduced in [20].We say that a bounded sequence ( x n : n ∈ N ) of elements from a normed space X belongs to D r ( X ) forsome r ≥ , if for each j ∈ { , , . . . , r } , ∞ Õ n = (cid:13)(cid:13) ∆ j x n (cid:13)(cid:13) rj < ∞ where ∆ x n = x n , ∆ j x n = ∆ j − x n + − ∆ j − x n , n ≥ . To simplify the notation, if X is the real line with the Euclidean norm, we shortly write D r = D r ( R ) . Givena compact set K ⊂ C and a vector space R , by D r ( K , R ) we denote the case when X is the space of allcontinuous mappings from K to R equipped with the supremum norm.Given a compact set K ⊂ R with non-empty interior, there is the unique probability measure ω K , called the equilibrium measure corresponding to K , minimizing the energy(2.4) I ( ν ) = − ∫ R ∫ R log | x − y | ν ( d x ) ν ( d y ) , among all probability measures ν supported on K . The measure ω K is absolutely continuous in the interiorof K with continuous density, see [15, Theorem IV.2.5, pp. 216].
3. Classes of sequences
In this article we are interested in Jacobi matrices having entries in one of the three classes defined interms of periodic sequences. Let us start by fixing some notation.By ( α n : n ∈ Z ) and ( β n : n ∈ Z ) we denote N -periodic sequences of real and positive numbers,respectively. For each k ≥ , let us define polynomials ( p [ k ] n : n ∈ N ) by relations p [ k ] ( x ) = , p [ k ] ( x ) = x − β k α k ,α n + k − p [ k ] n − ( x ) + β n + k p [ k ] n ( x ) + α n + k p [ k ] n + ( x ) = x p [ k ] n ( x ) , n ≥ . Let B n ( x ) = (cid:18) − α n − α n x − β n α n (cid:19) , and X n ( x ) = N + n − Ö j = n B j ( x ) , n ∈ Z where for a sequence of square matrices ( C n : n ≤ n ≤ n ) we set n Ö k = n C k = C n C n − · · · C n . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 7 By A we denote the Jacobi matrix corresponding to © « β α . . .α β α . . . α β α . . . α β ... ... ... . . . ª®®®®®®¬ . Let ω be the equilibrium measure corresponding to σ ess ( A ) . Since σ ess ( A ) is a finite union of compact in-tervals with non-empty interiors (see e.g. [2, Theorem 5.2.4 and Theorem 5.4.2]), ω is absolutely continuous.Moreover, for x in the interior of σ ess ( A ) we obtain(3.1) ω ′ ( x ) = N N − Õ i = |[ X i ( x )] , | π p − discr X i ( x ) · α i − . Indeed, by [22, Proposition 3] (see also Lemma 3.2 below), [ X i ( x )] , = − α i − α i p [ i + ] N − ( x ) . Since X i + ( x ) = (cid:0) B i ( x ) (cid:1) (cid:0) X i ( x ) (cid:1) (cid:0) B i ( x ) (cid:1) − , we have discr X i ( x ) = discr X ( x ) . Therefore, N N − Õ i = |[ X i ( x )] , | π p − discr X i ( x ) · α i − = π p − discr X ( x ) N N − Õ i = (cid:12)(cid:12) p [ i + ] N − ( x ) (cid:12)(cid:12) α i . Let us recall that the first formula on page 214 of [27] reads N − Õ i = (cid:12)(cid:12) p [ i + ] N − ( x ) (cid:12)(cid:12) α i = (cid:12)(cid:12) tr X ′ ( x ) (cid:12)(cid:12) , hence(3.2) N N − Õ i = |[ X i ( x )] , | π p − discr X i ( x ) · α i − = (cid:12)(cid:12) tr X ′ ( x ) (cid:12)(cid:12) π N p − discr X ( x ) . Now, in view of [27, the formula (3.2)] we obtain (3.1).3.1.
Asymptotically periodic.Definition 3.1.
The Jacobi matrix A associated to ( a n : n ∈ N ) and ( b n : n ∈ N ) has asymptotically N -periodic entries , if there are two N -periodic sequences ( α n : n ∈ Z ) and ( β n : n ∈ Z ) of positive and realnumbers, respectively, such that(a) lim n →∞ (cid:12)(cid:12) a n − α n (cid:12)(cid:12) = ,(b) lim n →∞ (cid:12)(cid:12) b n − β n (cid:12)(cid:12) = .Let us recall the following lemma. Lemma 3.2 ([22, Proposition 3]) . Let ( p n : n ∈ N ) be a sequence of orthonormal polynomials associatedto ( a n : n ∈ N ) and ( b n : n ∈ N ) . Then for all n ≥ and k ≥ , n + k − Ö j = k B j ( x ) = − a k − a k p [ k + ] n − ( x ) p [ k ] n − ( x )− a k − a k p [ k + ] n − ( x ) p [ k ] n ( x ) ! . GRZEGORZ ŚWIDERSKI AND BARTOSZ TROJAN
Proposition 3.3.
Suppose that A has asymptotically N -periodic entries. Then for each i ∈ { , , . . . , N − } and n ≥ , lim k →∞ (cid:13)(cid:13)(cid:13)(cid:13) k N + i + n Ö j = k N + i B j ( x ) − i + n Ö j = i B j ( x ) (cid:13)(cid:13)(cid:13)(cid:13) = , locally uniformly with respect to x ∈ C .Proof. Since for each i ∈ { , , . . . , N − } , we have lim k →∞ (cid:13)(cid:13) B k N + i ( x ) − B i ( x ) (cid:13)(cid:13) = uniformly on compact subsets of C , the conclusion follows by the continuity of B n . (cid:3) For a Jacobi matrix having asymptotically N -periodic entries, we set X n ( x ) = N + n − Ö j = n B j ( x ) . Let us denote by X i the limit of ( X j N + i : j ∈ N ) . Then, by Proposition 3.3, we conclude that X i = X i for all i ∈ { , , . . . , N − } . Proposition 3.4.
Suppose that a Jacobi matrix A has asymptotically N -periodic entries. Then for each i ∈ { , , . . . , N − } , and n ≥ , lim k →∞ p [ k N + i ] n ( x ) = p [ i ] n ( x ) , (3.3a) lim k →∞ a k N + i α i (cid:0) p [ k N + i ] n (cid:1) ′ ( x ) = (cid:0) p [ i ] n (cid:1) ′ ( x ) , (3.3b) lim k →∞ (cid:18) a k N + i α i (cid:19) (cid:0) p [ k N + i ] n (cid:1) ′′ ( x ) = (cid:0) p [ i ] n (cid:1) ′′ ( x ) , (3.3c) locally uniformly with respect to x ∈ C .Proof. Lemma 3.2 together with Proposition 3.3 easily gives (3.3a). Since lim k →∞ a k N + i α i = , the uniform convergence in (3.3a) entails (3.3b) and (3.3c). (cid:3) Corollary 3.5.
Suppose that a Jacobi matrix A has asymptotically N -periodic entries. Then for each i ∈ { , , . . . , N − } , lim k →∞ tr (cid:0) X k N + i ( x ) (cid:1) = tr (cid:0) X ( x ) (cid:1) , (3.4a) lim k →∞ a k N + i α i tr (cid:0) X ′ k N + i ( x ) (cid:1) = tr (cid:0) X ′ ( x ) (cid:1) , (3.4b) lim k →∞ (cid:18) a k N + i α i (cid:19) tr (cid:0) X ′′ k N + i ( x ) (cid:1) = tr (cid:0) X ′′ ( x ) (cid:1) , (3.4c) locally uniformly with respect to x ∈ C .Proof. Since for x ∈ C , X i + ( x ) = (cid:0) B i ( x ) (cid:1) (cid:0) X i ( x ) (cid:1) (cid:0) B i ( x ) (cid:1) − , we have(3.5) tr (cid:0) X ( s ) i ( x ) (cid:1) = tr (cid:0) X ( s ) ( x ) (cid:1) , for all s ≥ . By Lemma 3.2, tr (cid:0) X k N + i ( x ) (cid:1) = p [ k N + i ] N ( x ) − a k N + i − a k N + i p [ k N + i + ] N − ( x ) , SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 9 which together with Proposition 3.4, implies that lim k →∞ tr (cid:0) X k N + i ( x ) (cid:1) = tr (cid:0) X i ( x ) (cid:1) uniformly on compact subsets of C . Hence, by (3.5) we easily obtain (3.4a).For s ∈ { , } we write a k N + i − a k N + i (cid:16) a k N + i α i (cid:17) s (cid:0) p [ k N + i + ] N − (cid:1) ( s ) ( x ) = a k N + i − a k N + i (cid:16) a k N + i α i (cid:17) s (cid:16) α i + a k N + i + (cid:17) s (cid:16) a k N + i + α i + (cid:17) s (cid:0) p [ k N + i + ] N − (cid:1) ( s ) ( x ) , hence, by Proposition 3.4, we obtain lim k →∞ a k N + i − a k N + i (cid:16) a k N + i α i (cid:17) s (cid:0) p [ k N + i + ] N − (cid:1) ( s ) ( x ) = α i − α i (cid:0) p [ i + ] N − (cid:1) ( s ) ( x ) . Therefore, by Lemma 3.2, lim k →∞ (cid:16) a k N + i α i (cid:17) s tr (cid:0) X ( s ) k N + i ( x ) (cid:1) = (cid:0) p [ i ] N (cid:1) ( s ) ( x ) − α i − α i (cid:0) p [ i + ] N − (cid:1) ( s ) ( x ) = tr (cid:0) X ( s ) i ( x ) (cid:1) , which finishes the proof. (cid:3) Periodic modulations.Definition 3.6.
We say that the Jacobi matrix A associated to ( a n : n ∈ N ) and ( b n : n ∈ N ) has N -periodically modulated entries, if there are two N -periodic sequences ( α n : n ∈ Z ) and ( β n : n ∈ Z ) ofpositive and real numbers, respectively, such that(a) lim n →∞ a n = ∞ , (b) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) a n − a n − α n − α n (cid:12)(cid:12)(cid:12)(cid:12) = , (c) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) b n a n − β n α n (cid:12)(cid:12)(cid:12)(cid:12) = . Suppose that A is a Jacobi matrix with N -periodically modulated entries. Observe that, by setting ˜ a n = a n α n , and ˜ b n = b n α n , we obtain lim n →∞ ˜ a n − ˜ a n = , and ˜ b n ˜ a n = b n a n . Hence, A is N -periodic modulation of the Jacobi matrix corresponding to the sequences ( ˜ a n : n ∈ N )( ˜ b n : n ∈ N ) in the usual sense. Proposition 3.7.
If a Jacobi matrix A has N -periodically modulated entries and i ∈ N , then lim n →∞ α n + i α n a n a n + i = . In particular, lim n →∞ a n a n + N = . Proof.
Since lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) a n a n + − α n α n + (cid:12)(cid:12)(cid:12)(cid:12) = , one has lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) a n a n + i − α n α n + i (cid:12)(cid:12)(cid:12)(cid:12) = lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) i − Ö k = a n + k a n + k + − i − Ö k = α n + k α n + k + (cid:12)(cid:12)(cid:12)(cid:12) = . Hence, for some c > , lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) a n a n + i α n + i α n − (cid:12)(cid:12)(cid:12)(cid:12) = lim n →∞ α n + i α n (cid:12)(cid:12)(cid:12)(cid:12) a n a n + i − α n α n + i (cid:12)(cid:12)(cid:12)(cid:12) ≤ c lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) a n a n + i − α n α n + i (cid:12)(cid:12)(cid:12)(cid:12) = , and the proposition follows. (cid:3) Proposition 3.8.
Suppose that A has N -periodically modulated entries. Then for each i ∈ { , , . . . , N − } and n ≥ , lim k →∞ (cid:13)(cid:13)(cid:13)(cid:13) k N + i + n Ö j = k N + i B j ( x ) − i + n Ö j = i B j ( ) (cid:13)(cid:13)(cid:13)(cid:13) = , locally uniformly with respect to x ∈ C .Proof. Since for each i ∈ { , , . . . , N − } , we have lim k →∞ (cid:13)(cid:13) B k N + i ( x ) − B i ( ) (cid:13)(cid:13) = uniformly on compact subsets of C , the conclusion follows by the continuity of B n . (cid:3) For a Jacobi matrix with N -periodically modulated entries, we set X n ( x ) = N + n − Ö j = n B j ( x ) . Let us denote by X i the limit of ( X j N + i : j ∈ N ) . Then, by Proposition 3.8, we have X i ( x ) = X i ( ) for all i ∈ { , , . . . , N − } and x ∈ C . Proposition 3.9.
Suppose that a Jacobi matrix A has N -periodically modulated entries. Then for each i ∈ { , , . . . , N − } , and n ≥ , lim k →∞ p [ k N + i ] n ( x ) = p [ i ] n ( ) , (3.6a) lim k →∞ a k N + i α i (cid:0) p [ k N + i ] n (cid:1) ′ ( x ) = (cid:0) p [ i ] n (cid:1) ′ ( ) , (3.6b) lim k →∞ (cid:18) a k N + i α i (cid:19) (cid:0) p [ k N + i ] n (cid:1) ′′ ( x ) = (cid:0) p [ i ] n (cid:1) ′′ ( ) , (3.6c) locally uniformly with respect to x ∈ C .Proof. By Lemma 3.2 and Proposition 3.8 we obtain that for every i ≥ and n ≥ ,(3.7) lim k →∞ p [ k N + i ] n ( x ) = p [ i ] n ( ) uniformly on compact subsets of C , which is (3.6a). Next, let us recall that (see e.g. [22, Proposition 2])(3.8) p ′ n ( x ) = a n − Õ m = (cid:16) p m ( x ) p [ ] n − ( x ) − p n ( x ) p [ ] m − ( x ) (cid:17) p m ( x ) , therefore for every n ∈ N , (cid:0) p [ k N + i ] n (cid:1) ′ ( x ) = a k N + i n − Õ m = (cid:16) p [ k N + i ] m ( x ) p [ k N + i + ] n − ( x ) − p [ k N + i ] n ( x ) p [ k N + i + ] m − ( x ) (cid:17) p [ k N + i ] m ( x ) . Hence,(3.9) a k N + i α i (cid:0) p [ k N + i ] n (cid:1) ′ ( x ) = α i n − Õ m = (cid:16) p [ k N + i ] m ( x ) p [ k N + i + ] n − ( x ) − p [ k N + i ] n ( x ) p [ k N + i + ] m − ( x ) (cid:17) p [ k N + i ] m ( x ) . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 11
By (3.7) the right-hand side of (3.9) tends to α i n − Õ m = (cid:16) p [ i ] m ( ) p [ i + ] n − ( ) − p [ i ] n ( ) p [ i + ] m − ( ) (cid:17) p [ i ] m ( ) . Therefore, by (3.8) we conclude (3.6b). For the proof of (3.6c), let us observe that from (3.9) we get(3.10) (cid:16) a k N + i α i (cid:17) (cid:0) p [ k N + i ] n (cid:1) ′′ ( x ) = α i n − Õ m = (cid:16) p [ k N + i ] m ( x ) p [ k N + i + ] n − ( x ) − p [ k N + i ] n ( x ) p [ k N + i + ] m − ( x ) (cid:17) a k N + i α i (cid:0) p [ k N + i ] m (cid:1) ′ ( x ) + α i n − Õ m = a k N + i α i (cid:16) p [ k N + i ] m p [ k N + i + ] n − − p [ k N + i ] n p [ k N + i + ] m − (cid:17) ′ ( x ) p [ k N + i ] m ( x ) . Therefore, by (3.6b) and (3.7), the first sum in (3.10) approaches to(3.11) α i n − Õ m = (cid:16) p [ i ] m ( ) p [ i + ] n − ( ) − p [ i ] n ( ) p [ i + ] m − ( ) (cid:17) (cid:0) p [ i ] m (cid:1) ′ ( ) . In view of (3.6b), for each m ∈ N , lim k →∞ a k N + i α i (cid:0) p [ k N + i + ] m (cid:1) ′ ( x ) = lim k →∞ a k N + i α i α i + a k N + i + a k N + i + α i + (cid:0) p [ k N + i + ] m (cid:1) ′ ( x ) = (cid:0) p [ i + ] m (cid:1) ′ ( ) . (3.12)Hence, by (3.6b), (3.12) and (3.7), we obtain lim k →∞ a k N + i α i (cid:16) p [ k N + i ] m p [ k N + i + ] n − − p [ k N + i ] n p [ k N + i + ] m − (cid:17) ′ ( x ) = (cid:16) p [ k N + i ] m p [ k N + i + ] n − − p [ k N + i ] n p [ k N + i + ] m − (cid:17) ′ ( ) . Consequently, the second sum in (3.10) tends to(3.13) α i n − Õ m = (cid:16) p [ i ] m p [ i + ] n − − p [ i ] n p [ i + ] m − (cid:17) ′ ( ) p [ i ] m ( ) . Finally, putting (3.11) and (3.13) into (3.10), by (3.8), we obtain (3.6c). This completes the proof. (cid:3)
By reasoning analogous to the proof of Corollary 3.5 we obtain the following corollary.
Corollary 3.10.
Suppose that a Jacobi matrix A has N -periodically modulated entries. Then for each i ∈ { , , . . . , N − } , lim k →∞ tr (cid:0) X k N + i ( x ) (cid:1) = tr (cid:0) X ( ) (cid:1) , (3.14a) lim k →∞ a k N + i α i tr (cid:0) X ′ k N + i ( x ) (cid:1) = tr (cid:0) X ′ ( ) (cid:1) , (3.14b) lim k →∞ (cid:18) a k N + i α i (cid:19) tr (cid:0) X ′′ k N + i ( x ) (cid:1) = tr (cid:0) X ′′ ( ) (cid:1) , (3.14c) locally uniformly with respect to x ∈ C . A blend of bounded and unbounded parameters.Definition 3.11.
The Jacobi matrix A associated with sequences ( a n : n ∈ N ) and ( b n : n ∈ N ) isa N -periodic blend if there are an asymptotically N -periodic Jacobi matrix ˜ A associated with sequences ( ˜ a n : n ∈ N ) and ( ˜ b n : n ∈ N ) , and a sequence of positive numbers ( ˜ c n : n ∈ N ) , such that(i) lim n →∞ ˜ c n = ∞ , and lim m →∞ ˜ c m + ˜ c m = , (ii) a k ( N + ) + i = ˜ a k N + i if i ∈ { , , . . . , N − } , ˜ c k if i = N , ˜ c k + if i = N + , (iii) b k ( N + ) + i = ( ˜ b k N + i if i ∈ { , , . . . , N − } , if i ∈ { N , N + } . Proposition 3.12.
For i ∈ { , , . . . , N − } , lim j →∞ B j ( N + ) + i ( x ) = B i ( x ) locally uniformly with respect to x ∈ R . Moreover, lim j →∞ B (( j + )( N + ) ( x ) B j ( N + ) + N + ( x ) B ( j ( N + ) + N ( x ) = C( x ) , locally uniformly with respect to x ∈ R , where C( x ) = (cid:18) − α N − α − x − β α (cid:19) . Proof.
The argument is contained in the proof of [24, Corollary 9]. (cid:3)
For a Jacobi matrix being N -periodic blend, we set X n ( x ) = N + n + Ö j = n B j ( x ) . In view of Proposition 3.12, for i ∈ { , , . . . , N } , the sequence ( X j ( N + ) + i : j ∈ N ) converges to X i locallyuniformly on C where(3.15) X i ( x ) = (cid:18) i − Ö j = B j ( x ) (cid:19) C( x ) (cid:18) N − Ö j = i B j ( x ) (cid:19) . We set Λ = (cid:8) x ∈ R : (cid:12)(cid:12) tr X ( x ) (cid:12)(cid:12) < (cid:9) . Theorem 3.13.
There are non-empty open and disjoint intervals ( I j : 1 ≤ j ≤ N ) such that Λ = N Ø j = I j . Moreover, for x ∈ Λ , ω ′ ( x ) = N π p − discr X ( x ) (cid:18) N − Õ i = |[X i ( x )] , | α i − + |[X N ( x )] , | α N − (cid:19) where ω is the equilibrium measure corresponding to Λ .Proof. Let us begin with the case N = . Then X ( x ) = − x − β / α / ! . Therefore, by (3.1) one obtains ω ′ ( x ) = π p − discr X ( x ) |[X ( x )] , | α / , which ends the proof for N = . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 13
Suppose that N ≥ . For k ∈ Z and i ∈ { , , . . . , N − } , we set ˜ α k N + i = α i · ( √ if i ∈ { , N − } , otherwise , and ˜ β k N + i = β i · ( if i = , otherwise.Let ˜ B j ( x ) = − ˜ α j − ˜ α j x − ˜ β j ˜ α j ! and ˜ X n ( x ) = N + n − Ö j = n ˜ B j ( x ) . By (3.1),(3.16) ˜ ω ′ ( x ) = N π q − discr ˜ X ( x ) N − Õ i = |[ ˜ X i ( x )] , | ˜ α i − where ˜ ω is the equilibrium measure corresponding to the closure of ˜ Λ = (cid:8) x ∈ R : discr ˜ X ( x ) < (cid:9) . In particular, ˜ Λ is the union of N non-empty open and disjoint intervals, see [2, Theorem 5.4.2]. Notice that − (cid:18) √ (cid:19) ˜ B (cid:18) √
00 1 (cid:19) = C . We next show the following claim.
Claim 3.14. If N ≥ , then X = − (cid:18) √
00 1 (cid:19) ˜ X (cid:18) √ (cid:19) , (3.17) X j = − ˜ X j , for i = , , . . . , N − , (3.18) X N = − (cid:18) √ (cid:19) ˜ X (cid:18) √
00 1 (cid:19) . (3.19)For N = , the identities (3.17) and (3.19) can be checked by a direct computations. For N ≥ , we firstobserve that B = ˜ B (cid:18) √ (cid:19) , (3.20) B j = ˜ B j , for i = , , . . . , N − , (3.21) B N − = (cid:18) √ (cid:19) ˜ B N − . (3.22)Consequently, X = C N − Ö j = B j = − (cid:18) √
00 1 (cid:19) ˜ X (cid:18) √ (cid:19) . Similarly, one can show (3.18) and (3.19).Now, using (3.17), we easily get discr X = discr ˜ X , which implies that Λ = ˜ Λ . Hence, ˜ ω = ω . Moreover, by Claim 3.14, we obtain(3.23) |[ ˜ X ( x )] , | = √ |[X N ( x )] , | , |[ ˜ X ( x )] , | = √ |[X ( x )] , | , |[ ˜ X i ( x )] , | = |[X i ( x )] , | for i = , , . . . , N − Therefore, for x ∈ Λ , formula (3.16) gives ω ′ ( x ) = N π q − discr ˜ X ( x ) N − Õ i = |[ ˜ X i ( x )] , | ˜ α i − = N π p − discr X ( x ) (cid:18) |[X N ( x )] , (cid:12)(cid:12) α N − + N − Õ i = |[X i ( x )] , | α i − (cid:19) which finishes the proof. (cid:3) Corollary 3.15. N − Õ i = |[X i ( x )] , | α i − + |[X N ( x )] , | α N − = | tr X ′ ( x )| . Proof.
In view of [27, formula (3.2)], N − Õ i = |[ ˜ X i ( x )] , | ˜ α i − = (cid:12)(cid:12) tr ˜ X ′ ( x ) (cid:12)(cid:12) . By Claim 3.14, tr ˜ X ′ ( x ) = tr X ′ ( x ) , which together with (3.23) concludes the proof. (cid:3) Remark 3.16.
We want to emphasize that Theorem 3.13 says that Λ is the disjoint union of exactly N non-empty open intervals. This should be compared with the discussion at beginning of Section 5 in [1]. Proposition 3.17.
Suppose that a Jacobi matrix A is N -periodic blend. Then for each i ∈ { , , . . . , N } and n ≥ , lim k →∞ p [ k ( N + ) + i ] n ( x ) = p [ i ] n ( x ) , (3.24a) lim k →∞ a k ( N + ) + i α i (cid:0) p [ k ( N + ) + i ] n (cid:1) ′ ( x ) = (cid:0) p [ i ] n (cid:1) ′ ( x ) , (3.24b) lim k →∞ (cid:18) a k ( N + ) + i α i (cid:19) (cid:0) p [ k ( N + ) + i ] n (cid:1) ′′ ( x ) = (cid:0) p [ i ] n (cid:1) ′′ ( x ) , (3.24c) locally uniformly with respect to x ∈ C .Proof. Lemma 3.2 together with Proposition 3.3 easily gives (3.24a). Since lim k →∞ a k ( N + ) + i α i = , the uniform convergence in (3.24a) entails (3.24b) and (3.24c). (cid:3) Corollary 3.18.
Suppose that a Jacobi matrix A is N -periodic blend. Then for each i ∈ { , , . . . , N } , lim k →∞ tr (cid:0) X k ( N + ) + i ( x ) (cid:1) = tr (cid:0) X ( x ) (cid:1) , (3.25a) lim k →∞ a k ( N + ) + i α i tr (cid:0) X ′ k ( N + ) + i ( x ) (cid:1) = tr (cid:0) X ′ ( x ) (cid:1) , (3.25b) lim k →∞ (cid:18) a k ( N + ) + i α i (cid:19) tr (cid:0) X ′′ k ( N + ) + i ( x ) (cid:1) = tr (cid:0) X ′′ ( x ) (cid:1) , (3.25c) locally uniformly with respect to x ∈ C . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 15
4. Christoffel functions for D r , r ≥ General case.
In this section we determine the asymptotic behavior of the Christoffel–Darboux kernel(2.3) on the diagonal. We start by showing the following lemma.
Lemma 4.1.
Let ( γ n : n ≥ ) be a sequence of positive numbers and ( θ n : n ≥ ) be a sequence of continuousfunctions on some compact set K ⊂ R d with values in ( , π ) . Suppose that there is a function θ : K → ( , π ) such that lim n →∞ θ n ( x ) = θ ( x ) uniformly with respect to x ∈ K . Then there is c > such that for all x ∈ K and n ∈ N , (4.1) (cid:12)(cid:12)(cid:12)(cid:12) n Õ k = γ k exp (cid:18) i k Õ j = θ j ( x ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c (cid:16) n − Õ k = (cid:12)(cid:12) γ k + − γ k (cid:12)(cid:12) + γ k + (cid:12)(cid:12) θ k + ( x ) − θ ( x ) (cid:12)(cid:12)(cid:17) . In particular, if ∞ Õ k = γ k = ∞ , and lim n →∞ γ n − γ n = , then (4.2) lim n →∞ n Õ k = γ k Í nj = γ j exp (cid:18) i k Õ j = θ j ( x ) (cid:19) = uniformly with respect to x ∈ K .Proof. Let us observe that n Õ k = γ k exp (cid:16) i ( k + ) θ ( x ) + i η k ( x ) (cid:17) = n Õ k = γ k e i η k ( x ) (cid:0) s k + ( x ) − s k ( x ) (cid:1) where s k ( x ) = k Õ j = e ij θ ( x ) = e ik θ ( x )/ sin (cid:0) ( k + ) θ ( x )/ (cid:1) sin (cid:0) θ ( x )/ (cid:1) , and η k ( x ) = k Õ j = (cid:0) θ j ( x ) − θ ( x ) (cid:1) . Hence, by the summation by parts we get(4.3) n Õ k = γ k exp (cid:16) i ( k + ) θ ( x ) + i η k ( x ) (cid:17) = γ n e i η n ( x ) s n + ( x ) − γ e i η ( x ) s ( x ) + n − Õ k = (cid:16) γ k + e i η k + ( x ) − γ k e i η k ( x ) (cid:17) s k + ( x ) . Since sup k ∈ N sup x ∈ K | s k ( x )| < ∞ , the first term in (4.3) is bounded by a constant multiply of γ n . Moreover, (cid:12)(cid:12) γ n e i η n ( x ) s n + ( x ) (cid:12)(cid:12) ≤ c n − Õ k = (cid:12)(cid:12) γ k + − γ k (cid:12)(cid:12) because γ n ≤ n − Õ k = (cid:12)(cid:12) γ k + − γ k (cid:12)(cid:12) + γ . Similarly we treat the second term in (4.3). Lastly, the third term in (4.3) can be bounded by a constantmultiply of n − Õ k = | γ k + − γ k | + n − Õ k = γ k + | η k + ( x ) − η k ( x )| which together with the identity η k + ( x ) − η k ( x ) = θ k + ( x ) − θ ( x ) , entails that (4.3) is bounded by a constant multiply of n − Õ k = | γ k + − γ k | + n − Õ k = γ k + | θ k + ( x ) − θ ( x )| proving (4.1).Lastly, we observe that by the Stolz–Cesáro theorem, lim n →∞ Í n − k = | γ k + − γ k | Í nk = γ k = lim n →∞ | γ n − γ n − | γ n = . Similarly, we obtain lim n →∞ n − Õ j = γ j + Í nk = γ k sup x ∈ K | θ j + ( x ) − θ ( x )| = lim n →∞ γ n · sup x ∈ K | θ n ( x ) − θ ( x )| γ n = lim n →∞ sup x ∈ K | θ n ( x ) − θ ( x )| , thus lim n →∞ n − Õ j = γ j + Í nk = γ k | θ j + ( x ) − θ ( x )| = uniformly with respect to x ∈ K proving (4.2), and the lemma follows. (cid:3) The following theorem has been proved in [24].
Theorem 4.2. [24, Theorem C]
Let N and r be positive integer and i ∈ { , , . . . , N − } . Suppose that K isa compact interval contained in Λ = (cid:26) x ∈ R : lim j →∞ discr X j N + i ( x ) exists and is negative (cid:27) . Assume that lim j →∞ a ( j + ) N + i − a j N + i − = and (cid:0) X j N + i : j ∈ N (cid:1) ∈ D r (cid:0) K , GL ( , R ) (cid:1) . Suppose that X is the limit of ( X j N + i : j ∈ N ) . Then there is a probability measure ν such that ( p n : n ∈ N ) are orthonormal in L ( R , ν ) , which is purely absolutely continuous with continuous and positive density ν ′ on K . Moreover, there are M > and a real continuous function η : K → R such that for all k ≥ M , lim k →∞ sup x ∈ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ a ( k + ) N + i − p k N + i ( x ) − s |[X( x )] , | πν ′ ( x ) p − discr X( x ) sin (cid:16) k Õ j = M + θ j N + i ( x ) + η ( x ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = where (4.4) sup x ∈ K (cid:12)(cid:12) θ n ( x ) − arccos (cid:0) tr X ( x ) (cid:1) (cid:12)(cid:12) = . Proposition 4.3.
Under the hypotheses of Theorem 4.2, we have (4.5) inf n ∈ N a n > . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 17
Proof.
First, let us consider the case when the moment problem for ν is indeterminate, that is(4.6) ∞ Õ n = a n < ∞ . But (4.6) implies that lim n →∞ a n = ∞ , which easily gives (4.5).Assume now that the moment problem for ν is determinate. By Theorem 4.2, the measure ν has non-trivialabsolutely continuous part. To obtain a contradiction, suppose that there is a strictly increasing sequence ( L j : j ∈ N ) such that lim j →∞ a L j = . Without loss of generality we may assume that(4.7) ∞ Õ j = a L j < ∞ . The Jacobi matrix (2.1) can be written in the following block-form A = © « J a L . . . a L J a L . . . a L J a L . . . a L J ... ... ... . . . ª®®®®®®¬ where J i are some finite dimensional Jacobi matrices. Let B = diag ( J , J , J , . . . ) . By A and B we denote the restrictions of A and B to the maximal domains, respectively, that is Dom ( A ) = (cid:8) x ∈ ℓ : A x ∈ ℓ (cid:9) , and Dom ( B ) = (cid:8) x ∈ ℓ : B x ∈ ℓ (cid:9) . The determinacy of the moment problem for ν is equivalent to A being self-adjoint. Moreover,(4.8) ν ( · ) = h E A ( · ) δ , δ i ℓ where E A is the spectral resolution of A and δ is the sequence having on the zero position and zeroelsewhere. In view of (4.7), the operator A − B is self-adjoint and belongs to the trace class. Hence, by theKato–Rosenblum theorem (see, e.g., [16, Theorem 9.29]) σ ac ( A ) = σ ac ( B ) . Since B is unitary equivalent to an operator acting by the multiplication by a real-valued sequence, B hasonly discrete spectrum. Therefore, σ ac ( A ) = ∅ , and consequently, by (4.8), the measure ν has no non-trivial absolutely continuous part, which leads to thecontradiction. (cid:3) For i ∈ { , , . . . , N − } and n ∈ N we set K i ; n ( x , y ) = n Õ j = p j N + i ( x ) p j N + i ( y ) , x , y ∈ R , and ρ i ; n = n Õ j = a j N + i . Let us recall that the Carleman condition (2.2) implies that there is the unique probability measure µ suchthat ( p n : n ∈ N ) are orthonormal in L ( R , µ ) . Theorem 4.4.
Let N and r be positive integers and i ∈ { , , . . . , N − } . Suppose that K is a compactinterval with non-empty interior contained in Λ = (cid:26) x ∈ R : lim j →∞ discr X j N + i ( x ) exists and is negative (cid:27) . Assume that lim j →∞ a ( j + ) N + i − a j N + i − = and (cid:0) X j N + i : j ∈ N (cid:1) ∈ D r (cid:0) K , GL ( , R ) (cid:1) . Suppose that X is the limit of (cid:0) X j N + i : j ∈ N (cid:1) . If ∞ Õ j = a j N + i − = ∞ , then (4.9) K i ; n ( x , x ) = |[X( x )] , | π µ ′ ( x ) p − discr X( x ) ρ i − n + + E i ; n ( x ) where lim n →∞ ρ i − n + sup x ∈ K (cid:12)(cid:12) E i ; n ( x ) (cid:12)(cid:12) = . Proof.
Fix a compact interval K with non-empty interior contained in Λ . In view of Theorem 4.2, there is M > such that for all k ≥ M , a ( k + ) N + i − p k N + i ( x ) = |[X( x )] , | π µ ′ ( x ) p − discr X( x ) sin (cid:16) η ( x ) + k Õ j = M + θ j N + i ( x ) (cid:17) + E k N + i ( x ) where lim k →∞ sup x ∈ K | E k N + i ( x )| = . Since ( x ) = − cos ( x ) , we have n Õ k = M p k N + i ( x ) = |[X( x )] , | π µ ′ ( x ) p − discr X( x ) n Õ k = M a ( k + ) N + i − (cid:18) − cos (cid:16) η ( x ) + k Õ j = M + θ j N + i ( x ) (cid:17) (cid:19) + n Õ k = M a ( k + ) N + i − E k N + i ( x ) . Notice that the Stolz–Cesàro theorem gives lim n →∞ ρ i − n + n Õ j = M a ( j + ) N + i − E j N + i ( x ) = lim n →∞ E nN + i ( x ) = uniformly with respect to x ∈ K . Since there is c > such that sup x ∈ K M − Õ k = p k N + i ( x ) ≤ c , by Lemma 4.1, we conclude that(4.10) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ i − n + K i ; n ( x , x ) − |[X( x )] , | π µ ′ ( x ) p − discr X( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = , SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 19 which completes the proof. (cid:3)
Remark 4.5.
In view of [24, Proposition 7], for each compact set K ⊂ R with non-empty interior there is c > such that for all n ∈ N , ∞ Õ j = n sup x ∈ K (cid:13)(cid:13) X ( j + ) N + i ( x ) − X j N + i ( x ) (cid:13)(cid:13) ≤ c ∞ Õ j = n (cid:12)(cid:12)(cid:12)(cid:12) a ( j + ) N + i − a ( j + ) N + i − a j N + i − a j N + i (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) b ( j + ) N + i a ( j + ) N + i − b j N + i a j N + i (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) a ( j + ) N + i − a j N + i (cid:12)(cid:12)(cid:12)(cid:12) . Moreover, the right-hand side is comparable to a constant multiply of ∞ Õ j = n sup x ∈ K (cid:13)(cid:13) B ( j + ) N + i ( x ) − B j N + i ( x ) (cid:13)(cid:13) . Application to the classes.
We are now ready to prove the main theorem of this section.
Theorem 4.6.
Let A be a Jacobi matrix with N -periodically modulated entries. Suppose that there is r ≥ such that for every i ∈ { , , . . . , N − } , (cid:18) a k N + i − a k N + i : k ∈ N (cid:19) , (cid:18) b k N + i a k N + i : k ∈ N (cid:19) , (cid:18) a k N + i : k ∈ N (cid:19) ∈ D r , and (4.11) ∞ Õ n = a n = ∞ . If | tr X ( )| < , then K n ( x , x ) = ω ′ ( ) µ ′ ( x ) ρ n + E n ( x ) where lim n →∞ ρ n E n ( x ) = locally uniformly with respect to x ∈ R , where ω is the equilibrium measure corresponding to σ ess ( A ) , with A being the Jacobi matrix associated to ( α n : n ∈ N ) and ( β n : n ∈ N ) , and ρ n = n Õ j = α j a j . Proof.
Let K be a compact interval in R with non-empty interior. Observe that, by Remark 4.5, for each i ∈ { , , . . . , N − } the sequence ( X j N + i : j ≥ ) belongs to D r (cid:0) K , GL ( , R ) (cid:1) . Moreover, by Proposition 3.8we have lim j →∞ X j N + i ( x ) = X i ( ) , which together with discr X i ( ) < implies that Λ = R . Since for each n , n ′ ∈ N , by Proposition 3.7, wehave lim j →∞ a j N + n ′ a j N + n = α n ′ α n , by the Stolz–Cesàro theorem lim j →∞ ρ i ′ ; j ρ j N + i = lim j →∞ a j N + i ′ Í Nk = α i + k a j N + i + k = N α i ′ . (4.12) Consequently, the Carleman condition (4.11) implies that lim k →∞ k Õ j = a j N + i = ∞ , for each i ∈ { , , . . . , N − } . Thus, by Theorem 4.4, we obtain(4.13) lim n →∞ sup x ∈ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ i ; n K i ; n ( x , x ) − |[ X i ( )] , | π µ ′ ( x ) p − discr X i ( ) · α i α i − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = . Fix i ∈ { , , . . . , N − } and consider n = k N + i for k ∈ N . We write K k N + i ( x , x ) = N − Õ i ′ = K i ′ ; k ( x , x ) + N − Õ i ′ = i + (cid:0) K i ′ ; k − ( x , x ) − K i ′ ; k ( x , x ) (cid:1) . Observe that sup x ∈ K (cid:12)(cid:12) K i ′ ; k ( x , x ) − K i ′ ; k − ( x , x ) (cid:12)(cid:12) = sup x ∈ K p k N + i ′ ( x ) ≤ c , hence, by (4.13),(4.14) lim k →∞ ρ k N + i K k N + i ( x , x ) = N − Õ i ′ = |[ X i ′ ( )] , | π µ ′ ( x ) p − discr X i ′ ( ) · α i ′ α i ′ − · lim k →∞ ρ i ′ ; k ρ k N + i . Since X i + ( x ) = (cid:0) B i ( x ) (cid:1) (cid:0) X i ( x ) (cid:1) (cid:0) B i ( x ) (cid:1) − , we immediately get(4.15) discr X i ′ ( x ) = discr X ( x ) . Finally, putting (4.12) and (4.15) into (4.14), we obtain lim k →∞ ρ k N + i K k N + i ( x , x ) = π µ ′ ( x ) N − Õ i ′ = |[ X i ′ ( )] , | p − discr X i ′ ( ) N α i ′ − = N π µ ′ ( x ) p − discr X ( ) N − Õ i ′ = |[ X i ′ ( )] , | α i ′ − which together with (3.1) completes the proof. (cid:3) The following theorem has essentially the same proof as Theorem 4.6.
Theorem 4.7.
Let A be a Jacobi matrix with asymptotically N -periodic entries. Suppose that there is r ≥ such that for every i ∈ { , , . . . , N − } , (4.16) (cid:18) a k N + i − a k N + i : k ∈ N (cid:19) , (cid:18) b k N + i a k N + i : k ∈ N (cid:19) , (cid:18) a k N + i : k ∈ N (cid:19) ∈ D r . Let K be a compact interval with non-empty interior contained in Λ = (cid:8) x ∈ R : (cid:12)(cid:12) tr X ( x ) (cid:12)(cid:12) < (cid:9) . Then K n ( x , x ) = ω ′ ( x ) µ ′ ( x ) ρ n + E n ( x ) with lim n →∞ ρ n sup x ∈ K (cid:12)(cid:12) E n ( x ) (cid:12)(cid:12) = SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 21 where ω is the equilibrium measure corresponding to σ ess ( A ) , with A being the Jacobi matrix associated to ( α n : n ∈ N ) and ( β n : n ∈ N ) , and ρ n = n Õ j = α j a j . Remark 4.8.
If the Jacobi matrix A has asymptotically N -periodic entries, then the condition (4.16) isequivalent to (cid:0) a n : n ∈ N (cid:1) , (cid:0) b n : n ∈ N (cid:1) ∈ D r . To see this, let us observe that inf n ∈ N a n > . Hence, by [24, Corollary 2] (cid:18) a n : n ∈ N (cid:19) ∈ D r , thus by [24, Corollary 1] (cid:18) a n − a n : n ∈ N (cid:19) , (cid:18) b n a n : n ∈ N (cid:19) ∈ D r . Analogously, one can prove the opposite implication.
Remark 4.9.
If the Jacobi matrix A has asymptotically N -periodic entries, then(4.17) lim n →∞ ρ n n = . Indeed, by the Stolz–Cesàro theorem, we have lim n →∞ ρ n n = lim n →∞ α n a n = . Let us recall that in this case supp ( µ ) is compact and µ ′ is continuous and positive in the interior of supp ( µ ) .Thus, in view of (4.17), Theorem 4.7 follows from [25, Theorem 1]. We want to point out that our approachis completely different. Theorem 4.10.
Let A be a Jacobi matrix that is N -periodic blend. Suppose that there is r ≥ such that forevery i ∈ { , , . . . , N − } , (cid:18) a j ( N + ) + i : j ∈ N (cid:19) , (cid:18) b j ( N + ) + i a j ( N + ) + i : j ∈ N (cid:19) ∈ D r , and (cid:18) a j ( N + ) + N : j ∈ N (cid:19) , (cid:18) a j ( N + ) + N + : j ∈ N (cid:19) , (cid:18) a j ( N + ) + N a j ( N + ) + N + : j ∈ N (cid:19) ∈ D r . Let K be a compact subset with non-empty interior contained in Λ = (cid:8) x ∈ R : (cid:12)(cid:12) tr X ( x ) (cid:12)(cid:12) < (cid:9) where X is the limit of ( X j ( N + ) + : j ∈ N ) . Then K n ( x , x ) = ω ′ ( x ) µ ′ ( x ) ρ n + E n ( x ) where ω is the equilibrium measure corresponding to Λ , and lim n →∞ ρ n sup x ∈ K (cid:12)(cid:12) E n ( x ) (cid:12)(cid:12) = , with ρ n = N − Õ i = Õ ≤ m ≤ n m ≡ i mod ( N + ) α i a m . Proof.
Let K be a compact set with non-empty interior contained in Λ . Fix i ∈ { , , . . . , N } . Let us recallthat for each the sequence ( X j ( N + ) + i : j ∈ N ) converges to X i . Moreover, we claim the following holds. Claim 4.11.
The sequence ( X j ( N + ) + i : j ∈ N ) belongs to D r (cid:0) K , GL ( , R ) (cid:1) . For the proof, let us observe that if i ∈ { , , . . . , N − } , inf j ∈ N a j ( N + ) + i > , thus, by [24, Corollary 2], (cid:0) a j ( N + ) + i : j ∈ N (cid:1) ∈ D r , and consequently, by [24, Corollary 1(i)], (cid:0) b j ( N + ) + i : j ∈ N (cid:1) ∈ D r . Moreover, (cid:18) a j ( N + ) + N · a j ( N + ) + N + : j ∈ N (cid:19) ∈ D r . Since lim j →∞ a j ( N + ) + N a j ( N + ) + N + = , by [24, Corollary 1], we get (cid:18) a j ( N + ) + N + a j ( N + ) + N : j ∈ N (cid:19) ∈ D r . Now, the conclusion follows from the proof of [24, Corollary 9].Since lim k →∞ k Õ j = a j ( N + ) + i = ∞ , and, in view of (3.15), discr X i = discr X , therefore, by Theorem 4.4, we obtain(4.18) K i ; k ( x , x ) = |[X i ( x )] , | π µ ′ ( x ) p − discr X i ( x ) ρ i − k + E i ; k ( x ) where lim k →∞ ρ i − k sup x ∈ K (cid:12)(cid:12) E i ; k ( x ) (cid:12)(cid:12) = . Next, we show(4.19) lim k →∞ sup x ∈ K ρ N − k (cid:12)(cid:12) K k ( x , x ) − K N ; k ( x , x ) (cid:12)(cid:12) = . and(4.20) lim k →∞ sup x ∈ K ρ N − k (cid:12)(cid:12) K N + k ( x , x ) (cid:12)(cid:12) = . First, we prove the following claim.
Claim 4.12.
There is c > such that for all k ∈ N , (4.21) sup x ∈ K (cid:12)(cid:12) p k ( N + ) + N + ( x ) (cid:12)(cid:12) ≤ c a k ( N + ) + N + , and (4.22) sup x ∈ K (cid:12)(cid:12) p k ( N + ) + N ( x ) + p ( k + )( N + ) ( x ) (cid:12)(cid:12) ≤ c (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) − a k ( N + ) + N a k ( N + ) + N + (cid:12)(cid:12)(cid:12)(cid:12) + a k ( N + ) + N + (cid:19) . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 23
For the proof, let us notice that [24, Theorem A] with i = , implies sup x ∈ K a k ( N + ) (cid:0) p k ( N + ) ( x ) + p k ( N + ) + ( x ) (cid:1) ≤ c . Since ( a k ( N + ) : k ∈ N ) is bounded, by [24, Corollary 9] with i = ,(4.23) sup k ∈ N sup x ∈ K (cid:12)(cid:12) p k ( N + ) + ( x ) (cid:12)(cid:12) ≤ c , consequently we get(4.24) sup k ∈ N sup x ∈ K (cid:12)(cid:12) p k ( N + ) ( x ) (cid:12)(cid:12) ≤ c . Analogous reasoning for i = N shows that(4.25) sup k ∈ N sup x ∈ K (cid:12)(cid:12) p k ( N + ) + N ( x ) (cid:12)(cid:12) ≤ c . Recall that we have the following recurrence relation p k ( N + ) + N + ( x ) = x − b ( k + )( N + ) a k ( N + ) + N + p ( k + )( N + ) ( x ) − a ( k + )( N + ) a k ( N + ) + N + p ( k + )( N + ) + ( x ) . Therefore, by (4.23) and (4.24), we easily get (4.21). Moreover, we have p ( k + )( N + ) ( x ) = xa k ( N + ) + N + p k ( N + ) + N + ( x ) − a k ( N + ) + N a k ( N + ) + N + p k ( N + ) + N ( x ) , thus (cid:12)(cid:12) p ( k + )( N + ) ( x ) + p k ( N + ) + N ( x ) (cid:12)(cid:12) ≤ | x | a k ( N + ) + N + (cid:12)(cid:12) p k ( N + ) + N + ( x ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) − a k ( N + ) + N a k ( N + ) + N + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p k ( N + ) + N ( x ) (cid:12)(cid:12) , which together with (4.21) and (4.25) entails (4.22).Now using Claim 4.12 together with (4.24) and (4.25), we easily see that (cid:12)(cid:12) K k ( x , x ) − K N ; k ( x , x ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) k Õ j = p j ( N + ) ( x ) − k Õ j = p j ( N + ) + N ( x ) (cid:12)(cid:12)(cid:12) ≤ c k Õ j = a j ( N + ) + N + + (cid:12)(cid:12)(cid:12)(cid:12) − a j ( N + ) + N a j ( N + ) + N + (cid:12)(cid:12)(cid:12)(cid:12) . Since ( a k ( N + ) + N − : k ∈ N ) is bounded, by the Stolz–Cesàro theorem, we obtain lim k →∞ ρ N − k k Õ j = a j ( N + ) + N + = lim k →∞ a k ( N + ) + N − a k ( N + ) + N + = , and lim k →∞ ρ N − k k Õ j = (cid:12)(cid:12)(cid:12)(cid:12) − a j ( N + ) + N a j ( N + ) + N + (cid:12)(cid:12)(cid:12)(cid:12) = lim k →∞ a k ( N + ) + N − (cid:12)(cid:12)(cid:12)(cid:12) − a k ( N + ) + N a k ( N + ) + N + (cid:12)(cid:12)(cid:12)(cid:12) = , which gives (4.19). To prove (4.20), we reason analogously. Namely, by Claim 4.12, we have (cid:12)(cid:12) K N + k ( x , x ) (cid:12)(cid:12) ≤ c k Õ j = a j ( N + ) + N + , thus, by the Stolz–Cesàro theorem, we get lim k →∞ sup x ∈ K ρ N − k (cid:12)(cid:12) K N + k ( x , x ) (cid:12)(cid:12) ≤ c lim k →∞ ρ N − k k Õ j = a j ( N + ) + N + = lim k →∞ a k ( N + ) + N − a k ( N + ) + N + = . Notice that, by [24, Corollary 9] and (4.21), we have(4.26) sup n ∈ N sup x ∈ K (cid:12)(cid:12) p n ( x ) (cid:12)(cid:12) ≤ c . Next we show the following statement.
Claim 4.13.
For each i ∈ { , , . . . , N + } and i ′ ∈ { , , . . . , N − } , (4.27) lim k →∞ ρ i ′ ; k ρ k ( N + ) + i = N α i ′ . For the proof, let us consider i ∈ { , , . . . , N − } . By the Stolz–Cesàro theorem, we have lim k →∞ ρ i ′ ; k ρ k ( N + ) + i = lim k →∞ a k ( N + ) + i ′ α i + a k ( N + ) + i + + . . . + α N − a k ( N + ) + N − + α a ( k + )( N + ) + . . . + α i a ( k + )( N + ) + i = N α i ′ . For i = N − we obtain lim k →∞ ρ i ′ ; k ρ k ( N + ) + N − = lim k →∞ a k ( N + ) + i ′ α a ( k + )( N + ) + . . . + α N − a ( k + )( N + ) + N − = N α i ′ . Finally, we observe that ρ ( k + )( N + ) + N − − ρ k ( N + ) + N − = ρ ( k + )( N + ) + N − ρ k ( N + ) + N = ρ ( k + )( N + ) + N + − ρ k ( N + ) + N + , which entails (4.27) for i ∈ { N , N + } .Now, writing K k ( N + ) + i ( x , x ) = N + Õ i ′ = K i ′ ; k ( x , x ) + N + Õ i ′ = i + (cid:0) K i ′ ; k − ( x , x ) − K i ′ ; k ( x , x ) (cid:1) , by (4.19), (4.20) and (4.26), we obtain K k ( N + ) + i ( x , x ) = N − Õ i ′ = K i ′ ; k ( x , x ) + K N ; k ( x , x ) + o (cid:0) ρ k ( N + ) + i (cid:1) uniformly with respect to x ∈ K . Using (4.18) together with (4.27), we arrive at K n ( x , x ) = N π µ ′ ( x ) p − discr X ( x ) (cid:18) N − Õ i = |[X i ( x )] , | α i − + |[X N ( x )] , | α N − (cid:19) ρ n + o (cid:0) ρ n (cid:1) uniformly with respect to x ∈ K . To finish the proof, it is sufficient to invoke Theorem 3.13. (cid:3) Remark 4.14. If A is a Jacobi matrix that is N -periodic blend then(4.28) lim n →∞ ρ n n = NN + . Indeed, for each i ∈ { , , . . . , N + } , by Claim 4.13, we have lim k →∞ ρ k ( N + ) + i k ( N + ) + i = lim k →∞ ρ k ( N + ) + i ρ k · ρ k k ( N + ) + i = N α · lim k →∞ ρ k k ( N + ) + i . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 25
Now, by the Stolz–Cesàro theorem, we obtain lim k →∞ ρ k k ( N + ) + i = lim k →∞ ( N + ) a k N = ( N + ) α , and the formula (4.28) follows.Let us recall that supp ( µ ) is not compact. In [1, Theorem 5] there was provided examples of Jacobiparameters from this class such that the set of the accumulation points of supp ( µ ) is equal to the compactset Λ . From [24, Corollary 9] the density µ ′ is continuous and positive in Λ . Hence, in view of (4.28), thehypothesis on the compactness of supp ( µ ) from [25, Theorem 1] cannot be replaced by compactness of theset of its accumulation points.4.3. Ignjatović conjecture.
In this section we show the relation between Theorem 4.6 and the conjecturedue to Ignjatović [3, Conjecture 1].
Conjecture 4.15 (Ignjatović, 2016) . Suppose that ( C ) lim n →∞ a n = ∞ ; ( C ) lim n →∞ ∆ a n = ; ( C ) There exist n , m such that a n + m > a n holds for all n ≥ n and all m ≥ m ; ( C ) ∞ Õ n = a n = ∞ ; ( C ) There exists κ > such that Í ∞ n = a κ n < ∞ ; ( C ) ∞ Õ n = | ∆ a n | a n < ∞ ; ( C ) ∞ Õ n = (cid:12)(cid:12) ∆ a n (cid:12)(cid:12) a n < ∞ .If − < lim n →∞ b n a n < , then for any x ∈ R , the limit lim n →∞ (cid:18) n Õ j = a j (cid:19) − n Õ j = p j ( x ) , exists and is positive. In [3, Corollary 3] the conclusion was shown to hold in the case b n ≡ . Later, in [21, Corollary 3], theresult was extended to a more general class of sequences with lim n →∞ b n a n = , and it was shown that lim n →∞ (cid:18) n Õ j = a j (cid:19) − n Õ j = p j ( x ) = π µ ′ ( x ) . Our results imply the following corollary.
Corollary 4.16.
Let N be a positive integer and let r ≥ . Suppose that for each i ∈ { , , . . . , N − } , (cid:18) a j N + i − a j N + i : j ∈ N (cid:19) , (cid:18) b j N + i a j N + i : j ∈ N (cid:19) , (cid:18) a j N + i : j ∈ N (cid:19) ∈ D r , and lim n →∞ a n − a n = , lim n →∞ b n a n = q , lim n →∞ a n = ∞ . If | q | < with (4.29) q < (cid:8) ( j π N ) : j = , , . . . , N − (cid:9) , and the Carleman condition is satisfied, then lim n →∞ (cid:18) n Õ j = a j (cid:19) − n Õ j = p j ( x ) = π p − q µ ′ ( x ) , locally uniformly with respect to x ∈ R .Proof. Let α n ≡ , β n ≡ q . By (3.1), ω ′ ( x ) = π p − ( x − q ) , x ∈ (− + q , + q ) Hence, the conclusion follows from Theorem 4.6 provided that we can show(4.30) | tr X ( )| < . To do so, let us observe that X ( ) = (cid:18) − − q (cid:19) N = − − q / − / ! N . Hence, by Lemma 3.2 X ( ) = (cid:18) − U N − (− q ) U N − (− q )− U N − (− q ) U N (− q ) (cid:19) , where ( U n : n ∈ N ) is the sequence of the Chebyshev polynomials of the second kind (see [13, formula(1.6)]). By [13, formula (1.7)] U N (− q ) − U N − (− q ) = (cid:0) N arccos (− q ) (cid:1) . Hence, (4.29) implies (4.30). The proof is complete. (cid:3)
Remark 4.17.
The case when (4.29) is violated is more complicated and demands stronger hypotheses. Werefer to [23] for more details.As it was shown in [21, Section 4] conditions (C ) – (C ) imply the hypotheses of Corollary 4.16 with b n ≡ , N = and r = . On the other hand, in Conjecture 4.15 no regularity assumptions on ( b n ) wasimposed, whereas in Corollary 4.16 we asked for (cid:18) b j N + i a j N + i : j ∈ N (cid:17) ∈ D r , for each i ∈ { , , . . . , N − } . The following example illustrates that some regularity assumption on ( b n ) isnecessary. Example 1.
Let a n = √ n + , and b n = ( n even otherwise.Then the measure µ is absolutely continuous on R \ [ , ] with continuous and positive density, is not amass point of µ , and(4.31) lim n →∞ (cid:18) n Õ j = a j (cid:19) − n Õ j = p j ( ) = ∞ . For the proof, we set X n ( x ) = B n + ( x ) B n ( x ) , SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 27 that is(4.32) X n ( x ) = − a n − a n x − b n a n − a n − a n x − b n + a n + − a n a n + + x − b n + a n + x − b n a n ! . Since a n + (cid:0) X n ( x ) + Id (cid:1) = a n + a n ( a n − a n − ) a n + a n ( x − b n )− a n − a n ( x − b n + ) ( a n + − a n ) + ( x − b n + ) x − b n a n ! , we obtain R = lim n →∞ a n + (cid:0) X n ( x ) + Id (cid:1) = (cid:18) x − − x (cid:19) . Consequently, Λ = (cid:8) x ∈ R : discr R ( x ) < (cid:9) = R \ [ , ] . Therefore, by [22, Theorem D] and [22, Proposition 11], the measure µ is purely absolutely continuous on Λ with positive continuous density proving the first assertion.Next, let us observe that, by (4.32), (cid:18) p n ( ) p n + ( ) (cid:19) = X n − ( ) (cid:18) p n − ( ) p n − ( ) (cid:19) = (cid:18) − a n − a n − a n − a n − a n − a n − a n (cid:19) (cid:18) p n − ( ) p n − ( ) (cid:19) . Thus(4.33) p n ( ) = (− ) n a a . . . a n − a a . . . a n − = (− ) n p ( n ) !2 n n ! , and(4.34) p n + ( ) = − a n p n ( ) − a n − a n p n − ( ) . By (4.33) and (4.34), ( p n + ( ) : n ∈ N ) satisfies(4.35) x n = − √ n + (− ) n p ( n ) !2 n n ! − r n n + x n − , n ≥ , x = − . It can be verified that x n = (− ) n + p ( n + )( n + ) ! ( n + ) !2 n + / is the only solution of (4.35). Hence, p n + ( ) = ( n + )( n + ) ! (cid:0) ( n + ) ! (cid:1) n + . Using the Stirling’s formula, we can find that(4.36) lim n →∞ p n + ( )√ n + = √ π . Now, by the Stolz–Cesàro theorem lim n →∞ (cid:18) n Õ j = √ j + (cid:19) − n − Õ j = p j + ( ) = lim n →∞ p n − ( x ) √ n + √ n + = √ π lim n →∞ √ n √ n + √ n + = ∞ proving (4.31). Moreover, by (4.36), the sequence (cid:0) p n ( ) : n ∈ N (cid:1) is not square summable. Hence, is nota mass point of µ .
5. Christoffel–Darboux kernel for D For D class, we can describe the speed of convergence in Theorem 4.2 and Theorem 4.4. First, let usshow a refined asymptotic of polynomials.5.1. Asymptotics of polynomials.Theorem 5.1.
Let N be a positive integer and i ∈ { , , . . . , N − } . Suppose that K is a compact intervalwith non-empty interior contained in Λ = (cid:26) x ∈ R : lim j →∞ discr X j N + i ( x ) exists and is negative (cid:27) . Assume that lim j →∞ a ( j + ) N + i − a j N + i − = and (cid:0) X j N + i : j ∈ N (cid:1) ∈ D (cid:0) K , GL ( , R ) (cid:1) . Let X denote the limit of ( X j N + i : j ∈ N ) . Then there is a probability measure ν such that ( p n : n ∈ N ) areorthonormal in L ( R , ν ) which is purely absolutely continuous with continuous and positive density ν ′ on K .Moreover, there are M > and a real continuous function η : K → R , such that for all k ≥ M , √ a ( k + ) N + i − p k N + i ( x ) = s |[X( x )] , | πν ′ ( x ) p − discr X( x ) sin (cid:16) k Õ j = M + θ j N + i ( x ) + η ( x ) (cid:17) + E k N + i ( x ) where sup x ∈ K (cid:12)(cid:12) E k N + i ( x ) (cid:12)(cid:12) ≤ c ∞ Õ j = k sup x ∈ K (cid:13)(cid:13) X ( j + ) N + i ( x ) − X j N + i ( x ) (cid:13)(cid:13) , and (5.1) θ n ( x ) = arccos (cid:18) tr X n ( x ) p det X n ( x ) (cid:19) . Proof.
Let us fix a compact interval K with non-empty interior. Since X is the uniform limit of ( X j N + i : j ∈ N ) , there are δ > and M > such that for all x ∈ K and k ≥ M , discr X k N + i ( x ) ≤ − δ < . Therefore, the matrix X k N + i ( x ) has two eigenvalues λ k N + i and λ k N + i where(5.2) λ n ( x ) = tr X n ( x ) + i p − discr X n ( x ) . Let us next observe that for k ≥ M , ℑ λ k N + i ( x ) = p − discr X k N + i ( x ) ≥ √ δ. SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 29
Moreover, X k N + i ( x ) = C k ( x ) D k ( x ) C − k ( x ) where C k ( x ) = (cid:18) λ k N + i λ k N + i (cid:19) , D k ( x ) = (cid:18) λ k N + i λ k N + i (cid:19) . Let φ k N + i = p ( k + ) N + i − λ k N + i · p k N + i Î kj = M + λ j N + i . We claim the following holds true.
Claim 5.2.
There is c > such that for all m ≥ n ≥ M , and x ∈ K , (cid:12)(cid:12) φ mN + i ( x ) − φ nN + i ( x ) (cid:12)(cid:12) ≤ c ∞ Õ j = n (cid:12)(cid:12) λ ( j + ) N + i ( x ) − λ j N + i ( x ) (cid:12)(cid:12) . We start by writing p mN + i = * C m − (cid:16) m − Ö j = n D j C − j C j − (cid:17) C − n − (cid:18) p nN + i − p nN + i (cid:19) , (cid:18) (cid:19) + . Let us now introduce two auxiliary functions q m = * C ∞ (cid:16) m − Ö j = n D j (cid:17) C − n − (cid:18) p nN + i − p nN + i (cid:19) , (cid:18) (cid:19) + , and ψ m = q m + − λ mN + i · q m Î mj = M + λ j N + i . Notice that p mN + i − q m = (cid:28) Y m (cid:18) p nN + i − p nN + i (cid:19) , (cid:18) (cid:19) (cid:29) where Y m = C ∞ (cid:16) m − Ö j = n D j C j C − j − − m − Ö j = n D j (cid:17) C − n − + ( C ∞ − C n − ) (cid:16) m − Ö j = n D j (cid:17) C − n − . In view of [24, Propositon 1], we have k Y m k . (cid:18) m − Ö j = n k D j k (cid:19) (cid:18) ∞ Õ j = n − k ∆ C j k + k C ∞ − C m − k (cid:19) . (cid:18) m − Ö j = n k D j k (cid:19) ∞ Õ j = n − k ∆ C j k , thus k Y m k . m − Ö j = n | λ j N + i | · ∞ Õ j = n − (cid:12)(cid:12) λ ( j + ) N + i − λ j N + i (cid:12)(cid:12) . Next, by [24, Claim 2], there is c > such that for all n ≥ M and x ∈ K , q p nN + i ( x ) + p nN + i − ( x ) Î n − j = M + | λ j N + i ( x )| ≤ c , and consequently, for all m ≥ n ≥ M , (cid:12)(cid:12)(cid:12)(cid:12) p mN + i − q m Î m − j = M + λ j N + i (cid:12)(cid:12)(cid:12)(cid:12) . ∞ Õ j = n − (cid:12)(cid:12) λ ( j + ) N + i − λ j N + i (cid:12)(cid:12) . In particular, we obtain(5.3) (cid:12)(cid:12) φ mN + i − ψ m (cid:12)(cid:12) . ∞ Õ j = n − (cid:12)(cid:12) λ ( j + ) N + i − λ j N + i (cid:12)(cid:12) . Next, we notice that q m + − λ mN + i · q m = * C ∞ (cid:16) D m − λ mN + i Id (cid:17) (cid:16) m − Ö j = n D j (cid:17) C − n − (cid:18) p nN + i − p nN + i (cid:19) , (cid:18) (cid:19) + . Since Î mj = n λ j N + i (cid:16) D m − λ mN + i Id (cid:17) (cid:16) m − Ö j = n D j (cid:17) = λ mN + i (cid:18) λ mN + i − λ mN + i
00 0 (cid:19) , we obtain (cid:12)(cid:12) ψ m − ψ n (cid:12)(cid:12) . (cid:12)(cid:12)(cid:12)(cid:12) λ mN + i λ mN + i − λ nN + i λ nN + i (cid:12)(cid:12)(cid:12)(cid:12) . ∞ Õ j = n (cid:12)(cid:12) λ ( j + ) N + i − λ j N + i (cid:12)(cid:12) , which together with (5.3) implies that for all m ≥ n > M and x ∈ K , (cid:12)(cid:12) φ mN + i ( x ) − φ nN + i ( x ) (cid:12)(cid:12) . ∞ Õ j = n (cid:12)(cid:12) λ ( j + ) N + i ( x ) − λ j N + i ( x ) (cid:12)(cid:12) . In particular, the sequence ( φ mN + i : m ∈ N ) converges. Let us denote by ϕ its limit. Since polynomials p n are having real coefficients, by taking imaginary part we arrive at (cid:12)(cid:12)(cid:12)(cid:12) p − discr X nN + i ( x ) p nN + i ( x ) Î nj = M + (cid:12)(cid:12) λ j N + i ( x )| − | ϕ ( x )| sin (cid:16) n Õ j = M + arg λ j N + i ( x ) + arg ϕ ( x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) . ∞ Õ j = n (cid:12)(cid:12) λ ( j + ) N + i − λ j N + i (cid:12)(cid:12) . Because det X j N + i = ( j + ) N + i − Ö k = j N + i det B k = a j N + i − a ( j + ) N + i − , we obtain n Ö j = M + (cid:12)(cid:12) λ j N + i (cid:12)(cid:12) = n Ö j = M + det X j N + i = a ( M + ) N + i − a ( n + ) N + i − . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 31
Therefore, by [24, Theorem 6], (cid:12)(cid:12)(cid:12)(cid:12) √ a ( n + ) N + i − p − discr X nN + i ( x ) p nN + i ( x )− p − discr X( x ) s |[X( x )] , | πν ′ ( x ) sin (cid:16) n Õ j = M + arg λ j N + i ( x ) + ϕ ( x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) . ∞ Õ j = n (cid:12)(cid:12) λ ( j + ) N + i − λ j N + i (cid:12)(cid:12) . Observe that, by (5.2), (cid:12)(cid:12)(cid:12)(cid:12) p − discr X nN + i ( x ) − p − discr X( x ) (cid:12)(cid:12)(cid:12)(cid:12) . ∞ Õ j = n (cid:12)(cid:12) λ ( j + ) N + i − λ j N + i (cid:12)(cid:12) , thus (cid:12)(cid:12)(cid:12)(cid:12) √ a ( n + ) N + i − p nN + i ( x ) − s |[X( x )] , | πν ′ ( x ) p − discr X( x ) sin (cid:16) n Õ j = M + arg λ j N + i ( x ) + ϕ ( x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) . ∞ Õ j = n (cid:12)(cid:12) λ ( j + ) N + i − λ j N + i (cid:12)(cid:12) . Since (cid:12)(cid:12) λ ( j + ) N + i − λ j N + i (cid:12)(cid:12) . (cid:13)(cid:13) X ( j + ) N + i − X j N + i (cid:13)(cid:13) , we finish the proof. (cid:3) Remark 5.3.
Under the assumption of Theorem 5.1, we have θ k N + i ( x ) = arccos (cid:16) tr X( x ) (cid:17) + E k N + i ( x ) where sup x ∈ K (cid:12)(cid:12) E k N + i ( x ) (cid:12)(cid:12) ≤ c ∞ Õ j = k sup x ∈ K (cid:13)(cid:13) X ( j + ) N + i ( x ) − X j N + i ( x ) (cid:13)(cid:13) . Indeed, since for m ≥ n ≥ M , (cid:12)(cid:12)(cid:12)(cid:12) tr X nN + i √ det X nN + i − tr X mN + i √ det X mN + i (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) λ nN + i | λ nN + i | − λ mN + i | λ mN + i | (cid:12)(cid:12)(cid:12)(cid:12) . ∞ Õ j = n (cid:12)(cid:12) λ ( j + ) N + i − λ j N + i (cid:12)(cid:12) , we get (cid:12)(cid:12)(cid:12) θ n ( x ) − arccos (cid:16) tr X( x ) (cid:17)(cid:12)(cid:12)(cid:12) . ∞ Õ j = n (cid:12)(cid:12) λ ( j + ) N + i − λ j N + i (cid:12)(cid:12) , which proves our statement.5.2. Christofel functions.
We are now in the position to prove the main theorem of this section.
Theorem 5.4.
Let N be a positive integer and i ∈ { , , . . . , N − } . Suppose that K is a compact intervalwith non-empty interior and contained in Λ = (cid:26) x ∈ R : lim j →∞ discr X j N + i ( x ) exists and is negative (cid:27) . Assume that lim j →∞ a ( j + ) N + i − a j N + i − = and (cid:0) X j N + i : j ∈ N (cid:1) ∈ D (cid:0) K , GL ( , R ) (cid:1) . If ∞ Õ j = a j N + i − = ∞ , then K i ; n ( x , x ) = |[X( x )] , | π µ ′ ( x ) p − discr X( x ) ρ i − n + + E i ; n ( x ) , where X is the limit of ( X j N + i : j ∈ N ) , and sup x ∈ K (cid:12)(cid:12) E i ; n ( x ) (cid:12)(cid:12) ≤ c n Õ k = (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) a ( k + ) N + i − − a k N + i − (cid:12)(cid:12)(cid:12)(cid:12) + a ( k + ) N + i − Õ j ≥ k sup x ∈ K (cid:13)(cid:13) X ( j + ) N + i ( x ) − X j N + i ( x ) (cid:13)(cid:13)(cid:19) . Proof.
Let K ⊂ Λ be a compact interval with non-empty interior. By Theorem 5.1, there are c > and M ∈ N such that for all k ≥ M , a ( k + ) N + i − p k N + i ( x ) = |[X( x )] , | π µ ′ ( x ) p − discr X( x ) sin (cid:16) η ( x ) + k Õ j = M θ j N + i ( x ) (cid:17) + E k N + i ( x ) where sup x ∈ K (cid:12)(cid:12) E k N + i ( x ) (cid:12)(cid:12) ≤ c Õ j ≥ k sup x ∈ K (cid:13)(cid:13) X ( j + ) N + i ( x ) − X j N + i ( x ) (cid:13)(cid:13) . In view of the identity ( x ) = − cos ( x ) , we get n Õ k = M p k N + i ( x ) = |[X( x )] , | π µ ′ ( x ) p − discr X( x ) n Õ k = M a ( k + ) N + i − (cid:18) − cos (cid:16) η ( x ) + k Õ j = M θ j N + i ( x ) (cid:17) (cid:19) + n Õ k = M a ( k + ) N + i − E k N + i ( x ) . Since there is c > such that sup x ∈ K M − Õ k = p k N + i ( x ) ≤ c , by Lemma 4.1 and Remark 5.3, we obtain (cid:12)(cid:12)(cid:12)(cid:12) K i ; n ( x , x ) − |[X( x )] , | π µ ′ ( x ) p − discr X( x ) ρ i − n + (cid:12)(cid:12)(cid:12)(cid:12) ≤ c n Õ k = (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) a ( k + ) N + i − − a k N + i − (cid:12)(cid:12)(cid:12)(cid:12) + a ( k + ) N + i − Õ j ≥ k sup x ∈ K (cid:13)(cid:13) X ( j + ) N + i ( x ) − X j N + i ( x ) (cid:13)(cid:13)(cid:19) which completes the proof. (cid:3) Theorem 5.5.
Let A be a Jacobi matrix with N -periodically modulated entries. Suppose that for each i ∈ { , , . . . , N − } , (cid:18) a j N + i − a j N + i : j ∈ N (cid:19) , (cid:18) b j N + i a j N + i : j ∈ N (cid:19) , (cid:18) a j N + i : j ∈ N (cid:19) ∈ D , and ∞ Õ n = a n = ∞ . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 33 If | tr X ( )| < then K n ( x , x ) = ω ′ ( ) µ ′ ( x ) ρ n + E n ( x ) where ω is the equilibrium measure corresponding to σ ess ( A ) with A being the Jacobi matrix associated to ( α n : n ∈ N ) and ( β n : n ∈ N ) , ρ n = n Õ j = α j a j , and for each compact interval K ⊂ R with non-empty interior sup x ∈ K (cid:12)(cid:12) E n ( x ) (cid:12)(cid:12) ≤ c n + N Õ m = a m Õ j ≥ sup x ∈ K (cid:13)(cid:13) B m + ( j + ) N ( x ) − B m + j N ( x ) (cid:13)(cid:13) . Proof.
Let K be a compact interval with non-empty interior and contained in R . By Remark 4.5, for each i ∈ { , , . . . , N − } , the sequence ( X j N + i : j ∈ N ) belongs to D (cid:0) K , GL ( , R ) (cid:1) , thus, by Proposition 3.8, wehave lim j →∞ X j N + i ( x ) = X i ( ) uniformly with respect to x ∈ K . Since X i + ( x ) = (cid:0) B i ( x ) (cid:1) (cid:0) X i ( x ) (cid:1) (cid:0) B i ( x ) (cid:1) − , we have discr X i ( x ) = discr X ( x ) . By Theorem 5.4, K i ; n ( x , x ) = |[ X i ( )] , | π µ ′ ( x ) p − discr X ( ) ρ i − n + + E i ; n ( x ) where sup x ∈ K (cid:12)(cid:12) E i ; n ( x ) (cid:12)(cid:12) ≤ c n Õ k = a ( k + ) N + i − Õ j ≥ k sup x ∈ K (cid:13)(cid:13) B ( j + ) N + i ( x ) − B j N + i ( x ) (cid:13)(cid:13) . For k ∈ N and i ∈ { , , . . . , N − } we write K k N + i ( x , x ) = N − Õ i ′ = K i ′ ; k ( x , x ) + N − Õ i ′ = i + (cid:0) K i ′ ; k − ( x , x ) − K i ′ ; k ( x , x ) (cid:1) , Since (cid:12)(cid:12) K i ′ ; k − ( x , x ) − K i ′ ; k ( x , x ) (cid:12)(cid:12) = p k N + i ′ ( x ) ≤ c a k N + i ′ , we obtain(5.4) K k N + i ( x , x ) = π µ ′ ( x ) p − discr X ( x ) N − Õ i ′ = |[ X i ′ ( x )] , | · ρ i ′ − k + + E k N + i ( x ) , where sup x ∈ K (cid:12)(cid:12) E n ( x ) (cid:12)(cid:12) ≤ c n Õ m = a m + N − Õ j ≥ sup x ∈ K (cid:13)(cid:13) B m + ( j + ) N ( x ) − B m + j N ( x ) (cid:13)(cid:13) . We next claim the following holds true.
Claim 5.6.
For each i , i ′ ∈ { , , . . . , N − } , (5.5) (cid:12)(cid:12)(cid:12) α i ρ i ; k − α i ′ ρ i ′ ; k (cid:12)(cid:12)(cid:12) ≤ c ( k + ) N Õ m = a m Õ j ≥ sup x ∈ K (cid:13)(cid:13) B m + j N ( x ) − B m + ( j + ) N ( x ) (cid:13)(cid:13) . For the proof let us observe that, by Proposition 3.7, we have (cid:12)(cid:12)(cid:12)(cid:12) α i − α i − a k N + i − a k N + i (cid:12)(cid:12)(cid:12)(cid:12) ≤ Õ j ≥ k (cid:12)(cid:12)(cid:12)(cid:12) a j N + i − a j N + i − a ( j + ) N + i − a ( j + ) N + i (cid:12)(cid:12)(cid:12)(cid:12) ≤ c Õ j ≥ k sup x ∈ K (cid:13)(cid:13) B j N + i ( x ) − B ( j + ) N + i ( x ) (cid:13)(cid:13) . Therefore, (cid:12)(cid:12)(cid:12) α i ′ ρ i ′ ; k − α i ′′ ρ i ′′ ; k (cid:12)(cid:12)(cid:12) ≤ N − Õ i = k Õ j = (cid:12)(cid:12)(cid:12)(cid:12) α i − a j N + i − − α i a j N + i (cid:12)(cid:12)(cid:12)(cid:12) ≤ N − Õ i = k Õ j = α i a j N + i − (cid:12)(cid:12)(cid:12)(cid:12) α i − α i − a j N + i − a j N + i (cid:12)(cid:12)(cid:12)(cid:12) , which implies (5.5).Now, using Claim 5.6, we can write (cid:12)(cid:12) N α i ρ i ; k + − ρ ( k + ) N (cid:12)(cid:12) ≤ N − Õ i ′ = (cid:12)(cid:12) α i ρ i ; k + − α i ′ ρ i ′ ; k + (cid:12)(cid:12) ≤ c ( k + ) N Õ m = a m Õ j ≥ sup x ∈ K (cid:13)(cid:13) B m + j N ( x ) − B m + ( j + ) N ( x ) (cid:13)(cid:13) . Hence, by (5.4), we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K k N + i ( x , x ) − N π µ ′ ( x ) p − discr X ( ) N − Õ i ′ = |[ X i ′ ( )] , | α i ′ − ρ k N + i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( k + ) N Õ m = a m Õ j ≥ sup x ∈ K (cid:13)(cid:13) B m + j N ( x ) − B m + ( j + ) N ( x ) (cid:13)(cid:13) , which together with (3.1), concludes the proof. (cid:3) Theorem 5.7.
Let A be a Jacobi matrix with asymptotically N -periodic entries. Suppose that for each i ∈ { , , . . . , N − } , (cid:18) a j N + i − a j N + i : j ∈ N (cid:19) , (cid:18) b j N + i a j N + i : j ∈ N (cid:19) , (cid:18) a j N + i : j ∈ N (cid:19) ∈ D , Let K be a compact interval with non-empty interior contained in Λ = (cid:8) x ∈ R : (cid:12)(cid:12) tr X ( x ) (cid:12)(cid:12) < (cid:9) . Then K n ( x , x ) = ω ′ ( x ) µ ′ ( x ) ρ n + E n ( x ) where ω is the equilibrium measure corresponding to σ ess ( A ) with A being the Jacobi matrix associated to ( α n : n ∈ N ) and ( β n : n ∈ N ) , ρ n = n Õ j = α j a j , and sup x ∈ K (cid:12)(cid:12) E n ( x ) (cid:12)(cid:12) ≤ c n + N Õ m = a m Õ j ≥ sup x ∈ K (cid:13)(cid:13) B m + ( j + ) N ( x ) − B m + j N ( x ) (cid:13)(cid:13) . (5.6) SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 35
In the following two examples we want to compare the estimate (5.6) with some known results.
Example 2 (Generalized Jacobi) . Let h be a real-analytic positive function on the neighborhood of [− , ] .Let µ be a probability measure supported on [− , ] with the density µ ′ ( x ) = c · h ( x )( x + ) γ ( x − ) γ , x ∈ (− , ) , where γ , γ > − , and c is the normalizing constant. Then (see [6, Theorem 1.10]) a n = + c n + O (cid:0) n − (cid:1) , b n = c n + O (cid:0) n − (cid:1) . Therefore, by Theorem 5.5 we obtain ρ n K n ( x , x ) = π √ − x µ ′ ( x ) + O (cid:0) n − (cid:1) . Hence, we obtain the same rate as in [7, Theorem 1.1(a)].
Example 3 (Pollaczek-type) . Let µ be a probability measure supported on [− , ] with the density µ ′ ( x ) = c · exp (cid:0) − ( − x ) γ (cid:1) , x ∈ (− , ) , where γ ∈ ( , ) , and c is the normalizing constant. Then (see, [29, Corollary 4]) a n = + c n − /( + γ ) + O (cid:0) n − (cid:1) , b n = . Hence, Theorem 5.5 implies ρ n K n ( x , x ) = π √ − x µ ′ ( x ) + O (cid:0) n − (cid:1) . It should be compared with [28, Theorem 1(i)]5.3.
Auxiliary results.Lemma 5.8.
Let ( γ k : k ≥ ) be a sequence of positive numbers such that ∞ Õ k = γ k = ∞ , and lim n →∞ γ n − γ n = . Assume that ( θ n : n ≥ ) is a sequence of continuous functions on some open set U ⊂ R d with values in ( , π ) . Suppose that there is θ : U → ( , π ) such that lim n →∞ θ n ( x ) = θ ( x ) locally uniformly with respect to x ∈ U . Let ( r n : n ∈ N ) be a sequence of positive numbers such that lim n →∞ r n = ∞ . For x ∈ U , and a , b ∈ R , we set x n = x + ar n , and y n = x + br n . Then for each compact subset K ⊂ U , L > , and any function σ : U → R , (5.7) lim n →∞ n Õ k = γ k Í nj = γ j cos (cid:16) σ ( x n ) + σ ( y n ) + k Õ j = (cid:0) θ j ( x n ) + θ j ( y n ) (cid:1) (cid:17) = uniformly with respect to x ∈ K , and a , b ∈ [− L , L ] . Proof.
Let us fix K a compact subset of U and L > . Select R > so that (cid:18) x − LR , x + LR (cid:19) ⊂ U for all x ∈ K , and let N ∈ N be such that r n ≥ R for all n ≥ N . For ( x , a , b ) ∈ U × (− L , L ) , we set ˜ θ j ( x , a , b ) = θ j (cid:18) x + aR (cid:19) + θ j (cid:18) x + bR (cid:19) , and ˜ θ ( x , a , b ) = θ (cid:18) x + aR (cid:19) + θ (cid:18) x + bR (cid:19) . Then lim j →∞ ˜ θ j ( x , a , b ) = ˜ θ ( x , a , b ) uniformly with respect to ( x , a , b ) ∈ K × [− L , L ] . By Lemma 4.1, there is c > such that(5.8) (cid:12)(cid:12)(cid:12)(cid:12) n Õ k = γ k exp (cid:16) i k Õ j = ˜ θ j ( x , a , b ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c n − Õ k = (cid:12)(cid:12) γ k + − γ k (cid:12)(cid:12) + γ k + (cid:12)(cid:12) ˜ θ k + ( x , a , b ) − ˜ θ ( x , a , b ) (cid:12)(cid:12) for all x ∈ K , a , b ∈ [− L , L ] , and n ∈ N . Since ˜ θ j (cid:18) x , Rar n , Rbr n (cid:19) = θ j ( x n ) + θ j ( y n ) , by (5.8),(5.9) (cid:12)(cid:12)(cid:12)(cid:12) n Õ k = γ k exp (cid:16) i k Õ j = θ j ( x n ) + θ j ( y n ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c n − Õ k = (cid:12)(cid:12) γ k + − γ k (cid:12)(cid:12) + γ k + (cid:16)(cid:12)(cid:12) θ k + ( x n ) − θ ( x n ) (cid:12)(cid:12) + (cid:12)(cid:12) θ k + ( y n ) − θ ( y n ) (cid:12)(cid:12)(cid:17) for all x ∈ K , a , b ∈ [− L , L ] , and n ≥ N . Finally, cos (cid:16) k Õ j = (cid:0) θ j ( x n ) + θ j ( y n ) (cid:1) + σ ( x n ) + σ ( y n ) (cid:17) = cos (cid:16) k Õ j = (cid:0) θ j ( x n ) + θ j ( y n ) (cid:1) (cid:17) cos (cid:16) σ ( x n ) + σ ( y n ) (cid:17) − sin (cid:16) k Õ j = (cid:0) θ j ( x n ) + θ j ( y n ) (cid:1) (cid:17) sin (cid:16) σ ( x n ) + σ ( y n ) (cid:17) , which together with (5.9) implies that there are c > and N ∈ N , such that for any function σ : U → R andall x ∈ K , a , b ∈ [− L , L ] and n ≥ N ,(5.10) (cid:12)(cid:12)(cid:12)(cid:12) n Õ k = γ k cos (cid:16) σ ( x n ) + σ ( y n ) + k Õ j = (cid:0) θ j ( x n ) + θ j ( y n ) (cid:1) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c n − Õ k = (cid:12)(cid:12) γ k + − γ k (cid:12)(cid:12) + γ k + (cid:16)(cid:12)(cid:12) θ k + ( x n ) − θ ( x n ) (cid:12)(cid:12) + (cid:12)(cid:12) θ k + ( y n ) − θ ( y n ) (cid:12)(cid:12)(cid:17) . Finally, (5.7) follows from (5.10) by the Stolz–Cesàro theorem. (cid:3)
Theorem 5.9.
Let U be an open subset of R . Let ( γ k : k ≥ ) be a sequence of positive numbers such that (5.11) ∞ Õ k = γ k = ∞ , and lim k →∞ γ k − γ k = . Assume that ( θ k : k ≥ ) is a sequence of C ( U ) functions with values in ( , π ) such that for each compactset K ⊂ U there are functions θ : K → ( , π ) and ψ : K → ( , ∞) , and c > such that(a) lim n →∞ sup x ∈ K (cid:12)(cid:12) θ n ( x ) − θ ( x ) (cid:12)(cid:12) = , SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 37 (b) lim n →∞ sup x ∈ K (cid:12)(cid:12) γ − n · θ ′ n ( x ) − ψ ( x ) (cid:12)(cid:12) = , (c) sup n ∈ N sup x ∈ K (cid:12)(cid:12) γ − n · θ ′′ n ( x ) (cid:12)(cid:12) ≤ c . For x ∈ U and a , b ∈ R we set x n = x + a Í nk = γ k , and y n = x + b Í nk = γ k . Then for any continuous function σ : U → R , lim n →∞ n Õ k = γ k Í nj = γ j sin (cid:16) n Õ j = θ j ( x n ) + σ ( x n ) (cid:17) sin (cid:16) n Õ j = θ j ( y n ) + σ ( y n ) (cid:17) = sin (cid:0) ( b − a ) ψ ( x ) (cid:1) ( b − a ) ψ ( x ) locally uniformly with respect to x ∈ U , and a , b ∈ R .Proof. Let us fix a compact set K ⊂ U and L > . We write · sin (cid:16) n Õ j = θ j ( x n ) + σ ( x n ) (cid:17) sin (cid:16) n Õ j = θ j ( y n ) + σ ( y n ) (cid:17) = cos (cid:16) n Õ j = (cid:0) θ j ( x n ) − θ j ( y n ) (cid:1) + σ ( x n ) − σ ( y n ) (cid:17) − cos (cid:16) n Õ j = (cid:0) θ j ( x n ) + θ j ( y n ) (cid:1) + σ ( x n ) + σ ( y n ) (cid:17) . Moreover, cos (cid:16) n Õ j = (cid:0) θ j ( x n ) − θ j ( y n ) (cid:1) + σ ( x n ) − σ ( y n ) (cid:17) = cos (cid:16) n Õ j = θ j ( x n ) − θ j ( y n ) (cid:17) cos (cid:16) σ ( x n ) − σ ( y n ) (cid:17) − sin (cid:16) n Õ j = θ j ( x n ) − θ j ( y n ) (cid:17) sin (cid:16) σ ( x n ) − σ ( y n ) (cid:17) , thus, by the continuity of σ and Corollary 5.8, it is enough to show that lim n →∞ n Õ k = γ k Í nj = γ j cos (cid:16) n Õ j = (cid:0) θ j ( x n ) − θ j ( y n ) (cid:1) (cid:17) = sin (cid:0) ( b − a ) ψ ( x ) (cid:1) ( b − a ) ψ ( x ) uniformly with respect to x ∈ K , and a , b ∈ [− L , L ] . We first prove the following claim. Claim 5.10.
There is c > such that for all j ∈ N and n ∈ N , (5.12) (cid:12)(cid:12)(cid:12)(cid:12) θ j ( y n ) − θ j ( x n ) − ( b − a ) θ ′ j ( x ) (cid:16) n Õ ℓ = γ ℓ (cid:17) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ c (cid:16) n Õ ℓ = γ ℓ (cid:17) − sup x ∈ K | θ ′′ j ( x )| . For the proof, let us write Taylor’s polynomial for θ j centered at x , that is, θ j ( y ) = θ j ( x ) + θ ′ j ( x )( y − x ) + E j ( x ; y ) where | E j ( x ; y )| ≤ | y − x | sup w ∈[ x , y ] (cid:12)(cid:12) θ ′′ j ( w ) (cid:12)(cid:12) . Therefore, θ j ( y n ) − θ j ( x n ) = θ ′ j ( x )( y n − x n ) + E j ( x ; y n ) − E j ( x ; x n ) , which leads to (5.12). Let us now observe that, by the mean value theorem and Claim 5.10, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos (cid:16) k Õ j = θ j ( y n ) − θ j ( x n ) (cid:17) − cos (cid:16) ( b − a ) (cid:16) n Õ ℓ = γ ℓ (cid:17) − k Õ j = θ ′ j ( x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k Õ j = (cid:12)(cid:12)(cid:12) θ j ( y n ) − θ j ( x n ) − ( b − a ) (cid:16) n Õ ℓ = γ ℓ (cid:17) − θ ′ j ( x ) (cid:12)(cid:12)(cid:12) ≤ c (cid:16) n Õ ℓ = γ ℓ (cid:17) − k Õ j = sup x ∈ K | θ ′′ j ( x )| . Hence, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n Õ k = γ k Í nj = γ j cos (cid:16) k Õ j = θ j ( x n ) − θ j ( y n ) (cid:17) − cos (cid:16) ( b − a ) (cid:16) n Õ ℓ = γ ℓ (cid:17) − k Õ j = θ ′ j ( x ) (cid:17) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c (cid:16) n Õ ℓ = γ ℓ (cid:17) − n Õ k = γ k k Õ j = sup x ∈ K | θ ′′ j ( x )| . Now, by the Stolz–Cesàro theorem, we have lim n →∞ γ n Í nj = γ j = lim n →∞ γ n − γ n − γ n = , thus, by repeated application of the Stolz–Cesàro theorem we obtain lim n →∞ (cid:16) n Õ ℓ = γ ℓ (cid:17) − n Õ k = γ k k Õ j = sup x ∈ K | θ ′′ j ( x )| =
13 lim n →∞ (cid:16) n Õ ℓ = γ ℓ (cid:17) − n Õ j = sup x ∈ K | θ ′′ j ( x )| =
16 lim n →∞ γ − n (cid:16) n Õ ℓ = γ ℓ (cid:17) − sup x ∈ K | θ ′′ n ( x )| . In view of (c), it is enough to show that lim n →∞ n Õ k = γ k Í nj = γ j cos (cid:16) ( b − a ) (cid:16) n Õ ℓ = γ ℓ (cid:17) − k Õ j = θ ′ j ( x ) (cid:17) = sin (cid:0) ( b − a ) ψ ( x ) (cid:1) ( b − a ) ψ ( x ) . For the proof, we write n Õ k = γ k Í nj = γ j cos (cid:16) ( b − a ) (cid:16) n Õ ℓ = γ ℓ (cid:17) − k Õ j = θ ′ j ( x ) (cid:17) = n Õ k = ∞ Õ m = (− ) m ( m ) ! ( b − a ) m γ k (cid:16) n Õ ℓ = γ ℓ (cid:17) − m − (cid:16) k Õ j = θ ′ j ( x ) (cid:17) m = ∞ Õ m = (− ) m ( m ) ! ( b − a ) m (cid:16) n Õ ℓ = γ ℓ (cid:17) − m − n Õ k = γ k (cid:16) k Õ j = θ ′ j ( x ) (cid:17) m . We now claim the following.
Claim 5.11.
For each m ∈ N , (5.13) lim n →∞ (cid:16) n Õ ℓ = γ ℓ (cid:17) − m − n Õ k = γ k (cid:16) k Õ j = θ ′ j ( x ) (cid:17) m = m + (cid:0) ψ ( x ) (cid:1) m . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 39
By (b) and the Stolz–Cesàro theorem, we get lim n →∞ Í nj = θ ′ j ( x ) Í n ℓ = γ ℓ = ψ ( x ) . Since m + (cid:18) Í nj = θ ′ j ( x ) Í n ℓ = γ ℓ (cid:19) m ≤ γ n (cid:0) Í nj = θ ′ j ( x ) (cid:1) m (cid:0) Í n ℓ = γ ℓ (cid:1) m + − (cid:0) Í n − ℓ = γ ℓ (cid:1) m + ≤ m + (cid:18) Í nj = θ ′ j ( x ) Í n − ℓ = γ ℓ (cid:19) m , we get lim n →∞ γ n (cid:0) Í nj = θ ′ j ( x ) (cid:1) m (cid:0) Í n ℓ = γ ℓ (cid:1) m + − (cid:0) Í n − ℓ = γ ℓ (cid:1) m + = m + (cid:0) ψ ( x ) (cid:1) m . Therefore, another application of the Stolz–Cesàro theorem leads to (5.13).Let us notice that for some c > , n Õ k = γ k (cid:16) k Õ j = θ ′ j ( x ) (cid:17) m ≤ c m (cid:16) n Õ k = γ k (cid:17) m + , thus, we have the estimate (cid:12)(cid:12)(cid:12)(cid:12) (− ) m ( m ) ! ( b − a ) m (cid:16) n Õ ℓ = γ ℓ (cid:17) − m − n Õ k = γ k (cid:16) k Õ j = θ ′ j ( x ) (cid:17) m (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( m ) ! ( b − a ) m c m . Hence, by the dominated convergence theorem and Claim 5.11, we can compute lim n →∞ ∞ Õ m = (− ) m ( m ) ! ( b − a ) m (cid:16) n Õ ℓ = γ ℓ (cid:17) − m − n Õ k = γ k (cid:16) k Õ j = θ ′ j ( x ) (cid:17) m = + ∞ Õ m = (− ) m ( m + ) ! (cid:0) ( b − a ) ψ ( x ) (cid:1) m = sin (cid:0) ( b − a ) ψ ( x ) (cid:1) ( b − a ) ψ ( x ) , which finishes the proof of the theorem. (cid:3) Christoffel–Darboux kernel.Proposition 5.12.
Let A be a Jacobi matrix with N -periodically modulated entries. Then for every compactsubset K ⊂ R we have (5.14) lim n →∞ sup x ∈ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a n α n θ ′ n ( x ) + tr X ′ ( ) N p − discr X ( ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = , and (5.15) sup x ∈ K | θ ′′ n ( x )| ≤ c (cid:16) α n a n (cid:17) for some c > .Proof. Let us fix i ∈ { , , . . . , N − } . Since(5.16) det X k N + i ( x ) = a k N + i − a ( k + ) N + i − , by Proposition 3.7, we conclude that(5.17) lim k →∞ det X k N + i ( x ) = . The chain rule applied to (5.1) leads to θ ′ k N + i ( x ) = − − (cid:18) tr X k N + i ( x ) p det X k N + i ( x ) (cid:19) ! − / tr X ′ k N + i ( x ) p det X k N + i ( x ) = − tr X ′ k N + i ( x ) p − discr X k N + i ( x ) , thus, by (5.17) and Corollary 3.10, we obtain (5.14). Consequently, in view of (5.16), θ ′′ k N + i ( x ) = − tr X ′′ k N + i ( x ) p − discr X k N + i ( x ) − (cid:0) tr X ′ k N + i ( x ) (cid:1) tr X k N + i ( x ) (cid:0) − discr X k N + i ( x ) (cid:1) / . Therefore, the estimate (5.15) is a consequence of Corollary 3.10. (cid:3)
Theorem 5.13.
Let A be a Jacobi matrix with N -periodically modulated entries. Suppose that for each i ∈ { , , . . . , N − } , (5.18) (cid:18) a j N + i − a j N + i : j ∈ N (cid:19) , (cid:18) b j N + i a j N + i : j ∈ N (cid:19) , (cid:18) a j N + i : j ∈ N (cid:19) ∈ D and (5.19) ∞ Õ j = a j = ∞ . If | tr X ( )| < , then lim n →∞ ρ n K n (cid:18) x + u ρ n , x + v ρ n (cid:19) = ω ′ ( ) µ ′ ( x ) · sin (cid:0) ( u − v ) πω ′ ( ) (cid:1) ( u − v ) πω ′ ( ) locally uniformly with respect to x , u , v ∈ R , where ρ n = n Õ j = α j a j , and ω is the equilibrium measure corresponding to σ ess ( A ) with A being the Jacobi matrix associated to ( α n : n ∈ N ) and ( β n : n ∈ N ) .Proof. Let us fix a compact set K ⊂ Λ with non-empty interior and L > . Let ˜ K ⊂ Λ be a compactset containing K in its interior. There is n > such that for all x ∈ K , n ≥ n , u ∈ [− L , L ] , and i ∈ { , , . . . , N − } , x + uN α i ρ i ; n , x + u ρ nN + i ∈ ˜ K . Given u , v ∈ [− L , L ] , we set x i ; n = x + uN α i ρ i ; n , x nN + i = x + u ρ nN + i , y i ; n = x + v N α i ρ i ; n , y nN + i = x + v ρ nN + i . Remark 4.5 together with (5.18) entails that ( X j N + i : j ∈ N ) belongs to D . Moreover, lim k →∞ X k N + i ( x ) = X i ( ) uniformly with respect to x ∈ K . Let us recall that discr X i ( x ) = discr X ( x ) . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 41
In view of (4.12), the Carleman condition (5.19) implies that lim k →∞ ρ i ; k = ∞ . Hence, by Theorem 5.1, there are c > and M ∈ N such that for all x , y ∈ ˜ K and k ≥ M ,(5.20) a ( k + ) N + i − p k N + i ( x ) p k N + i ( y ) = |[ X i ( )] , | π p − discr X ( y ) · p µ ′ ( x ) µ ′ ( y ) sin (cid:16) k Õ j = M + θ j N + i ( x ) + η ( x ) (cid:17) sin (cid:16) k Õ j = M + θ j N + i ( y ) + η ( y ) (cid:17) + E k N + i ( x , y ) where sup x , y ∈ ˜ K (cid:12)(cid:12) E k N + i ( x , y ) (cid:12)(cid:12) ≤ c Õ j ≥ k sup x ∈ ˜ K (cid:13)(cid:13) X ( j + ) N + i ( x ) − X j N + i ( x ) (cid:13)(cid:13) . Hence, we obtain n Õ k = M p k N + i ( x ) p k N + i ( y ) = |[ X i ( )] , | π p − discr X ( x ) · p µ ′ ( x ) µ ′ ( y )× n Õ k = M a ( k + ) N + i − sin (cid:16) k Õ j = M + θ j N + i ( x ) + η ( x ) (cid:17) sin (cid:16) k Õ j = M + θ j N + i ( y ) + η ( y ) (cid:17) + n Õ k = M a ( k + ) N + i − E k N + i ( x , y ) . Let γ k = N α i − a k N + i − , and ψ ( x ) = − tr X ′ ( ) N p − discr X ( ) . By Proposition 5.12 lim n →∞ sup x ∈ ˜ K (cid:12)(cid:12)(cid:12)(cid:12) a k N + i − α i − θ ′ k N + i ( x ) − ψ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = . Moreover, by (3.1) and (3.2) | ψ ( x )| = πω ′ ( ) . Therefore, by Theorem 5.9 we get lim n →∞ N α i ρ i ; n n Õ k = M N α i − a k N + i − sin (cid:16) k Õ j = M + θ j N + i ( x i − n ) + η ( x i − n ) (cid:17) sin (cid:16) k Õ j = M + θ j N + i ( y i − n ) + η ( y i − n ) (cid:17) = sin (cid:0) ( u − v ) πω ′ ( x ) (cid:1) ( u − v ) πω ′ ( x ) . Now, by uniformness and (4.12), for any i ′ ∈ { , , . . . , N − } , we obtain lim n →∞ N α i − ρ nN + i K i ; n ( x nN + i ′ , y nN + i ′ ) = |[ X i ( )] , | π µ ′ ( x ) p − discr X ( ) · lim n →∞ sin (cid:18) ( u − v ) ρ nN + i ′ N α i − ρ i − n πω ′ ( x ) (cid:19) ( u − v ) ρ nN + i ′ N α i − ρ i − n πω ′ ( x ) = |[ X i ( )] , | π µ ′ ( x ) p − discr X ( ) · sin (cid:0) ( u − v ) πω ′ ( x ) (cid:1) ( u − v ) πω ′ ( x ) (5.21)uniformly with respect to x ∈ K and u , v ∈ [− L , L ] . Here, we have also used that(5.22) sup m ∈ N sup x ∈ ˜ K | p m ( x )| ≤ c . Since K nN + i ( x , y ) = N − Õ i ′ = K i ′ ; n ( x , y ) + N − Õ i ′ = i + (cid:0) K i ′ ; n − ( x , y ) − K i ′ ; n ( x , y ) (cid:1) , by (5.22) and (5.21), we obtain lim n →∞ ρ nN + i K nN + i ( x nN + i , y nN + i ) = N N − Õ i ′ = N α i ′ − ρ nN + i K i ′ ; n ( x nN + i , y nN + i ) α i ′ − = sin (cid:0) ( u − v ) πω ′ ( x ) (cid:1) ( u − v ) πω ′ ( x ) · N π µ ′ ( x ) p − discr X ( ) N − Õ i ′ = |[ X i ′ ( )] , | α i ′ − which together with (3.1) concludes the proof. (cid:3) Proposition 5.14.
Let A be a Jacobi matrix that is N -periodic blend. Let K be a non-empty compact intervalcontained in Λ = (cid:8) x ∈ R : | tr X ( x )| < (cid:9) where X is the limit of ( X j ( N + ) + : j ∈ N ) . Then for each i ∈ { , , . . . , N } , (5.23) lim n →∞ sup x ∈ K (cid:12)(cid:12)(cid:12)(cid:12) a n ( N + ) + i α i θ ′ n ( N + ) + i ( x ) + tr X ′ ( x ) N p − discr X ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = , and (5.24) sup x ∈ K | θ ′′ n ( N + ) + i ( x )| ≤ c (cid:18) α i a n ( N + ) + i (cid:19) for some c > .Proof. Let us fix i ∈ { , , . . . , N } . Since(5.25) det X k ( N + ) + i ( x ) = a k ( N + ) + i − a ( k + )( N + ) + i − , we conclude that(5.26) lim k →∞ det X k ( N + ) + i ( x ) = . The chain rule applied to (5.1) leads to θ ′ k ( N + ) + i ( x ) = − − (cid:18) tr X k ( N + ) + i ( x ) p det X k ( N + ) + i ( x ) (cid:19) ! − / tr X ′ k ( N + ) + i ( x ) p det X k ( N + ) + i ( x ) = − tr X ′ k ( N + ) + i ( x ) p − discr X k ( N + ) + i ( x ) thus, by (5.26) and Corollary 3.18, we obtain (5.23). Consequently, in view of (5.25), θ ′′ k ( N + ) + i ( x ) = − tr X ′′ k ( N + ) + i ( x ) p − discr X k ( N + ) + i ( x ) − (cid:0) tr X ′ k ( N + ) + i ( x ) (cid:1) tr X k ( N + ) + i ( x ) (cid:0) − discr X k ( N + ) + i ( x ) (cid:1) / . Therefore, the estimate (5.24) is a consequence of Corollary 3.18. (cid:3)
Theorem 5.15.
Let A be a Jacobi matrix that is N -periodic blend. Suppose that for each i ∈ { , , . . . , N − } , (cid:18) a j ( N + ) + i : j ∈ N (cid:19) , (cid:18) b j ( N + ) + i a j ( N + ) + i : j ∈ N (cid:19) ∈ D and (cid:18) a j ( N + ) + N : j ∈ N (cid:19) , (cid:18) a j ( N + ) + N + : j ∈ N (cid:19) , (cid:18) a j ( N + ) + N a j ( N + ) + N + : j ∈ N (cid:19) ∈ D . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 43
Let K be a compact interval with non-empty interior contained in Λ = (cid:8) x ∈ R : | tr X ( x )| < (cid:9) where X is the limit of ( X j ( N + ) + : j ∈ N ) . Then lim n →∞ ρ n K n (cid:18) x + u ρ n , x + v ρ n (cid:19) = ω ′ ( x ) µ ′ ( x ) · sin (cid:0) ( u − v ) πω ′ ( x ) (cid:1) ( u − v ) πω ′ ( x ) locally uniformly with respect to x ∈ K , and u , v ∈ R , where ω is the equilibrium measure corresponding to Λ , and ρ n = N − Õ i = n Õ m = m ≡ i mod ( N + ) α m a m . Proof.
Let K ⊂ Λ be a compact interval with non-empty interior and let L > . Let ˜ K ⊂ Λ be a compactset containing K in its interior. There is n > such that for all x ∈ K , n ≥ n , i ∈ { , , . . . , N + } , and u ∈ [− L , L ] , x + u ρ n ( N + ) + i ∈ ˜ K . Given x ∈ K and u , v ∈ [− L , L ] , we set x n = x + u ρ n , and y n = x + v ρ n . For each i ∈ { , , . . . , N } and i ′ ∈ { , , . . . , N + } , by the reasoning analogous to the proof of Theorem5.13 one can show that(5.27) lim n →∞ N α i − ρ i ; n K i ; n ( x n ( N + ) + i ′ , y n ( N + ) + i ′ ) = |[X i ( x )] , | π µ ′ ( x ) p − discr X i ( x ) · sin (cid:0) ( u − v ) πω ′ ( x ) (cid:1) ( u − v ) πω ′ ( x ) uniformly with respect to x ∈ K and u , v ∈ [− L , L ] . By Claim 4.11, for each i ∈ { , , . . . , N } the sequence ( X j ( N + ) + i : j ∈ N ) belongs to D (cid:0) K , GL ( , R ) (cid:1) and converges to X i satisfying discr X i = discr X . For i = , by (4.26) and Claim 4.12, (cid:12)(cid:12) K n ( x , y ) − K N ; n ( x , y ) (cid:12)(cid:12) ≤ c (cid:16) n Õ j = (cid:12)(cid:12) p j ( N + ) ( x ) + p j ( N + ) + N ( x ) (cid:12)(cid:12) + (cid:12)(cid:12) p j ( N + ) ( y ) + p j ( N + ) + N ( y ) (cid:12)(cid:12)(cid:17) ≤ c k Õ j = a j ( N + ) + N + + (cid:12)(cid:12)(cid:12)(cid:12) − a j ( N + ) + N a j ( N + ) + N + (cid:12)(cid:12)(cid:12)(cid:12) . Therefore,(5.28) lim n →∞ ρ N − n sup x , y ∈ ˜ K (cid:12)(cid:12) K n ( x , y ) − K N ; n ( x , y ) (cid:12)(cid:12) = . Similarly, for i = N + , one can show(5.29) lim n →∞ ρ N − n sup x , y ∈ ˜ K (cid:12)(cid:12) K N + n ( x , y ) (cid:12)(cid:12) = . Now, let i ∈ { , , . . . , N + } . We write K n ( N + ) + i ( x , y ) = N + Õ i ′ = K i ′ ; n ( x , y ) + N + Õ i ′ = i + (cid:0) K i ′ ; n ( x , y ) − K i ′ ; n − ( x , y ) (cid:1) . By (4.26), (5.28) and (5.29), we obtain K n ( N + ) + i ( x , y ) = N − Õ i ′ = K i ′ ; n ( x , y ) + K N ; n ( x , y ) + o (cid:0) ρ N − n (cid:1) uniformly with respect to x , y ∈ ˜ K . Therefore, by (5.27) and (4.12), lim n →∞ ρ n ( N + ) + i K n ( N + ) + i ( x n ( N + ) + i , y n ( N + ) + i ) = N − Õ i ′ = lim n →∞ N α i ′ − ρ i ′ ; n K i ′ ; n ( x n ( N + ) + i , y n ( N + ) + i ) + n →∞ N α N − ρ N − n K N − n ( x n ( N + ) + i , y n ( N + ) + i ) = sin (cid:0) ( u − v ) πω ′ ( x ) (cid:1) ( u − v ) πω ′ ( x ) · π µ ′ ( x ) p − discr X ( x ) (cid:18) N − Õ i ′ = |[X i ′ ( x )] , | α i ′ − + |[X N ( x )] , | α N − (cid:19) which together with Theorem 3.13 finishes the proof. (cid:3) References [1] A. Boutet de Monvel, J. Janas, and S. Naboko,
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E-mail address : [email protected] Bartosz Trojan, Instytut Matematyczny, Polskiej Akademii Nauk, ul. Śniadeckich 8, 00-696 Warszawa, Poland
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