Asymptotic behaviour of Christoffel-Darboux kernel via three-term recurrence relation II
aa r X i v : . [ m a t h . SP ] A p r ASYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL VIA THREE-TERMRECURRENCE RELATION II
GRZEGORZ ŚWIDERSKI AND BARTOSZ TROJANAbstract. We study orthogonal polynomials with periodically modulated Jacobi parameters in the case when lies on the soft edge of the spectrum of the corresponding periodic Jacobi matrix. We determine when theorthogonality measure is absolutely continuous and we provide a constructive formula for it in terms of the limitof Turán determinants. We next consider asymptotics of the solutions of associated second order differenceequation. Finally, we study scaling limits of the Christoffel–Darboux kernel. 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 on the sequence of monomials ( x n : n ∈ N ) the Gram–Schmidt orthogonalization processone 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 . One ofthe central object of this article is the
Christoffel–Darboux kernel K n , which is defined as K n ( x , y ) = n Õ j = p j ( x ) p j ( y ) . The classical topic in analysis is studying the asymptotic behavior of orthogonal polynomials ( p n ) , whichallows to find the asymptotic of Christoffel–Darboux kernel. To motivate the interest in Christoffel–Darbouxkernel see surveys [11] and [15].The case when the measure µ has compact support is well understood. For the asymptotics of thepolynomials see e.g. the monograph [17] where the classical potential theory is the basic tool. Nowadays,the so-called Riemann–Hilbert method is commonly used to derive precise asymptotics of the orthogonalpolynomials as well as its Jacobi parameters, see e.g. [1, 9] and the book [2]. However, this method demandsstronger regularity conditions than those imposed in [17]. One of the most general result concerning the
Mathematics Subject Classification.
Primary: 42C05, 47B36.
Key words and phrases.
Orthogonal polynomials, asymptotics, Turán determinants, Christoffel functions, scaling limits.
Christoffel–Darboux kernel has been proven in [24]. Namely, if I is an open interval contained in supp ( µ ) ,so that µ is absolutely continuous on I with continuous positive density µ ′ , then(1.3) 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 [17, Definition 3.1.2]).In the formula (1.3), ω ′ denotes the density of the equilibrium measure corresponding to the support of µ ,see (2.1) 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 ) .The best understood class of measures with unbounded support is the class of exponential weights . Inthe monograph [10] asymptotics of the polynomials as well as their Jacobi parameters were studied under anumber of regularity conditions imposed on the function Q ( x ) = − log µ ′ ( x ) . Concerning the Christoffel–Darboux kernel, it was recently proven in [6] that under some conditions(1.4) 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 . Let us comment that ρ n is comparable to n , if the sequences ( a n ) and ( a − n ) are bounded. In particular,the conditions imposed on Q imply that the density of µ is an even, everywhere positive and continuouslydifferentiable function.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) for a given sequences ( a n ) and ( b n ) such that a n > and b n ∈ R . Inview of the Favard’s theorem (see, e.g. [14, Theorem 5.10]), there is a probability measure ν such that ( p n ) isorthonormal in L ( R , ν ) . The measure ν is unique, if and only if there is exactly one measure with the samemoments as ν . In such a case we call ν determinate and denote it by µ . Otherwise ν is indeterminate . Forexample, the determinacy of ν is implied by the Carleman condition (1.5) ∞ Õ n = a n = ∞ (see, e.g. [14, Corollary 6.19]). However, the condition (1.5) is not sufficient. Let us recall that theorthogonality measure has compact support, if and only if the Jacobi parameters are bounded.In the setup when the Jacobi parameters are central objects, the questions concerning asymptotic behaviorof orthogonal polynomials and the Christoffel–Darboux kernel make a perfect sense. Additionally, if themeasure ν is determinate, one can ask how to approximate it.In this article we are exclusively interested in unbounded Jacobi parameters. We shall mostly consider theclass of periodically modulated sequences. This class has been introduced in [7] and systematically studiedsince then. To be more precise, let N be a positive integer. We say that Jacobi parameters ( a n ) and ( b n ) are N -periodically modulated if there are two N -periodic sequences ( α n : n ∈ Z ) and ( β n : n ∈ Z ) of positiveand 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) = . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 3
It turns out that properties of µ corresponding to N -periodically modulated Jacobi parameters are intimatelyrelated to the trace of the matrix X ( x ) = N Ö j = − α j − α j x − β j α j ! . More precisely, under some regularity assumptions imposed on the Jacobi parameters, the measure µ is purelyabsolutely continuous on R with positive continuous density when | tr X ( )| < (see [18, 23]), whereas µ ispurely discrete when | tr X ( )| > (see [7]). In the boundary case | tr X ( )| = , we have two possibilities:either the matrix X ( ) is diagonalizable (which implies that X ( ) = σ Id for some σ ∈ {− , } ), or it issimilar to a non-trivial Jordan block. In this article we are concerned with the first case. The second one ismore challenging and we leave it for future research.In the case | tr X ( )| < the asymptotic behavior of ( p n : n ∈ N ) has already been studied in [23], andthe scaling limit of K n has been recently obtained in [21]. In particular, it has been shown that under someregularity assumptions one has(1.6) 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 , where(1.7) ω ′ ( ) = | tr X ′ ( )| N q − (cid:0) tr X ( ) (cid:1) , and ρ n = n Õ j = α j a j . Notice that by taking α n ≡ , β n ≡ , N = , and u = v = , we can reproduce (1.4).Let us emphasize that | tr X ( )| = when X ( ) = σ Id . This situation lies on the boundary of theprevious case. In particular, the formula (1.7) is not well-defined, and consequently, it is not clear whetherany analogue of (1.6) holds true. Moreover, the behavior of the corresponding measure is different than inthe case | tr X ( )| < . In fact, under some regularity conditions, there is an explicit compact interval I ⊂ R such that the measure µ is purely absolutely continuous on R \ I with continuous positive density, and it isdiscrete on I (see [19]). Moreover, in our forthcoming article [22] we have shown that in fact the supportof µ has no accumulation points in the interior of I . All of this suggest that the asymptotic behavior of thepolynomials and the Christoffel–Darboux kernel might be different in this setup. A very natural examplesatisfying our theorems are the following Jacobi parameters a n = ( n + ) κ + (cid:0) (− ) n + (cid:1) , b n ≡ , for κ ∈ ( , ) and N = . In view of [3], the measure µ is absolutely continuous on R and has support equalto R \ (− , ) .Before we go further, let us introduce some terminology. A sequence ( u n ∈ N ) is generalized eigenvector associated with x ∈ R , if it satisfies the recurrence relation(1.8) xu n = a n u n + + b n u n + a n − u n − , n ≥ , with some initial condition ( u , u ) , ( , ) . The relation (1.8) can be rewritten as(1.9) (cid:18) u n u n + (cid:19) = B n ( x ) (cid:18) u n − u n (cid:19) where B n is -step transfer matrix defined as B n ( x ) = (cid:18) − a n − a n x − b n a n (cid:19) . GRZEGORZ ŚWIDERSKI AND BARTOSZ TROJAN
To study N -periodic modulations we consider N -step transfer matrix defined as X n = n + N − Ö j = n B j . By GL ( , R ) we denote × real invertible matrices equipped with the spectral norm. For a matrix Y = (cid:18) y , y , y , y , (cid:19) we set [ Y ] i , j = y i , j ; its discriminant is defined as discr Y = ( tr Y ) − Y . Given a compact subset K ⊂ R ,we say that the sequence ( Y n : n ∈ N ) of mappings Y n : K → GL ( , R ) belongs to D (cid:0) K , GL ( , R ) (cid:1) , if ∞ Õ n = sup x ∈ K k Y n + ( x ) − Y n ( x )k < ∞ . It turns out that some properties of the measure µ depend on the asymptotic behavior of generalizedeigenvectors. For example, µ is determinate, if and only if there is a generalized eigenvector correspondingto x ∈ R which is not square-summable. Moreover, subordinacy theory (see e.g. [8]) implies that, if µ isdeterminate and I ⊂ R is an open interval such that for any generalized eigenvectors ( u n ) , ( v n ) associatedwith x ∈ I , sup n ≥ Í nk = | u k | Í nk = | v k | < ∞ , then µ is absolutely continuous on I , and I ⊂ supp ( µ ) . This motivates the study of the asymptotic behaviorof generalized eigenvectors. Theorem A.
Let N be a positive integer, and i ∈ { , , . . . , N − } . Suppose that lim j →∞ a j N + i − = ∞ . Let Λ i = n x ∈ R : lim j →∞ discr R j N + i ( x ) exists and is negative o where R n ( x ) = a n + N − (cid:0) X n ( x ) − σ Id (cid:1) for some σ ∈ {− , } . Suppose that K ⊂ Λ i is a compact interval with non-empty interior such that (cid:0) X j N + i : j ∈ N (cid:1) , (cid:0) R j N + i : j ∈ N (cid:1) ∈ D (cid:0) K , GL ( , R ) (cid:1) . Then there is a constant c > such that for every generalized eigenvector ( u n : n ∈ N ) associated with x ∈ K , and all n ≥ , (1.10) c − (cid:0) u + u (cid:1) ≤ a nN + i − (cid:0) u nN + i − + u nN + i (cid:1) ≤ c (cid:0) u + u (cid:1) . Theorem A is a generalization of [19, Theorem C], and is proven in Section 3, see Theorem 1. If thehypotheses of Theorem A are satisfied for all i ∈ { , , . . . , N − } , the Carleman condition (1.5), with ahelp of subordinacy theory, implies that the bounds (1.10) entail the absolute continuity of the measure µ oneach compact subset K ⊂ Ñ N − i = Λ i . However, this method does not give any additional information about µ .Because of this and inspired by earlier work for bounded Jacobi parameters [4, 12], in [18] the first authorhas introduced a method in the case of unbounded Jacobi parameters that allows to approximate µ in termsof N -shifted Turán determinants. The later are defined as follows D Nn ( x ) = det (cid:18) p n + N − ( x ) p n − ( x ) p n + N ( x ) p n ( x ) (cid:19) = p n ( x ) p n + N − ( x ) − p n − ( x ) p n + N ( x ) . See also [19, 23] and the references therein.Our second result concerns the convergence of N -shifted Turán determinants. SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 5
Theorem B.
Suppose that the hypotheses of Theorem A are satisfied. Assume further that lim j →∞ (cid:0) a ( j + ) N + i − − a j N + i − (cid:1) = . Then there is a positive function g i , such that (1.11) lim j →∞ sup x ∈ K (cid:12)(cid:12)(cid:12) a ( j + ) N + i − (cid:12)(cid:12) D Nj N + i ( x ) (cid:12)(cid:12) − g i ( x ) (cid:12)(cid:12)(cid:12) = . Moreover, the measure µ is absolutely continuous on K with the density µ ′ ( x ) = p − h i ( x ) π g i ( x ) , x ∈ K where (1.12) h i ( x ) = lim n →∞ discr (cid:0) R nN + i ( x ) (cid:1) , x ∈ K . Again, Theorem B is a generalization of [19, Theorem D]. Its proof is in Section 4, see Theorem 2.In the next theorem we study asymptotics of the polynomials in more detail.
Theorem C.
Suppose that the hypotheses of Theorem B are satisfied. Then there are M ≥ and a continuousreal-valued function η such that for all n ≥ M , (1.13) lim n →∞ sup x ∈ K (cid:12)(cid:12)(cid:12)(cid:12) √ a nN + i − p nN + i ( x ) − vt (cid:12)(cid:12) [R i ( x )] , (cid:12)(cid:12) π µ ′ ( x ) p − h i ( x ) sin (cid:16) n Õ j = M + θ j ( x ) + η ( x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) = where h i is given by (1.12) , R i ( x ) = lim n →∞ R nN + i ( x ) , and θ j ( x ) = arccos (cid:18) tr X j N + i ( x ) p det X j N + i ( x ) (cid:19) . Let us mention that asymptotics of the polynomials in the case when ( X nN + i : n ∈ N ) is convergent tothe matrix X such that discr X < , has been obtained in [23]. Theorem C corresponds to the case when discr X = . For the proof of Theorem C we refer to Theorem 3.Finally, in the last theorem we study the analogues of (1.6) for periodic modulations such that X ( ) = σ Id for some σ ∈ {− , } . It is proven in Theorem 7. Theorem D.
Suppose that the hypotheses of Theorem B are satisfied for all i ∈ { , , . . . , N − } . Assumefurther that Jacobi parameters ( a n ) and ( b n ) are N -periodically modulated so that X ( ) = σ Id . Then (1.14) lim n →∞ ρ n K n (cid:18) x + u ρ n , x + v ρ n (cid:19) = υ ( x ) µ ′ ( x ) sin (cid:0) ( u − v ) υ ( x ) (cid:1) ( u − v ) υ ( x ) locally uniformly with respect to ( x , u , v ) ∈ Λ × R , where ρ n = n Õ j = α j a j , and υ ( x ) = π N α N − | h ′ ( x )| p − h ( x ) . Observe that in (1.6) the factor ω ′ ( ) is constant whereas in (1.14) the factor υ usually depends on x ∈ Λ .For cases when υ ( x ) = ω ′ ( ) for each x , see Theorem 6. By taking u = v = in Theorem D we obtainsufficient conditions under which Ignjatović’s conjecture is valid, see Corollary 2 for details. In general theconjecture is false.Let us present some ideas of the proofs. In view of [20, Theorem 1] it is enough to prove the uniformconvergence of generalized N -shifted Turán determinants a n + N − det (cid:18) u n + N − u n − u n + N u n (cid:19) . GRZEGORZ ŚWIDERSKI AND BARTOSZ TROJAN
We do so by careful analysis of ( X nN + i : n ∈ N ) following the method developed in [19], see Section 3for details. To prove Theorem B, we use the convergence of generalized N -shifted Turán determinantstogether with the approximation method described in [18], see Section 4. To prove Theorem C we follow themethod recently introduced in [23] for the case when the limit of the sequence ( X nN + i : n ∈ N ) has negativediscriminant. In the current setup the analysis is much more subtle and involved. Finally, to prove Theorem Dwe use the asymptotics from Theorem C. It results in estimating the following oscillatory sum 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 sum we prove two auxiliary results (see Lemma 1 and Lemma 2) that are valid for sequencesnot necessarily belonging to D . In Section 7, we show a number of necessary algebraic identities which arespecific to the periodic modulations in the setup X ( ) = σ Id .The article is organized as follows: In Section 2 we fix some basic notation. Section 3 is devoted to provingTheorem A (see Theorem 1). In the next section we describe the approximation procedure (see Proposition1) which is a tool in proving Theorem B (see Theorem 2). In Section 5, we study the asymptotic behavior oforthogonal polynomials. Behavior of the Christoffel function in residue classes is analyzed in Section 6. Inthe next section we define periodic modulations and introduce a function υ . Section 8 is dedicated to studythe Christoffel function for periodic modulations. Finally, in Section 9 we investigate asymptotic behaviorof the Christoffel–Darboux kernel. Let us emphasize that Lemma 1 and Lemma 2 are sufficiently general toallow studying other types of scaling limits of Christoffel–Darboux kernels. In particular, in Theorem 9 it isshown how Lemma 2 may be applied. 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 whose value may vary from occurrence to occurrence. Acknowledgment.
The first author was partially supported by the Foundation for Polish Science (FNP) andby long term structural funding – Methusalem grant of the Flemish Government. Preliminaries
Given Jacobi parameters ( a n : n ∈ N ) and ( b n : n ∈ N ) and k ∈ N , we define polynomials ( p [ k ] n : n ∈ N ) by relations p [ k ] ( x ) = , p [ k ] ( x ) = x − b k a k , xp [ k ] n ( x ) = a n + k − p [ k ] n − ( x ) + b n + k p [ k ] n ( x ) + a n + k p [ k ] n + ( x ) , n ≥ . We usually omit the superscript when k = . In particular, ( p n ( x ) : n ∈ N ) are the generalized eigenvectorsassociated with x satisfying the initial condition p ( x ) = , p ( x ) = x − b a . 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(2.1) 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 on the interiorof K with continuous density, see [13, Theorem IV.2.5, pp. 216]. SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 7
Let r be a positive integer. We say that a 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 − , j ≥ . For a positive integer N , we say that ( x n : n ∈ N ) ∈ D Nr ( V ) if for each i ∈ { , , . . . , N − } ( x nN + i : n ∈ N ) ∈ D r ( V ) . If Y is a matrix Y = (cid:18) y , y , y , y , (cid:19) , we set [ Y ] i , j = y i , j . The symmetrization and the discriminant of Y are defined as sym ( Y ) = Y + Y ∗ , and discr Y = ( tr Y ) − Y , respectively. Here Y ∗ is the Hermitian transpose of the matrix Y .
3. Turán determinants
Let N be a positive integer and let ( u n : n ∈ N ) be a generalized eigenvector associated with α ∈ R \ { } and x ∈ R . We define N -shifted generalized Turán determinant by the formula(3.1) D n ( α, x ) = u n u n + N − − u n − u n + N . Theorem 1.
Let N be a positive integer and i ∈ { , , . . . , N − } . Suppose that lim j →∞ a j N + i − = ∞ . Let Λ = (cid:8) x ∈ R : lim j →∞ discr R j N + i ( x ) exists and is negative (cid:9) where R n ( x ) = a n + N − ( X n ( x ) − σ Id ) for some σ ∈ {− , } . Suppose that K ⊂ Λ is a compact interval with non-empty interior and Ω is a compactand connected subset of R \ { } . If (cid:0) X j N + i : j ∈ N (cid:1) , (cid:0) R j N + i : j ∈ N (cid:1) ∈ D (cid:0) K ; GL ( , R ) (cid:1) , then there is g a real continuous function without zeros on Ω × K and a constant c > such that sup α ∈ Ω sup x ∈ K (cid:12)(cid:12) a ( k + ) N + i − D k N + i ( α, x ) − g ( α, x ) (cid:12)(cid:12) ≤ c ∞ Õ j = k (cid:16) sup x ∈ K (cid:13)(cid:13) R ( j + ) N + i ( x ) − R j N + i ( x ) (cid:13)(cid:13) + sup x ∈ K (cid:13)(cid:13) X ( j + ) N + i ( x ) − X j N + i ( x ) (cid:13)(cid:13)(cid:17) . GRZEGORZ ŚWIDERSKI AND BARTOSZ TROJAN
Proof.
We follow the method developed in [20, Theorem 3], namely we write S ( k + ) N + i − S mN + i − = k Ö j = m ( + F j N + i − ) where for x ∈ R and α ∈ R \ { } we have set S n ( α, x ) = a n + N − D n ( α, x ) , and F j N + i − = S ( j + ) N + i − − S j N + i − S j N + i − . Therefore, for the proof it is sufficient to show(3.2) ∞ Õ j = m sup α ∈ Ω sup x ∈ K | F j N + i − ( α, x )| < ∞ . In view of [19, Lemma 3] we have(3.3) (cid:12)(cid:12) S n + N ( α, x ) − S n ( α, x ) (cid:12)(cid:12) ≤ a n + N − (cid:13)(cid:13)(cid:13)(cid:13) a n + N − a n + N − R n + N ( x ) − a n + N − a n − R n ( x ) (cid:13)(cid:13)(cid:13)(cid:13) (cid:0) u n + N − + u n + N (cid:1) . We next consider a quadratic form on R defined as Q xn ( v ) = a n + N − (cid:10) E X n ( x ) v , v (cid:11) where x ∈ R , and E = (cid:18) −
11 0 (cid:19) . In particular, we have(3.4) S n ( α, x ) = a n + N − Q xn (cid:18) u n + N − u n + N (cid:19) . We claim the following holds true.
Claim 1.
There is c > such that for all j ∈ N , x ∈ K , and v ∈ R , (3.5) c − ( v + v ) ≤ Q xj N + i ( v ) ≤ c ( v + v ) . For the proof, let us observe that Q xj N + i ( v ) = a ( j + ) N + i − a j N + i − (cid:10) E R j N + i ( x ) v , v (cid:11) . Since the sequence ( R j N + i : j ∈ N ) belongs to D (cid:0) K , GL ( , R ) (cid:1) , it is convergent. Let R i ( x ) denote its limit.We next notice that lim j →∞ det X j N + i ( x ) = lim j →∞ det (cid:0) a − j N + i − R j N + i + σ Id (cid:1) = σ = . On the other hand, det X j N + i ( x ) = a j N + i − a ( j + ) N + i − . Consequently, we obtain lim j →∞ a ( j + ) N + i − a j N + i − = , and hence lim j →∞ Q xj N + i ( v ) = (cid:10) E R i ( x ) v , v (cid:11) . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 9
Since det sym ( E R i ) = − discr R i , we conclude that for each x ∈ K , det sym ( E R i ( x )) > , which easily leads to (3.5).Using now Claim 1 together with (3.3) and (3.4), we obtain(3.6) sup α ∈ Ω sup x ∈ K (cid:12)(cid:12) F j N + i − ( α, x ) (cid:12)(cid:12) ≤ c sup x ∈ K (cid:13)(cid:13)(cid:13)(cid:13) a ( j + ) N + i − a ( j + ) N + i − R ( j + ) N + i ( x ) − a ( j + ) N + i − a j N + i − R j N + i ( x ) (cid:13)(cid:13)(cid:13)(cid:13) . By [20, the formula (6)], there is a constant c > , such that ∞ Õ j = m sup K (cid:13)(cid:13)(cid:13)(cid:13) a ( j + ) N + i − a ( j + ) N + i − R ( j + ) N + i − a ( j + ) N + i − a j N + i − R j N + i (cid:13)(cid:13)(cid:13)(cid:13) ≤ c ∞ Õ j = m (cid:18) sup K (cid:13)(cid:13) R ( j + ) N + i − R j N + i (cid:13)(cid:13) + (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:19) . Since for each r , s ∈ { , } , (cid:12)(cid:12)(cid:12)(cid:2) X ( j + ) N + i ( x ) (cid:3) r , s − (cid:2) X j N + i ( x ) (cid:3) r , s (cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13) X ( j + ) N + i ( x ) − X j N + i ( x ) (cid:13)(cid:13) , we obtain ∞ Õ j = m sup K (cid:12)(cid:12) det X ( j + ) N + i − det X j N + i (cid:12)(cid:12) ≤ c ∞ Õ j = m sup K (cid:13)(cid:13) X ( j + ) N + i − X j N + i (cid:13)(cid:13) . Hence, ∞ Õ j = m sup K (cid:13)(cid:13)(cid:13)(cid:13) a ( j + ) N + i − a ( j + ) N + i − R ( j + ) N + i − a ( j + ) N + i − a j N + i − R j N + i (cid:13)(cid:13)(cid:13)(cid:13) ≤ c ∞ Õ j = n (cid:16) sup K (cid:13)(cid:13) R ( j + ) N + i − R j N + i (cid:13)(cid:13) + sup K (cid:13)(cid:13) X ( j + ) N + i − X j N + i (cid:13)(cid:13)(cid:17) , which together with (3.6) implies (3.2) and the theorem follows. (cid:3)
4. Approximation procedure
In this section we present a method that allows to prove a formula for the density of an orthogonalitymeasure. It is a further development of [18] and [23].Let ( p n : n ∈ N ) be a sequence of polynomials corresponding to sequences ( a n : n ∈ N ) and ( b n : n ∈ N ) . We set(4.1) D n ( x ) = p n ( x ) p n + N − ( x ) − p n − ( x ) p n + N ( x ) . Then D n ( x ) = D n (cid:16) (cid:16) , x − b a (cid:17) , x (cid:17) where D n is given by (3.1).Given L ∈ N , we consider the truncated sequences ( a Ln : n ∈ N ) and ( b Ln : n ∈ N ) defined by(4.2a) a Ln = ( a n if ≤ n < L + N , a L + i if L + N ≤ n , and n − L ≡ i mod N , and(4.2b) b Ln = ( b n if ≤ n < L + N , b L + i if L + N ≤ n , and n − L ≡ i mod N , where i ∈ { , , . . . , N − } . Let ( D Ln : n ∈ N ) be the sequence (4.1) associated to the polynomials ( p Ln : n ∈ N ) that are corresponding to the sequences a L and b L . Then D Ln ( x ) = (cid:28) E X Ln ( x ) (cid:18) p Ln − ( x ) p Ln ( x ) (cid:19) , (cid:18) p Ln − ( x ) p Ln ( x ) (cid:19) (cid:29) where(4.3) X Ln ( x ) = n + N − Ö j = n © « − a Lj − a Lj x − b Lj a Lj ª®¬ and E = (cid:18) −
11 0 (cid:19) . Observe that there is the unique measure µ L orthonormalizing the polynomials ( p Ln : n ∈ N ) . Proposition 1.
Given σ ∈ {− , } we set (4.4) R n = a n + N − (cid:0) X n − σ Id (cid:1) , and R Ln = a Ln + N − (cid:0) X Ln − σ Id (cid:1) . Suppose that ( L j : j ∈ N ) is an increasing sequence of integers such that (4.5) lim j →∞ a L j − = ∞ , and lim j →∞ ( a L j + N − − a L j − ) = . If K is a compact subset of R , such that (4.6) sup j ∈ N sup K k R L j k < ∞ , then lim j →∞ sup K (cid:13)(cid:13)(cid:13) R L j − R L j L j + N (cid:13)(cid:13)(cid:13) = . Proof.
Since R L − R LL + N = a L + N − (cid:0) X L − X LL + N (cid:1) , in view of [23, Corollary 4] we have(4.7) (cid:13)(cid:13) R L − R LL + N (cid:13)(cid:13) ≤ k X L k a L + N − a L − (cid:12)(cid:12) a L + N − − a L − (cid:12)(cid:12) . By (4.4), we have k X n k ≤ + k R n k a n + N − , thus, (4.5) and (4.6) imply that(4.8) sup j ∈ N sup K k X L j k < ∞ . Lastly, by (4.5) we have(4.9) lim j →∞ a L j + N − a L j − = . Therefore, by using (4.8), (4.9) and (4.5) in (4.7), the conclusion follows. (cid:3)
Theorem 2.
Let N be a positive integer and σ ∈ {− , } . We set R n = a n + N − (cid:0) X n − σ Id (cid:1) . Suppose ( L j : ∈ N ) is an increasing sequence of positive integers such that (4.10) lim j →∞ a L j − = ∞ , and (4.11) lim j →∞ (cid:0) a L j + N − − a L j − (cid:1) = . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 11
Let K be a compact subset of Λ = n x ∈ R : lim j →∞ discr R L j ( x ) exists and is negative o such that (4.12) sup j ∈ N sup x ∈ K k R L j ( x )k < ∞ . Suppose that there is a positive function g : K → R such that (4.13) lim j →∞ sup x ∈ K (cid:12)(cid:12)(cid:12) a L j + N − (cid:12)(cid:12) D L j ( x ) (cid:12)(cid:12) − g ( x ) (cid:12)(cid:12)(cid:12) = . If ν is any weak accumulation point of the sequence ( µ L j : j ∈ N ) , then ν is a probability measure such that ( p n : n ∈ N ) are orthogonal in L ( R , ν ) , which is absolutely continuous on K with the density ν ′ ( x ) = p − h ( x ) π g ( x ) , x ∈ K where (4.14) h ( x ) = lim j →∞ discr R L j ( x ) , x ∈ K . Proof.
Let us fix a positive integer L . We set(4.15) R Ln = a Ln + N − (cid:0) X Ln − σ Id (cid:1) , and Λ L = n x ∈ R : discr (cid:0) R LL + N ( x ) (cid:1) < o , and S Ln ( x ) = (cid:0) a Ln + N − (cid:1) D Ln ( x ) , n ≥ . By (4.15), discr (cid:0) X LL + N ( x ) (cid:1) = a L + N − discr (cid:0) R LL + N ( x ) (cid:1) . Hence, by [18, Theorem 3], (see also [4, Theorem 6]), for each x ∈ Λ L ,the limit lim k →∞ (cid:12)(cid:12) S LL + k N ( x ) (cid:12)(cid:12) exists,and defines a positive continuous function g L : Λ L → R . Moreover, the orthogonality measure µ L isabsolutely continuous on Λ L with the density µ ′ L ( x ) = a L + N − q − discr (cid:0) X LL + N ( x ) (cid:1) π g L ( x ) = q − discr (cid:0) R LL + N ( x ) (cid:1) π g L ( x ) . (4.16)Next, we observe that by estimates (21) and (22) from [20] (cid:12)(cid:12)(cid:12) S Ln + N ( x ) − S Ln ( x ) (cid:12)(cid:12)(cid:12) ≤ a Ln + N − (cid:16) (cid:0) p Ln + N − ( x ) (cid:1) + (cid:0) p Ln + N − ( x ) (cid:1) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) a Ln + N − a Ln + N − R Ln + N ( x ) − a Ln + N − a Ln − R Ln ( x ) (cid:13)(cid:13)(cid:13)(cid:13) , which together with (4.2a) and (4.2b) entails that S Ln + N ( x ) = S Ln ( x ) for all n ≥ L + . Hence, for all x ∈ Λ L ,(4.17) g L ( x ) = (cid:12)(cid:12) S LL + N ( x ) (cid:12)(cid:12) . Let us fix a compact subset K ⊂ Λ . Since discr (cid:0) R L j ( x ) (cid:1) is a polynomial of degree at most N , theconvergence in (4.14) is uniform on K . Thus, by Proposition 1,(4.18) lim j →∞ sup x ∈ K (cid:12)(cid:12)(cid:12) discr (cid:16) R L j L j + N ( x ) (cid:17) − h ( x ) (cid:12)(cid:12)(cid:12) = . Moreover, K ⊂ Λ L j for all j sufficiently large. Now, setting S n ( x ) = a n + N − D n ( x ) , by [23, Proposition 5], we obtain (cid:12)(cid:12) S L j L j + N ( x ) − S L j ( x ) (cid:12)(cid:12) = a L j + N − (cid:12)(cid:12) D L j L j + N ( x ) − D L j ( x ) (cid:12)(cid:12) ≤ a L j + N − (cid:0) p L j + N − ( x ) + p L j + N ( x ) (cid:1) k X L j ( x )k (cid:12)(cid:12) a L j + N − − a L j − (cid:12)(cid:12) . Let us notice that S L ( x ) = a L + N − a L − (cid:28) E R L ( x ) (cid:18) p L + N − ( x ) p L + N ( x ) (cid:19) , (cid:18) p L + N − ( x ) p L + N ( x ) (cid:19) (cid:29) . Since K is a compact subset of Λ , there are j and δ > such that for all j ≥ j and x ∈ K we have(4.19) det (cid:16) sym (cid:0) E R L j ( x ) (cid:1) (cid:17) = −
14 discr (cid:0) R L j ( x ) (cid:1) ≥ δ. By (4.10) and (4.11) lim j →∞ a L j + N − a L j − = , which together with (4.19) implies that there are j and c > such that for all j ≥ j and x ∈ K , (cid:12)(cid:12) S L j ( x ) (cid:12)(cid:12) ≥ c − a L j + N − (cid:0) p L j + N − ( x ) + p L j + N ( x ) (cid:1) . Hence, (cid:12)(cid:12) S L j L j + N ( x ) − S L j ( x ) (cid:12)(cid:12) ≤ c | S L j ( x )| · k X L j ( x )k (cid:12)(cid:12) a L j + N − − a L j − (cid:12)(cid:12) . Since X L j = σ Id + a L j + N − R L j , by (4.10) and (4.12), we easily obtain sup j ∈ N sup x ∈ K k X L j ( x )k < ∞ . Therefore, by (4.13) and (4.17), sup x ∈ K (cid:12)(cid:12) g L j ( x ) − g ( x ) (cid:12)(cid:12) ≤ c (cid:12)(cid:12) a L j + N − − a L j − (cid:12)(cid:12) , thus(4.20) lim j →∞ sup x ∈ K (cid:12)(cid:12) g L j ( x ) − g ( x ) (cid:12)(cid:12) = . Finally, by (4.16), (4.18), and (4.20) we obtain(4.21) lim j →∞ sup x ∈ K (cid:12)(cid:12)(cid:12)(cid:12) µ ′ L j ( x ) − p − h ( x ) π g ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = , and the theorem is a consequence of [23, Proposition 4]. (cid:3) SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 13
Corollary 1.
Let the hypothesis of Theorem 1 be satisfied. If (4.22) lim j →∞ ( a ( j + ) N + i − − a j N + i − ) = , then the sequence ( µ j N + i : j ∈ N ) is weakly convergent to the probability measure µ which is absolutelycontinuous on K . Moreover, the sequence ( p n : n ∈ N ) is orthonormal in L ( R , µ ) , and (4.23) lim j →∞ sup x ∈ K (cid:12)(cid:12) µ ′ j N + i ( x ) − ν ′ ( x ) (cid:12)(cid:12) = . Proof.
In view of (4.22), there is c > such that for all k ≥ , a k N + i = k − Õ j = (cid:0) a ( j + ) N + i − a j N + i (cid:1) + a i ≤ c ( k + ) . Therefore, k N + i Õ n = a n ≥ k Õ k = a k N + i ≥ c k Õ k = k . Thus, the Carleman condition is satisfied, and consequently, there is the only one measure µ such that ( p n : n ∈ N ) are orthonormal in L ( R , µ ) . Lastly, (4.23) is a consequence of (4.21). (cid:3)
5. Asymptotics of orthogonal polynomials
Our next goal is to derive the asymptotic formula for the polynomials ( p n : n ∈ N ) . Theorem 3.
Let N be a positive integer, σ ∈ {− , } , and i ∈ { , , . . . , N − } . We set R n = a n + N − ( X n − σ Id ) . Suppose that lim j →∞ a j N + i − = ∞ . Let K be a compact interval with non-empty interior contained in Λ = (cid:8) x ∈ R : lim j →∞ discr R j N + i ( x ) exists and is negative (cid:9) . If ( X j N + i : j ∈ N ) , ( R j N + i : j ∈ N ) ∈ D (cid:0) K , GL ( , R ) (cid:1) , then there are M > and a continuous function ϕ such that (5.1) lim k →∞ sup x ∈ K (cid:12)(cid:12)(cid:12)(cid:12) a ( k + ) N + i − Î kj = M + λ j N + i ( x ) (cid:16) p ( k + ) N + i ( x ) − λ k N + i ( x ) p k N + i ( x ) (cid:17) − ϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = where λ n ( x ) = tr X n ( x ) + i p − discr X n ( x ) . Moreover, (5.2) s a ( k + ) N + i − a ( M + ) N + i − p k N + i ( x ) = | ϕ ( x )| p − discr R i ( x ) sin (cid:18) k Õ j = M + arg λ j N + i ( x ) + arg ϕ ( x ) (cid:19) + E k N + i ( x ) where R i is the limit of ( R j N + i : j ∈ N ) , and sup K | E k N + i | ≤ c ∞ Õ j = k (cid:16) sup K k X ( j + ) N + i − X j N + i k + sup K k R ( j + ) N + i − R j N + i k (cid:17) . Proof.
Let us fix a compact interval K ⊂ Λ with non-empty interior. Since R i is the uniform limit of ( R j N + i : j ∈ N ) , there are δ > and M > such that for all x ∈ K and k ≥ M , discr R k N + i ( x ) ≤ − δ < . Therefore, the matrix R k N + i has two eigenvalues ξ k and ξ k where(5.3) ξ k ( x ) = tr R k N + i ( x ) + i p − discr R k N + i ( x ) . Let us next observe that for k ≥ M , ℑ ξ k ( x ) = p − discr R k N + i ( x ) ≥ √ δ. Moreover, R k N + i ( x ) = C k ( x ) ˜ D k ( x ) C − k ( x ) where C k ( x ) = (cid:18) ξ k N + i ξ k N + i (cid:19) , and ˜ D k ( x ) = (cid:18) ξ k ξ k (cid:19) . Since X k N + i = σ Id + a ( k + ) N + i − R k N + i , we obtain X k N + i ( x ) = C k ( x ) D k ( x ) C − k ( x ) where D k = σ Id + a ( k + ) N + i − ˜ D k . In particular, X k N + i has two eigenvalues λ k N + i and λ k N + i where λ k N + i = σ + a ( k + ) N + i − ξ k . We next set φ k = p ( k + ) N + i − λ k N + i p k N + i Î kj = M + λ j N + i , and claim that the following holds true. Claim 2.
There is c > such that for all m ≥ n ≥ M , and x ∈ K , (cid:12)(cid:12) a ( m + ) N + i − φ m ( x ) − a ( n + ) N + i − φ n ( x ) (cid:12)(cid:12) ≤ c (cid:16) ∞ Õ j = n (cid:12)(cid:12) ξ j + ( x ) − ξ j ( 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)(cid:17) . 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 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) + , SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 15 and ψ m = q m + − λ mN + i q m Î mj = M + λ j N + i . Notice that a ( m + ) N + i − (cid:0) φ m − ψ m (cid:1) = (cid:28) Y m (cid:18) p nN + i − p nN + i (cid:19) , (cid:18) (cid:19) (cid:29) m Ö j = M + λ j N + i where Y m = ( C m − C ∞ ) D − m (cid:16) a ( m + ) N + i − (cid:0) D m − λ mN + i Id (cid:1) (cid:17) (cid:16) m Ö j = n D j C j C − j − (cid:17) C − n − + C ∞ D − m (cid:16) a ( m + ) N + i − (cid:0) D m − λ mN + i Id (cid:1) (cid:17) (cid:16) m Ö j = n D j C j C − j − − m Ö j = n D j (cid:17) C − n − . In view of [23, Propositon 1], we have k Y m k . (cid:18) m Ö j = n k D j k (cid:19)(cid:13)(cid:13)(cid:13) a ( m + ) N + i − (cid:0) D m − λ mN + i Id (cid:1)(cid:13)(cid:13)(cid:13)(cid:18) ∞ Õ j = n − k ∆ C j k + k C ∞ − C m k (cid:19) . Since(5.4) a ( m + ) N + i − (cid:0) D m − λ mN + i Id ) = (cid:18) i ℑ ξ m
00 0 (cid:19) , the right-hand side is convergent, hence bounded. Therefore, we have k Y m k . (cid:18) m Ö j = n k D j k (cid:19) ∞ Õ j = n − k ∆ C j k . m Ö j = n | λ j N + i | · ∞ Õ j = n − (cid:12)(cid:12) ξ j + − ξ j (cid:12)(cid:12) . Following the arguments used in the proof of [23, Claim 2], we conclude that there is c > so that for all n > M and x ∈ K ,(5.5) 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 ,(5.6) a ( m + ) N + i − (cid:12)(cid:12) φ m − ψ m (cid:12)(cid:12) . ∞ Õ j = n − (cid:12)(cid:12) ξ j + − ξ j (cid:12)(cid:12) . We next notice that a ( m + ) N + i − (cid:0) q m + − λ mN + i q m (cid:1) = * C ∞ (cid:16) a ( m + ) N + i − (cid:0) D m − λ mN + i Id (cid:1) (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 by (5.4) a ( m + ) N + i − Î mj = n λ j N + i (cid:16) D m − λ mN + i Id (cid:17) m − Ö j = n D j = i λ mN + i (cid:18) ℑ ξ m
00 0 (cid:19) , we obtain a ( m + ) N + i − (cid:12)(cid:12) ψ m − ψ n (cid:12)(cid:12) . (cid:12)(cid:12)(cid:12)(cid:12) ℑ ξ m λ mN + i − ℑ ξ n λ nN + i (cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ Õ j = n (cid:12)(cid:12)(cid:12)(cid:12) ℑ ξ j + λ ( j + ) N + i − ℑ ξ j λ j N + i (cid:12)(cid:12)(cid:12)(cid:12) . Observe that (cid:12)(cid:12)(cid:12)(cid:12) ℑ ξ j + λ ( j + ) N + i − ℑ ξ j λ j N + i (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) λ j N + i ℑ( ξ j + − ξ j ) − ( λ ( j + ) N + i − λ j N + i )ℑ ξ j λ ( j + ) N + i λ j N + i (cid:12)(cid:12)(cid:12)(cid:12) . | ξ j + − ξ j | + | λ ( j + ) N + i − λ j N + i | . Hence, a ( m + ) N + i − (cid:12)(cid:12) ψ m − ψ n (cid:12)(cid:12) . ∞ Õ j = n (cid:16)(cid:12)(cid:12) ξ j + − ξ j (cid:12)(cid:12) + (cid:12)(cid:12) λ ( j + ) N + i − λ j N + i (cid:12)(cid:12)(cid:17) , which together with (5.6) implies that for all m ≥ n > M and x ∈ K , (cid:12)(cid:12) a ( m + ) N + i − φ m ( x ) − a ( n + ) N + i − φ n ( x ) (cid:12)(cid:12) . ∞ Õ j = n (cid:16)(cid:12)(cid:12) ξ j + ( x ) − ξ j ( x ) (cid:12)(cid:12) + (cid:12)(cid:12) λ ( j + ) N + i ( x ) − λ j N + i ( x ) (cid:12)(cid:12)(cid:17) , proving Claim 2.In particular, Claim 2 entails that the sequence ( a ( m + ) N + i − φ m : m ∈ N ) converges. Let us denote by ϕ its limit. Hence, (cid:12)(cid:12) ϕ ( x ) − a ( n + ) N + i − φ n ( x ) (cid:12)(cid:12) . ∞ Õ j = n (cid:16)(cid:12)(cid:12) ξ j + ( x ) − ξ j ( x ) (cid:12)(cid:12) + (cid:12)(cid:12) λ ( j + ) N + i ( x ) − λ j N + i ( x ) (cid:12)(cid:12)(cid:17) Since polynomials p n are having real coefficients, by taking imaginary part we obtain (cid:12)(cid:12)(cid:12)(cid:12) a ( n + ) N + i − p − discr X nN + i ( x ) p nN + i ( x ) Î nj = M + | 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:16)(cid:12)(cid:12) ξ j + ( x ) − ξ j ( x ) (cid:12)(cid:12) + (cid:12)(cid:12) λ ( j + ) N + i ( x ) − λ j N + i ( x ) (cid:12)(cid:12)(cid:17) . Observe that det X j N + i = ( j + ) N + i − Ö k = j N + i det B k = a j N + i − a ( j + ) N + i − , thus 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 − . Moreover, discr X nN + i = ( λ nN + i − λ nN + i ) = a ( n + ) N + i − ( ξ n − ξ n ) = a ( n + ) N + i − discr R nN + i . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 17
Therefore, (cid:12)(cid:12)(cid:12)(cid:12)s a ( n + ) N + i − a ( M + ) N + i − p − discr R nN + i ( x ) p nN + 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:16)(cid:12)(cid:12) ξ j + − ξ j (cid:12)(cid:12) + (cid:12)(cid:12) λ ( j + ) N + i − λ j N + i (cid:12)(cid:12)(cid:17) . By (5.3), we can write (cid:12)(cid:12)(cid:12)(cid:12) p − discr R nN + i ( x ) − p − discr R i ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . ∞ Õ j = n (cid:12)(cid:12) ξ j + − ξ j (cid:12)(cid:12) , thus (cid:12)(cid:12)(cid:12)(cid:12)s a ( n + ) N + i − a ( M + ) N + i − p nN + i ( x ) − | ϕ ( x )| p − discr R i ( x ) sin (cid:18) n Õ j = M + θ j N + i ( x ) + arg ϕ ( x ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . ∞ Õ j = n (cid:16)(cid:12)(cid:12) ξ j + − ξ j (cid:12)(cid:12) + (cid:12)(cid:12) λ ( j + ) N + i − λ j N + i (cid:12)(cid:12)(cid:17) . Finally, we have (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) and (cid:12)(cid:12) ξ j + − ξ j (cid:12)(cid:12) . (cid:13)(cid:13) R ( j + ) N + i − R j N + i (cid:13)(cid:13) and the theorem follows. (cid:3) Our next task is to compute | ϕ ( x )| . To do this, once again, we use the truncated sequences defined in (4.2a)and (4.2b). Theorem 4.
Let N be a positive integer, σ ∈ {− , } , and i ∈ { , , . . . , N − } . Suppose that lim n →∞ a nN + i − = ∞ and lim n →∞ ( a ( n + ) N + i − − a nN + i − ) = . Let K be a compact interval with non-empty interior contained in Λ = n x ∈ R : lim n →∞ discr R nN + i ( x ) exists and is negative o where (5.7) R n = a n + N − ( X n − σ Id ) . If ( X nN + i : n ∈ N ) , ( R nN + i : n ∈ N ) ∈ D (cid:0) K ; GL ( , R ) (cid:1) , then the polynomials ( p n : n ∈ N ) are orthonormal with respect to the measure µ , which is purely absolutelycontinuous on K . Moreover, there is M > and a continuous real-valued function η such that √ a ( n + ) N + i − p nN + i ( x ) = vt (cid:12)(cid:12) [R i ( x )] , (cid:12)(cid:12) π µ ′ ( x ) p − discr R i ( x ) sin (cid:18) n Õ j = M + θ j ( x ) + η ( x ) (cid:19) + E nN + i ( x ) where R i is the limit of ( R nN + i : n ∈ N ) , θ j ( x ) = arccos (cid:18) tr X j N + i ( x ) p det X j N + i ( x ) (cid:19) and sup K | E nN + i | ≤ c ∞ Õ j = n (cid:16) sup K (cid:13)(cid:13) X ( j + ) N + i − X j N + i (cid:13)(cid:13) + sup K (cid:13)(cid:13) R ( j + ) N + i − R j N + i (cid:13)(cid:13)(cid:17) . Proof.
Since K ⊂ Λ , and discr R k N + i is a polynomial of degree at most N , there are δ > and M ≥ suchthat for all x ∈ K and n ≥ M , discr R nN + i ( x ) ≤ − δ. Let us consider L ∈ { L k : k ∈ N } where L k = k N + i . We set Λ L = (cid:8) x ∈ R : discr (cid:0) R LL + N ( x ) (cid:1) < (cid:9) where(5.8) R Ln = a Ln + N − (cid:0) X Ln − σ Id (cid:1) , and X Ln is defined by the formula (4.3). In view of (4.2a) and (4.2b), we have(5.9) X L k j N + i = ( X j N + i if ≤ j ≤ k , X L k L k + N if k < j . Moreover, by Proposition 1, there is L ≥ M such that K ⊂ Λ L for all L ≥ L . For x ∈ K we set ξ m ( x ) =
12 tr R mN + i ( x ) + i p − discr R mN + i ( x ) , (5.10) ξ Lm ( x ) =
12 tr R LmN + i ( x ) + i q − discr R LmN + i ( x ) , (5.11)and λ mN + i ( x ) = γ + a ( m + ) N + i ξ m ( x ) , (5.12) λ LmN + i ( x ) = γ + a ( m + ) N + i ξ Lm ( x ) . (5.13)Lastly, we define φ mN + i ( x ) = p ( m + ) N + i ( x ) − λ mN + i ( x ) p mN + i ( x ) Î mj = M + λ j N + i ( x ) φ LmN + i ( x ) = p L ( m + ) N + i ( x ) − λ LmN + i ( x ) p LmN + i ( x ) Î mj = M + λ Lj N + i where ( p Ln : n ∈ N ) is the sequence of orthogonal polynomials corresponding to (4.2a) and (4.2b). Claim 3. lim k →∞ sup x ∈ K (cid:12)(cid:12)(cid:12) a L k + N − φ L k L k + N ( x ) − ϕ ( x ) (cid:12)(cid:12)(cid:12) = . First, let us observe that, by (5.9), we have φ L k + N = λ L k + N Î kj = M + λ j N + i (cid:28)(cid:18) X L k + N − λ L k + N Id (cid:19) (cid:18) p L k + N − p L k + N (cid:19) , (cid:18) (cid:19) (cid:29) and φ L k L k + N = λ L k L k + N Î kj = M + λ j N + i (cid:28)(cid:18) X L k L k + N − λ L k L k + N Id (cid:19) (cid:18) p L k + N − p L k + N (cid:19) , (cid:18) (cid:19) (cid:29) . Thus, a L k + N − φ L k + N − a L k L k + N − φ L k L k + N = Î kj = M + λ j N + i (cid:28) Y k (cid:18) p L k + N − p L k + N (cid:19) , (cid:18) (cid:19) (cid:29) , SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 19 where Y k = a L k + N − λ L k + N (cid:16) X L k + N − λ L k + N Id (cid:17) − a L k L K + N − λ L k L k + N (cid:0) X L k L k + N − λ L k L k + N Id (cid:1) . Hence, by (5.7), (5.8), (5.12) and (5.13) Y k = λ L k + N (cid:16) R L k + N − ξ L k + N Id (cid:17) − λ L k L k + N (cid:16) R L k L k + N − ξ L k L k + N Id (cid:17) , and consequently, k Y k k ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ L k + N − λ L k L k + N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:13)(cid:13)(cid:13) R L k + N (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12) λ L k L k + N (cid:12)(cid:12) · (cid:13)(cid:13)(cid:13) R L k + N − R L k L k + N (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ L k L k + N λ L k L k + N − ξ L k + N λ L k + N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Since ( R L k : k ∈ N ) is convergent, Proposition 1 together with (5.10)–(5.13) implies that lim k →∞ sup K k Y k k = . Hence, by (5.5) lim k →∞ sup K (cid:12)(cid:12)(cid:12) a L k + N − φ L k + N − a L k L k + N − φ L k L k + N (cid:12)(cid:12)(cid:12) = , and the conclusion follows by (5.1). Claim 4.
For x ∈ K , we have (cid:12)(cid:12) ϕ ( x ) (cid:12)(cid:12) = |[R i ( x )] , | p − discr R i ( x ) π µ ′ ( x ) a ( M + ) N + i − . In view of [23, Claim 5] we can write (cid:12)(cid:12) φ LL + N ( x ) (cid:12)(cid:12) = |[ X LL + N ( x )] , | q − discr (cid:0) X LL + N ( x ) (cid:1) π µ ′ L ( x ) a ( M + ) N + i − . Hence, (cid:12)(cid:12) a LL + N − φ LL + N ( x ) (cid:12)(cid:12) = |[ R LL + N ( x )] , | q − discr (cid:0) R LL + N ( x ) (cid:1) π µ ′ L ( x ) a ( M + ) N + i − which, by Proposition 1 and Corollary 1, approaches to |[R i ( x )] , | p − discr R i ( x ) π µ ′ ( x ) a ( M + ) N + i − as L tends to infinity, uniformly with respect to x ∈ K . Thus, the conclusion follows by Claim 3.Now, to finish the proof of the theorem it is enough to combine Claim 4 with (5.2). (cid:3)
6. Christoffel functions in residue classes
Lemma 1.
Let ( γ n : n ∈ N ) be a sequence of positive numbers such that ∞ Õ n = γ n = ∞ , and lim n →∞ γ n = . Suppose that there are a compact set K ⊂ R d , and ( ξ n : n ∈ N ) a sequence of real functions on K such that lim n →∞ sup x ∈ K (cid:12)(cid:12)(cid:12)(cid:12) ξ n ( x ) γ n − ψ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = for some function ψ : K → [ , ∞) satisfying c − ≤ ψ ( x ) ≤ c , for all x ∈ K . We set Ξ n ( x ) = n Õ j = ξ j ( x ) , and Γ n = n Õ j = γ j . Then for any f ∈ C ([ , ∞)) such that both f and f ′ are bounded on [ , ∞) , lim n →∞ Γ n n Õ k = γ k f ( Ξ k ( x )) = lim n →∞ Ξ n ( x ) ∫ Ξ n ( x ) f ( t ) d t uniformly with respect to x ∈ K provided that the right-hand side exists.Proof. Since f and / ψ are bounded on [ , ∞) and K respectively, by the Stolz–Cesáro theorem we get lim n →∞ Γ n n Õ k = (cid:12)(cid:12)(cid:12)(cid:12) γ k − ξ k ( x ) ψ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12) f (cid:0) Ξ k ( x ) (cid:1) (cid:12)(cid:12) = lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) ψ ( x ) − ξ n ( x ) γ n (cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12) f (cid:0) Ξ k ( x ) (cid:1) ψ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = uniformly with respect to x ∈ K . Therefore,(6.1) lim n →∞ Γ n n Õ k = γ k f (cid:0) Ξ k ( x ) (cid:1) = lim n →∞ Γ n n Õ k = ξ k ( x ) ψ ( x ) f (cid:0) Ξ k ( x ) (cid:1) . We next observe that, by the mean value theorem, we obtain (cid:12)(cid:12)(cid:12)(cid:12) n Õ k = ξ k ψ f ( Ξ k ) − ψ ∫ Ξ n f ( t ) d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ n Õ k = ψ ∫ Ξ k Ξ k − (cid:12)(cid:12) f ( Ξ k ) − f ( t ) (cid:12)(cid:12) d t ≤ sup t ∈[ , ∞) | f ′ ( t )| · n Õ k = ξ k ψ . Since by the Stolz–Cesáro theorem, lim n →∞ Γ n n Õ k = ξ k ( x ) ψ ( x ) = lim n →∞ ξ n ( x ) γ n ψ ( x ) = lim n →∞ γ n ψ ( x ) (cid:18) ξ n ( x ) γ n ψ ( x ) (cid:19) = uniformly with respect to x ∈ K , we conclude that lim n →∞ Γ n n Õ k = ξ k ψ f (cid:0) Ξ k (cid:1) = lim n →∞ ψ Γ n ∫ Ξ n f ( t ) d t , which together with (6.1) implies that(6.2) lim n →∞ Γ n n Õ k = γ k f (cid:0) Ξ k (cid:1) = lim n →∞ ψ Γ n ∫ Ξ n f ( t ) d t uniformly on K . In view of the Stolz–Cesáro theorem, we obtain(6.3) lim n →∞ Ξ n ( x ) ψ ( x ) Γ n = lim n →∞ ξ n ( x ) ψ ( x ) γ n = , which combined with (6.2) completes the proof. (cid:3) Lemma 2.
Assume that ( γ n : n ∈ N ) is a sequence of positive numbers such that ∞ Õ n = γ n = ∞ , and lim n →∞ γ n − γ n = . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 21
Let K be compact subset of R . Suppose that ( ξ n : n ∈ N ) is a sequence of real function on K such that lim n →∞ sup x ∈ K (cid:12)(cid:12)(cid:12) ξ n ( x ) γ n − ψ ( x ) (cid:12)(cid:12)(cid:12) = for some function ψ : K → [ , ∞) satisfying c − ≤ ψ ( x ) ≤ c , for all x ∈ K . We set Ξ n ( x ) = n Õ k = ξ k ( x ) , and Γ n = n Õ k = γ k . Then for any f ∈ C (cid:0) [ , sup K ψ ] (cid:1) , (6.4) lim n →∞ Γ n n Õ k = γ k f (cid:0) Γ − n Ξ n ( x ) (cid:1) = ψ ( x ) ∫ ψ ( x ) f ( t ) d t uniformly with respect to x ∈ K .Proof. First, let us observe that by the Stolz–Cesáro theorem(6.5) lim n →∞ Ξ n ( x ) Γ n = lim n →∞ ξ n ( x ) γ n = ψ ( x ) uniformly with respect to x ∈ K .Let U be an open set containing (cid:2) , sup K ψ (cid:3) and such that f ∈ C ( U ) . In view of (6.5), there is M suchthat for every n ≥ M , x ∈ K , and all k ∈ { M , , . . . , n } , Ξ k ( x ) Γ n ∈ U . Let n ≥ M . Notice that sup x ∈ K (cid:12)(cid:12)(cid:12)(cid:12) n Õ k = γ k f (cid:0) Γ − n Ξ k ( x ) (cid:1) − ψ ( x ) n Õ k = ξ k ( x ) f (cid:0) Γ − n Ξ k ( x ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c sup t ∈ U | f ( t )| · n Õ k = γ k sup x ∈ K (cid:12)(cid:12)(cid:12)(cid:12) ψ ( x ) − ξ k ( x ) γ k (cid:12)(cid:12)(cid:12)(cid:12) . Since, by the Stolz–Cesáro theorem lim n →∞ Γ n n Õ k = γ k sup x ∈ K (cid:12)(cid:12)(cid:12)(cid:12) ψ ( x ) − ξ k ( x ) γ k (cid:12)(cid:12)(cid:12)(cid:12) = lim n →∞ sup x ∈ K (cid:12)(cid:12)(cid:12)(cid:12) ψ ( x ) − ξ n ( x ) γ n (cid:12)(cid:12)(cid:12)(cid:12) = , we obtain(6.6) lim n →∞ Γ n n Õ k = γ k f (cid:0) Γ − n Ξ k ( x ) (cid:1) = lim n →∞ Γ n n Õ k = ξ k ( x ) f (cid:0) Γ − n Ξ n ( x ) (cid:1) uniformly with respect to x ∈ K . Next, we are going to replace the sum by the integral. By the mean valuetheorem, we can write (cid:12)(cid:12)(cid:12)(cid:12) Γ n n Õ k = ξ k ( x ) f (cid:0) Γ − n Ξ k ( x ) (cid:1) − ∫ Ξ n ( x )/ Γ n f ( t ) d t (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) n Õ k = ∫ Ξ k ( x )/ Γ n Ξ k − ( x )/ Γ n f (cid:0) Γ − n Ξ k ( x ) (cid:1) − f ( t ) d t (cid:12)(cid:12)(cid:12)(cid:12) ≤
12 sup t ∈ U | f ′ ( t )| Γ n n Õ k = ξ k ( x ) . By repeated application of the Stolz–Cesáro theorem, we get lim n →∞ Í nk = ξ k ( x ) Γ n = lim n →∞ ξ n ( x ) γ n ( Γ n + Γ n − ) ≤ ψ ( x ) lim n →∞ γ n Γ n = ψ ( x ) lim n →∞ γ n − γ n − γ n = uniformly with respect to x ∈ K . Therefore, lim n →∞ Γ n n Õ k = ξ k ( x ) f (cid:0) Γ − n Ξ k ( x ) (cid:1) = lim n →∞ ∫ Ξ n ( x )/ Γ n f ( t ) d t uniformly with respect to x ∈ K , which together with (6.6) implies that lim n →∞ Γ n n Õ k = γ k f (cid:0) Γ − n Ξ k ( x ) (cid:1) = ψ ( x ) lim n →∞ ∫ Ξ n ( x )/ Γ n f ( t ) d t uniformly with respect to x ∈ K . Now, by (6.5) we conclude (6.4). (cid:3) Proposition 2.
Let N be a positive integer, σ ∈ {− , } and i ∈ { , , . . . , N − } . Suppose that (6.7) lim j →∞ a j N + i − = ∞ . Let K be a compact interval with non-empty interior contained in Λ = (cid:26) x ∈ R : lim j →∞ discr R j N + i ( x ) exists and is negative (cid:27) where R n = a n + N − ( X n − σ Id ) . Assume that for each x ∈ K , the sequence ( R j N + i ( x ) : j ∈ N ) converges to R i ( x ) . Then lim j →∞ a ( j + ) N + i − · arccos (cid:18) σ tr X n ( x ) p det X n ( x ) (cid:19) = p − discr R i ( x ) uniformly with respect to x ∈ K .Proof. Let K be a compact subset of Λ . Since each entry in R n ( x ) is a polynomial of degree at most N , lim j →∞ R j N + i ( x ) = R i ( x ) uniformly with respect to x ∈ K . Hence, by (6.7), lim j →∞ X j N + i ( x ) = σ Id uniformly with respect to x ∈ K . In particular, lim j →∞ tr X j N + i ( x ) p det X j N + i ( x ) = σ. Since lim t → − arccos t √ − t = , we obtain lim j →∞ (cid:18) − (cid:18) tr X j N + i ( x ) p det X j N + i ( x ) (cid:19) (cid:19) − / arccos (cid:18) σ tr X n ( x ) p det X n ( x ) (cid:19) = . Finally, let us observe that vt − (cid:18) tr X n ( x ) p det X n ( x ) (cid:19) = p − discr X n ( x ) p det X n ( x ) = a n + N − p − discr R n ( x ) p det X n ( x ) . Hence, lim j →∞ a ( j + ) N + i − vt − (cid:18) tr X j N + i ( x ) p det X j N + i ( x ) (cid:19) = lim j →∞ p − discr R j N + i ( x ) p det X j N + i ( x ) = p − discr R i ( x ) , which concludes the proof. (cid:3) SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 23
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 . We can now state one of the main results of this article.
Theorem 5.
Let N be a positive integer, σ ∈ {− , } , and i ∈ { , , . . . , N − } . Suppose that (6.8) lim n →∞ a nN + i − = ∞ , and lim n →∞ ( a ( n + ) N + i − − a nN + i − ) = . Let K be a compact interval with non-empty interior contained in Λ = n x ∈ R : lim n →∞ discr R nN + i ( x ) exists and is negative o where R n = a n + N − ( X n − σ Id ) . If ( X j N + i : j ∈ N ) , ( R j N + i : j ∈ N ) ∈ D (cid:0) K ; GL ( , R ) (cid:1) then lim n →∞ ρ i − n K i ; n ( x , x ) = π µ ′ ( x ) |[R i ( x )] , | p − discr R i ( x ) uniformly with respect to x ∈ K , where R i is the limit of ( R j N + i : j ∈ N ) .Proof. Let K ⊂ Λ be a compact interval with non-empty interior. By Theorem 4, there is M > such thatfor all k ≥ M , a ( k + ) N + i − p k N + i ( x ) = |[R i ( x )] , | π µ ′ ( x ) p − discr R i ( x ) sin (cid:16) η ( x ) + k Õ j = M + θ j ( x ) (cid:17) + E k N + i ( x ) where(6.9) lim k →∞ sup x ∈ K | E k N + i ( x )| = . Therefore, n Õ k = M p k N + i ( x ) = |[R i ( x )] , | π µ ′ ( x ) p − discr R i ( x ) n Õ k = M a ( k + ) N + i − sin (cid:16) η ( x ) + k Õ j = M + θ j N + i ( x ) (cid:17) + n Õ k = M a ( k + ) N + i − E k N + i ( x ) . Observe that there is c > such that sup x ∈ K M − Õ k = p k N + i ( x ) ≤ c . By (6.8), a j N + i − = a i + j Õ k = a k N + i − − a ( k − ) N + i − ≤ c ( j + ) , thus lim n →∞ ρ i − n = ∞ . Next, by the Stolz–Cesáro theorem and (6.9), we obtain lim n →∞ ρ i − n n Õ k = M a ( k + ) N + i − E k N + i ( x ) = lim n →∞ a nN + i − a ( n + ) N + i − E nN + i ( x ) = , since (6.8) entails that lim n →∞ a nN + i − a ( n + ) N + i − = . Therefore,(6.10) lim n →∞ ρ i − n K i ; n ( x , x ) = |[R i ( x )] , | π µ ′ ( x ) p − discr R i ( x ) · lim n →∞ ρ i − n n Õ k = a ( k + ) N + i − sin (cid:16) η ( x ) + k Õ j = M + θ j N + i ( x ) (cid:17) . Since sin ( k π + x ) = sin ( x ) , by Proposition 2, we have sin (cid:16) η ( x ) + k Õ j = M + θ j N + i ( x ) (cid:17) = sin (cid:16) − η ( x ) + k Õ j = M + (cid:0) π − θ j N + i ( x ) (cid:1) (cid:17) . Therefore, by taking γ j = a j N + i , ψ ( x ) = p − discr R i ( x ) , and ξ j ( x ) = ( θ j N + i ( x ) if σ = ,π − θ j N + i ( x ) if σ = − , by Proposition 2 we obtain lim j →∞ γ j ξ j ( x ) = ψ ( x ) . Hence, in view of Lemma 1 we get lim j →∞ ρ i − n n Õ k = a ( k + ) N + i − sin (cid:16) ση ( x ) + k Õ j = ξ j ( x ) (cid:17) = lim n →∞ Ξ n ( x ) ∫ Ξ n ( x ) sin ( t ) d t = lim n →∞ − Ξ n ( x ) sin (cid:0) Ξ n ( x ) (cid:1) . Lastly, by (6.3), lim n →∞ Ξ n ( x ) = uniformly with respect to x ∈ K , thus lim j →∞ ρ i − n n Õ k = a ( k + ) N + i − sin (cid:16) − η ( x ) + k Õ j = ξ j ( x ) (cid:17) = , which together with (6.10) finishes the proof. (cid:3) SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 25
7. Periodic modulations
Definitions and basic properties.
We say that Jacobi parameters ( a n ) and ( b n ) are N -periodicallymodulated 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) = . By ( p n : n ∈ N ) we denote orthogonal polynomials associated with sequences ( α n : n ∈ N ) and ( β n : n ∈ N ) . We set X n ( x ) = n + N − Ö j = n B j ( x ) , where B j ( x ) = − α j − α j x − β j α j ! . In this article we are interested in the case when X ( ) = σ Id for some σ ∈ {− , } and for any i ∈ { , , . . . N − } , the limit(7.1) R i ( x ) = lim n →∞ a ( n + ) N + i − (cid:0) X nN + i ( x ) − σ Id (cid:1) exists. Proposition 3.
If for some i the limit (7.1) exists, then it exists for all i ∈ N . Moreover, (7.2) R i + ( x ) = α i α i − B i ( )R i ( x ) B − i ( ) . Proof.
It is enough to prove (7.2). Observe that a ( n + ) N + i (cid:0) X nN + i + ( x ) − σ Id (cid:1) = a ( n + ) N + i a ( n + ) N + i − B ( n + ) N + i ( x ) (cid:16) a ( n + ) N + i − (cid:0) X nN + i ( x ) − σ Id (cid:1) (cid:17) B − nN + i ( x ) . Computing limits of both sides gives lim n →∞ a ( n + ) N + i (cid:0) X nN + i + ( x ) − σ Id (cid:1) = α i α i − B i ( )R i ( x ) B − i ( ) . Hence, we obtain the existence of R i + and the formula (7.2) follows. (cid:3) We define(7.3) Λ = N − Ù i = (cid:8) x ∈ R : discr R i ( x ) < (cid:9) . In view of Proposition 3(7.4) Λ = (cid:8) x ∈ R : discr R ( x ) < (cid:9) . We set(7.5) υ ( x ) = N π N − Õ i = α i − (cid:12)(cid:12) [R i ( x )] , (cid:12)(cid:12)p − discr R i ( x ) , x ∈ Λ . Assume that there are N -periodic sequences ( s n : n ∈ N ) and ( z n : n ∈ N ) such that(7.6) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) α n − α n a n − a n − − s n (cid:12)(cid:12)(cid:12)(cid:12) = , lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) β n α n a n − b n − z n (cid:12)(cid:12)(cid:12)(cid:12) = . Then according to [19, Proposition 9] the limit (7.1) exists. Moreover, by [19, Corollary 1], there is a compactinterval I (possibly empty) such that Λ = R \ I . Proposition 4.
Let ( a n : n ∈ N ) and ( b n : n ∈ N ) be N -periodically modulated. Suppose that there aretwo N -periodic sequences ( s n : n ∈ N ) and ( z n : n ∈ N ) , such that lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) α n − α n a n − a n − − s n (cid:12)(cid:12)(cid:12)(cid:12) = , and lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) β n α n a n − b n − z n (cid:12)(cid:12)(cid:12)(cid:12) = . Then for each i ∈ { , , . . . , N − } , lim k →∞ (cid:0) a ( k + ) N + i − a k N + i (cid:1) = α i N − Õ j = s j + α j . Proof.
Since N − Õ j = (cid:18) a n + j + α n + j + − a n + j α n + j (cid:19) = a n + N α n + N − a n α n , by N -periodicity of α we obtain a n + N − a n = α n N − Õ j = α n + j (cid:18) α n + j α n + j + a n + j + − a n + j (cid:19) . Thus, for i ∈ { , , . . . , N − } , lim k →∞ (cid:0) a ( k + ) N + i − a k N + i (cid:1) = α i N − Õ j = s j + i + α i + j = α i N − Õ j = s j + α j , which finishes the proof. (cid:3) The function υ .Theorem 6. Let N be a positive integer and σ ∈ {− , } . Let ( a n : n ∈ N ) and ( b n : n ∈ N ) be N -periodically modulated Jacobi parameters so that X ( ) = σ Id . Suppose that (7.7) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) α n − α n a n − a n − (cid:12)(cid:12)(cid:12)(cid:12) = , and lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) β n α n a n − b n (cid:12)(cid:12)(cid:12)(cid:12) = . Then Λ = R \ { } , and (7.8) υ ( x ) = ω ′ ( ) , x ∈ Λ where ω ′ ( x ) is the version of the density of the equilibrium measure of E = (cid:8) x ∈ R : | tr X ( x )| ≤ (cid:9) which is continuous on int ( E ) .Proof. First of all, by [13, Theorem IV.2.5, pp. 216] (see also the proof of [16, Corollary 5.4.6]), the densityof ω is continuous on int ( E ) .Let ( ˜ a k : k ∈ N ) be a positive sequence tending to infinity. By (7.7), we can apply [19, Proposition 8 and9] to conclude that(7.9) α i − R i ( x ) = lim k →∞ ˜ a k (cid:18) X i (cid:16) x ˜ a k (cid:17) − σ Id (cid:19) . We set x k = x ˜ a k . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 27
By [19, Proposition 13] we see that belongs to int ( E ) , and for all sufficiently large k discr X i ( x k ) < . Next, by [21, formula (3.1)], we have(7.10) πω ′ ( x k ) = N N − Õ i = α i − |[ X i ( x k )] , | p − discr X i ( x k ) . Since |[ X i ( x k )] , | p − discr X i ( x k ) = ˜ a k |[ X i ( x k )] , | ˜ a k p − discr X i ( x k ) = (cid:12)(cid:12)(cid:2) ˜ a k (cid:0) X i ( x k ) − σ Id (cid:1) (cid:3) , (cid:12)(cid:12)q − discr (cid:0) ˜ a k (cid:0) X i ( x k ) − σ Id (cid:1) (cid:1) , by (7.9), we get lim k →∞ |[ X i ( x k )] , | p − discr X i ( x k ) = α i − |[R i ( x )] , | p − discr R i ( x ) . Thus, by (7.10) we arrive at lim k →∞ ω ′ ( x k ) = π N N − Õ i = α i − |[R i ( x )] , | p − discr R i ( x ) . Since ω ′ is continuous at , the conclusion follows. (cid:3) In the following theorem we show that there is a simple expression for υ . Theorem 7.
Let N be a positive integer and σ ∈ {− , } . Let ( a n : n ∈ N ) and ( b n : n ∈ N ) be N -periodically modulated Jacobi parameters so that X ( ) = σ Id . Suppose that for each i ∈ { , , . . . , N − } the limit R i = lim n →∞ a ( n + ) N + i − ( X nN + i − σ Id ) . exists. If lim n →∞ ( a n + N − a n ) = , then for all x ∈ Λ and i ∈ { , , . . . , N − } , υ ( x ) = lim n →∞ a nN + i α i (cid:12)(cid:12) tr R ′ nN + i ( x ) (cid:12)(cid:12) N π p − discr R nN + i ( x ) (7.11) = N πα i − (cid:12)(cid:12) ( discr R i ) ′ ( x ) (cid:12)(cid:12)p − discr R i ( x ) . (7.12) Proof.
In the proof we use the truncated sequences ( a Ln : n ∈ N ) and ( b Ln : n ∈ N ) defined by formulas(4.2a) and (4.2b), respectively. Let X Ln , R n and R Ln be defined in (4.3) and (4.4). Claim 5. lim L →∞ a L α L tr (cid:16) R L − R LN + L (cid:17) ′ = . Observe that Id − B − L ( x ) (cid:18) − a L + N − a L x − b L a L (cid:19) = (cid:18) − a L + N − a L − (cid:19) (cid:18) (cid:19) , thus X L ( x ) − X LL + N ( x ) = X L ( x ) (cid:18) Id − B − L ( x ) (cid:18) − a L + N − a L x − b L a L (cid:19) (cid:19) = (cid:18) − a L + N − a L − (cid:19) X L ( x ) (cid:18) (cid:19) . Hence, R L ( x ) − R LL + N ( x ) = a L + N − (cid:0) X L ( x ) − X LL + N ( x ) (cid:1) = a L + N − (cid:18) − a L + N − a L − (cid:19) X L ( x ) (cid:18) (cid:19) , which, by [19, Proposition 3], leads to tr (cid:16) R L − R LL + N (cid:17) = a L + N − a L − (cid:0) a L − − a L + N − (cid:1) (cid:18) − a L − a L p [ L + ] N − (cid:19) . By [21, Proposition 3.9], for each i ∈ { , , . . . , N − } , we have 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) ′ ( ) , thus lim k →∞ a k N + i α i tr (cid:16) R k N + i − R k N + ik N + N + i (cid:17) ′ = , which completes the proof of the claim.Next, we consider matrices X Lm and R Lm defined for m ∈ N as X Lm = X LL + N + m , R Lm = R LL + N + m . Clearly, both sequences (cid:0) X Lm : m ∈ N (cid:1) and (cid:0) R Lm : m ∈ N (cid:1) are N -periodic. Let Λ L = (cid:8) x ∈ R : discr R LL + N ( x ) < (cid:9) . Since discr X LL + N = a L + N − discr R LL + N , in view of [21, formula (3.2)], for x ∈ Λ L we have |( tr X L ) ′ ( x )| q − discr X L ( x ) = N − Õ j = a L + j |[ X Lj + ( x )] , | q − discr X Lj + ( x ) . Hence,(7.13) |( tr R L ) ′ ( x )| q − discr R L ( x ) = N − Õ j = a L + j |[R Lj + ( x )] , | q − discr R Lj + ( x ) . Let us now consider x ∈ Λ . By Proposition 1, x ∈ Λ L for sufficiently large L . Let L = nN + i , for n ∈ N and i ∈ { , , . . . , N − } . By Claim 5, we get(7.14) lim n →∞ a nN + i α i |( tr R nN + i ) ′ ( x )| p − discr R nN + i ( x ) = lim n →∞ a nN + i α i |( tr R nN + i ) ′ ( x )| q − discr R nN + i ( x ) . Now we need the following statement.
Claim 6. (7.15) lim n →∞ R nN + ij = R i + j . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 29
First, let us see that (7.15) together with (7.11) give lim n →∞ a nN + i α i |( tr R nN + i ) ′ ( x )| p − discr R nN + i ( x ) = N − Õ j = α i + j |[R i + j + ( x )] , | p − discr R i + j + ( x ) = N − Õ j = α j − |[R j ( x )] , | p − discr R j ( x ) , proving (7.11).Therefore, it remains to prove Claim 6. Observe that for i ′ ∈ { , , . . . , N − } , R Li ′ = R LL + N + i ′ = a L + i ′ − a L + N − (cid:18) L + N + i ′ − Ö k = L + N B Lk (cid:19) R LL + N (cid:18) L + N + i ′ − Ö k = L + N B Lk (cid:19) − . Hence, by Proposition 1, we get(7.16) lim n →∞ R nN + ii ′ ( x ) = α i + i ′ − α i − (cid:18) i + i ′ − Ö k = i B k ( ) (cid:19) R i ( x ) (cid:18) i + i ′ − Ö k = i B k ( ) (cid:19) − . Thus, by repeated application of Proposition 3 to the right-hand side of (7.16), we arrive at (7.15).We now turn to proving (7.12). Since discr X L = a L + N − discr R L , we easily get(7.17) ( discr X L ) ′ = a L + N − ( discr R L ) ′ . On the other hand, we have det X L ≡ , thus(7.18) ( discr X L ) ′ = X L ( tr X L ) ′ . Moreover,(7.19) tr X L = σ + a L + N − tr R L , thus for any x ∈ Λ there is L x > such that for all L ≥ L x one has | tr X L ( x )| > . Therefore, by (7.18) and(7.17) , we obtain ( tr X L ) ′ ( x ) =
12 tr X L ( x ) ( discr X L ) ′ ( x ) = a L + N − ·
12 tr X L ( x ) ( discr R L ) ′ ( x ) , which together with (7.19) gives a L ( tr R L ) ′ ( x ) q − discr R L ( x ) = a L a L + N − ·
12 tr X L ( x ) · ( discr R L ) ′ ( x ) q − discr R L ( x ) . Consequently, we get lim n →∞ a nN + i α i · |( tr R L ) ′ ( x )| q − discr R L ( x ) = α i − · |( discr R i ) ′ ( x )| p − discr R i ( x ) , and the conclusion follows by (7.14) and (7.11). (cid:3) Proposition 5.
For all x ∈ Λ , υ ( x ) > . Proof.
Suppose, contrary to our claim, that υ ( x ) = for some x ∈ Λ . Then by (7.5), [R ( x )] , = , and consequently, discr R ( x ) = (cid:16) [R ( x )] , − [R ( x )] , (cid:17) ≥ , which in view of (7.3) leads to contradiction. (cid:3) The following two examples demonstrates that the assumption (7.7) is necessary for the conclusion (7.8)to hold.
Example 1.
Let N = . Suppose that A has N -periodically modulated entries corresponding to α n ≡ , and β n ≡ . Assume that (7.6) is satisfied with s n = (− ) n , and z n ≡ . Then, by [19, Proposition 9], we have R ( x ) = (cid:18) − x − x (cid:19) , and R ( x ) = (cid:18) x − x − (cid:19) Hence, Λ = R \ [− , ] , and, by (7.5), υ ( x ) = | x | π √ x − , x ∈ Λ which agrees with the formula (7.12) (in view of Proposition 4, the hypotheses of Theorem 7 are satisfied). Example 2.
Let N = . Suppose that A has N -periodically modulated entries corresponding to α n ≡ , and β n ≡ . Assume that (7.6) is satisfied with s n ≡ , and z n = − (− ) n + . Then, by [19, Proposition 9], we have R ( x ) = (cid:18) x − x + (cid:19) , and R ( x ) = (cid:18) x − − x (cid:19) Hence, Λ = R \ [ , ] , and υ ( x ) = | x | + | x − | π √ x − x = | x − | π √ x − x , x ∈ Λ which agrees with the formula (7.12) (in view of Proposition 4, the hypotheses of Theorem 7 are satisfied).
8. Christoffel functions for periodic modulations
Theorem 8.
Let N be a positive integer and σ ∈ {− , } . Let ( a n : n ∈ N ) and ( b n : n ∈ N ) be N -periodically modulated Jacobi parameters so that X ( ) = σ Id . Suppose that (8.1) lim n →∞ ( a n + N − a n ) = . Set R n = a n + N − (cid:0) X n − σ Id (cid:1) . Let K ⊂ Λ be a compact interval with non-empty interior, where Λ is defined in (7.3) . Suppose that (8.2) ( X n : n ∈ N ) , ( R n : n ∈ N ) ∈ D N (cid:0) K , GL ( , R ) (cid:1) SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 31
Then lim n →∞ ρ n K n ( x , x ) = υ ( x ) µ ′ ( x ) uniformly with respect to x ∈ K , where υ is given by (7.5) and ρ n = n Õ j = α j a j . Proof.
Let us fix a compact interval K with non-empty interior contained in Λ . Consider i ∈ { , , . . . , N − } .By (8.1), there is c > such that for all k ∈ N , a k N + i − = a i − + k Õ j = (cid:0) a j N + i − − a ( j − ) N + i − (cid:1) ≤ c ( k + ) , hence lim k →∞ ρ i − k = ∞ . Thus, Theorem 5 easily leads to(8.3) K i ; n ( x , x ) = π µ ′ ( x ) |[R i ( x )] , | p − discr R i ( x ) ρ i − n + E i ; n ( x ) where lim n →∞ ρ i − n sup x ∈ K | E i ; n ( x )| = . Next, by [21, Proposition 3.7], for n , n ′ ∈ N , lim j →∞ a j N + n ′ a j N + n = α n ′ α n , thus, by the Stolz–Cesáro theorem, for each i , i ′ ∈ { , , . . . , N − } , 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 ′ . (8.4)Let us now consider n = k N + i where i ∈ { , , . . . , N − } . We write K k N + i ( x , x ) = N − Õ i ′ = K i ′ ; k ( x , x ) + N − Õ i ′ = i + ( K i ′ − k ( x , x ) − K i ′ ; k ( x , x )) . 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 (8.3), lim k →∞ ρ k N + i K k N + i ( x , x ) = N − Õ i ′ = π µ ′ ( x ) |[R i ′ ( x )] , | p − discr R i ′ ( x ) · lim k →∞ ρ i ′ − k ρ k N + i . Using now (8.4) and (7.5), we obtain lim k →∞ ρ k N + i K k N + i ( x , x ) = µ ′ ( x ) · N π N − Õ i ′ = |[R i ′ ( x )] , | p − discr R i ′ ( x ) · α i ′ − = υ ( x ) µ ′ ( x ) , which completes the proof. (cid:3) Remark 1.
If one assumes that (cid:18) α n − α n a n − a n − : n ∈ N (cid:19) , (cid:18) β n α n a n − b n : n ∈ N (cid:19) , (cid:18) a n : n ∈ N (cid:19) ∈ D N ( R ) , then condition (8.2) is satisfied for any compact K ⊂ R , see [19, Proposition 9].8.1. Applications to Ignjatović conjecture.
In this section we show how Theorem 8 leads to the conjecturedue to Ignjatović [5, Conjecture 1].
Conjecture 1 (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. Our results entail the following corollary.
Corollary 2.
Let N be a positive integer. Suppose that lim n →∞ a n − a n = , lim n →∞ b n a n = q , lim n →∞ a n = ∞ , lim n →∞ ( a n + N − a n ) = , for some (8.5) q ∈ (cid:8) ( j π N ) : j = , , . . . , N − (cid:9) . If (cid:0) a n − a n − : n ∈ N (cid:1) , (cid:0) b n − qa n : n ∈ N (cid:1) , (cid:18) a n : n ∈ N (cid:19) ∈ D N ( R ) , then lim n →∞ (cid:18) n Õ j = a j (cid:19) − n Õ j = p j ( x ) = υ ( x ) µ ′ ( x ) , locally uniformly with respect to x ∈ Λ , where Λ and υ are defined in (7.3) and (7.5) , respectively.Proof. Let α n ≡ , β n ≡ q . Observe that X ( ) = (cid:18) − − q (cid:19) N = − − q / − / ! N . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 33
Hence, by [21, 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 Chebyshev polynomials of the second kind defined as U n ( x ) = sin (cid:0) ( n + ) arccos ( x ) (cid:1) sin (cid:0) arccos ( x ) (cid:1) , x ∈ (− , ) from which we readily derive X ( ) = σ Id with σ = (− ) N + j . Now, in view of Remark 1, the conclusion is a consequence of Theorem 8. (cid:3) By using a different method the conclusion of Corollary 2 for q = , s n ≡ and z n ≡ (cf. formula (7.6))has been proven in [18, Corollary 3].Let us recall that in [21, Corollary 4.16] there was considered the case when (8.5) is not satisfied. In fact,under some hypotheses it was shown that(8.6) 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 . Let us stress that in the setup of Corollary 1 it is still possible tohave (8.6) locally uniformly with respect to x ∈ R \ { } , see Theorem 6. Nevertheless, as the next exampledemonstrates, it is not the case that one always obtains (8.6) provided that the left hand side exists. Example 3.
Let a n = √ n + , and b n = − (− ) n . Then lim n →∞ (cid:18) n Õ j = a j (cid:19) − n Õ j = p j ( x ) = | x − | π p x ( x − ) µ ′ ( x ) locally uniformly with respect to x ∈ R \ [ , ] . Indeed, since the hypothesis of Corollary 2 are satisfied for N = and q = , the conclusion follows from Example 2.
9. Universality limits of Christoffel–Darboux kernel
Proposition 6.
Let N be a positive integer and σ ∈ {− , } . Let ( a n : n ∈ N ) and ( b n : n ∈ N ) be N -periodically modulated Jacobi parameters so that X ( ) = σ Id . Suppose that for each i ∈ { , , . . . , N − } the limit R i = lim n →∞ a ( n + ) N + i − ( X nN + i − σ Id ) . exists. Let (9.1) θ n ( x ) = arccos (cid:18) tr X n ( x ) p det X n ( X ) (cid:19) , Then for each i ∈ { , , . . . , N − } , (9.2) lim n →∞ a nN + i α i (cid:12)(cid:12) θ ′ nN + i ( x ) (cid:12)(cid:12) = N πυ ( x ) , and (9.3) lim n →∞ a nN + i α i θ ′′ nN + i ( x ) = − α i − tr X ′′ i ( ) p − discr R i ( x ) − α i − σ (cid:0) N πυ ( x ) (cid:1) (cid:0) − discr R i ( x ) (cid:1) / locally uniformly with respect to x ∈ Λ , where Λ and υ are given by (7.3) and (7.5) , respectively. Proof.
Let us fix i ∈ { , , . . . , N − } . Since(9.4) det X k N + i ( x ) = a k N + i − a ( k + ) N + i − , we conclude that lim k →∞ det X k N + i ( x ) = det (cid:0) σ Id (cid:1) = . The chain rule applied to (9.1) 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 ) , (9.5)thus θ ′ k N + i ( x ) = − tr R ′ k N + i ( x ) p − discr R k N + i ( x ) where we have set R n = a n + N − ( X n − σ Id ) . Hence, the formula (9.2) is a consequence of (7.11).Next, by taking derivative of (9.5), we obtain(9.6) 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) / . Since a k N + i α i tr X ′′ k N + i ( x ) p − discr X k N + i ( x ) = α i a ( k + ) N + i − a k N + i (cid:0) a k N + i α i tr X ′′ k N + i ( x ) (cid:1) p − discr R k N + i ( x ) , by [21, Corollary 3.10], we get(9.7) lim k →∞ a k N + i α i tr X ′′ k N + i ( x ) p − discr X k N + i ( x ) = α i − tr X ′′ i ( ) p − discr R i ( x ) . Similarly, a k N + i α i (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) / = α i a ( k + ) N + i − a k N + i (cid:0) a k N + i α i tr R ′ k N + i ( x ) (cid:1) tr X k N + i ( x ) (cid:0) − discr R k N + i ( x ) (cid:1) / , and, by (7.11),(9.8) lim k →∞ a k N + i α i (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) / = α i − σ (cid:0) N πυ ( x ) (cid:1) (cid:0) − discr R i ( x ) (cid:1) / . Finally, combining (9.7) and (9.8) with (9.6) we obtain (9.3). This completes the proof. (cid:3)
Lemma 3.
Let ( γ k : k ∈ N ) be a sequence of positive numbers such that ∞ Õ k = γ k = ∞ , and lim k →∞ γ k − γ k = . Assume that ( ξ k : k ∈ N ) is a sequence of continuous functions on some open subset U ⊂ R d , with valuesin ( , π ) . Suppose that there is ψ : U → ( , π ) such that lim n →∞ ξ n ( x ) γ n = ψ ( x ) SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 35 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 , y n = x + br n . Then for each compact subset K ⊂ U , L > and any function σ : U → R , lim n →∞ n Õ k = γ k Í nj = γ j cos (cid:16) k Õ j = ξ j ( x n ) + ξ j ( y n ) + σ ( x n ) + σ ( y n ) (cid:17) = uniformly with respect to x ∈ K , and a , b ∈ [− L , L ] .Proof. Let us fix a compact set K , and let N ∈ N be such that r n ≥ R , for n ≥ N . For ( x , a , b ) ∈ U ×(− L , L ) ,we set ˜ ξ j ( x , a , b ) = ξ j (cid:18) x + aR (cid:19) + ξ j (cid:18) x + bR (cid:19) , ˜ ψ ( x , a , b ) = ψ (cid:18) x + aR (cid:19) + ψ (cid:18) x + bR (cid:19) . Thus(9.9) lim j →∞ γ j ˜ ξ j ( x , a , b ) = ˜ ψ ( x , a , b ) . In view of Lemma 1 we obtain lim n →∞ n Õ k = γ k Í nj = γ j cos (cid:16) k Õ j = ˜ ξ j ( x , a , b ) (cid:17) = lim n →∞ Ξ n ( x , a , b ) ∫ ˜ Ξ n ( x , a , b ) cos ( t ) d t . By (9.9), there is c > such that for all x ∈ K , and a , b ∈ [− L , L ] , ˜ ξ j ( x , a , b ) ≥ c γ j . Hence, Ξ n ( x , a , b ) ∫ ˜ Ξ n ( x , a , b ) cos ( t ) d t ≤ Ξ n ( x , a , b ) , which implies that lim n →∞ n Õ k = γ k Í nj = γ j cos (cid:16) k Õ j = ˜ ξ j ( x , a , b ) (cid:17) = lim n →∞ Ξ n ( x , a , b ) ∫ ˜ Ξ n ( x , a , b ) cos ( t ) d t = , uniformly with respect to x ∈ K , a , b ∈ [− L , L ] . (cid:3) Theorem 9.
Assume that ( ξ j : j ∈ N ) is a sequence of C ( U ) functions with values in ( , π ) such that foreach compact set K ⊂ U there are functions ξ : U → ( , ∞) and ψ : U → ( , ∞) , and c > so that(a) lim n →∞ sup x ∈ K (cid:12)(cid:12) γ − n · ξ n ( x ) − ξ ( x ) (cid:12)(cid:12) = , (b) lim n →∞ sup x ∈ K (cid:12)(cid:12) γ − n · ξ ′ n ( x ) − ψ ( x ) (cid:12)(cid:12) = , (c) sup n ∈ N sup x ∈ K (cid:12)(cid:12) γ − n · ξ ′′ n ( x ) (cid:12)(cid:12) ≤ c , where ( γ k : k ∈ N ) is a sequence of positive numbers such that ∞ Õ k = γ k = ∞ , and lim k →∞ γ k − γ k = . For x ∈ U and a , b ∈ [− L , L ] , we set x n = x + a Í nk = γ k , 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) k Õ j = ξ j ( x n ) + σ ( x n ) (cid:17) sin (cid:16) k Õ j = ξ 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. We write · sin (cid:16) k Õ j = ξ j ( x ) + σ ( x ) (cid:17) sin (cid:16) k Õ j = ξ j ( y ) + σ ( y ) (cid:17) = cos (cid:16) k Õ j = (cid:0) ξ j ( x ) − ξ j ( y ) (cid:1) + (cid:0) σ ( x ) − σ ( y ) (cid:1) (cid:17) − cos (cid:16) k Õ j = (cid:0) ξ j ( x ) + ξ j ( y ) (cid:1) + (cid:0) σ ( x ) + σ ( y ) (cid:1) (cid:17) . By Lemma 3, we conclude that lim n →∞ n Õ k = γ k Í nj = γ j cos (cid:16) k Õ j = (cid:0) ξ j ( x n ) + ξ j ( y n ) (cid:1) + (cid:0) σ ( x n ) + σ ( y n ) (cid:1) (cid:17) = . We next write cos (cid:16) k Õ j = (cid:0) ξ j ( x ) − ξ j ( y ) (cid:1) + (cid:0) σ ( x ) − σ ( y ) (cid:1) (cid:17) = cos (cid:16) k Õ j = (cid:0) ξ j ( x ) − ξ j ( y ) (cid:1) (cid:17) cos (cid:0) σ ( y ) − σ ( x ) (cid:1) − sin (cid:16) k Õ j = (cid:0) ξ j ( x ) − ξ j ( y ) (cid:1) (cid:17) sin (cid:0) σ ( y ) − σ ( x ) (cid:1) , thus, it is enough to prove that lim n →∞ n Õ k = γ k Í nj = γ j cos (cid:16) k Õ j = (cid:0) ξ j ( y n ) − ξ j ( x n ) (cid:1) (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 . Since (see, e.g., [21, Claim 5.10]) (cid:12)(cid:12)(cid:12) ξ j ( y n ) − ξ j ( x n ) − ( b − a ) ξ ′ j ( x ) (cid:16) n Õ ℓ = γ ℓ (cid:17) − (cid:12)(cid:12)(cid:12) ≤ c (cid:16) n Õ ℓ = γ ℓ (cid:17) − sup u ∈ K | ξ ′′ j ( t )| , we obtain (cid:12)(cid:12)(cid:12) cos (cid:16) k Õ j = (cid:0) ξ j ( x n ) − ξ j ( y n ) (cid:1) (cid:17) − cos (cid:16) ( b − a ) (cid:16) n Õ ℓ = γ ℓ (cid:17) − k Õ j = ξ ′ j ( x ) (cid:17) (cid:12)(cid:12)(cid:12) ≤ k Õ j = (cid:12)(cid:12)(cid:12) ξ j ( y n ) − ξ j ( x n ) − ( b − a ) ξ ′ j ( x ) (cid:16) n Õ j = γ ℓ (cid:17) − (cid:12)(cid:12)(cid:12) ≤ c (cid:16) n Õ ℓ = γ ℓ (cid:17) − k Õ j = sup u ∈ K | ξ ′′ j ( t )| . SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 37
Consequently, (cid:12)(cid:12)(cid:12)(cid:12) n Õ k = γ k Í nj = γ j (cid:18) cos (cid:16) k Õ j = ξ j ( x n ) − ξ j ( y n ) (cid:17) − cos (cid:16) ( b − a ) (cid:16) n Õ ℓ = γ ℓ (cid:17) − k Õ j = ξ ′ j ( x ) (cid:17) (cid:19)(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 u ∈ K | ξ ′′ j ( u )| . In view of the Stolz–Cesàro theorem, 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 arrive at lim n →∞ (cid:16) n Õ ℓ = γ ℓ (cid:17) − n Õ k = γ k k Õ j = sup u ∈ K | ξ ′′ j ( u )| =
13 lim n →∞ (cid:16) n Õ ℓ = γ ℓ (cid:17) − n Õ j = sup u ∈ K | ξ ′′ j ( u )| =
16 lim n →∞ γ − n (cid:16) n Õ ℓ = γ ℓ (cid:17) − sup u ∈ K | ξ ′′ n ( u )| = where the last equality follows by (c). Now, our task is to show that lim n →∞ n Õ k = γ k Í n ℓ = γ ℓ cos (cid:16) ( b − a ) (cid:16) n Õ j = γ ℓ (cid:17) − k Õ j = ξ ′ j ( x ) (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 . At this point we apply Lemma 2 to get lim n →∞ n Õ k = γ k Í n ℓ = γ ℓ cos (cid:16) ( b − a ) (cid:16) n Õ j = γ ℓ (cid:17) − k Õ j = ξ ′ j ( x ) (cid:17) = ( b − a ) ψ ( x ) ∫ ( b − a ) ψ ( x ) cos ( t ) d t = sin (cid:0) ( b − a ) ψ ( x ) (cid:1) ( b − a ) ψ ( x ) , and the theorem follows. (cid:3) Theorem 10.
Let N be a positive integer and σ ∈ {− , } . Let ( a n : n ∈ N ) and ( b n : n ∈ N ) be N -periodically modulated Jacobi parameters so that X ( ) = σ Id . Suppose that lim n →∞ ( a n + N − a n ) = and for each i ∈ { , , . . . , N − } the limit R i = lim n →∞ R nN + i ( x ) exists where R n = a n + N − (cid:0) X n − σ Id (cid:1) . Let K ⊂ Λ be a compact interval with non-empty interior, where Λ is defined in (7.3) . If ( X n : n ∈ N ) , ( R n : n ∈ N ) ∈ D N (cid:0) K , GL ( , R ) (cid:1) , then lim n →∞ ρ n K n (cid:18) x + u ρ n , x + v ρ n (cid:19) = υ ( x ) µ ′ ( x ) sin (cid:0) ( u − v ) υ ( x ) (cid:1) ( u − v ) υ ( x ) locally uniformly with respect to x ∈ Λ and u , v ∈ R , where ρ n = n Õ j = α j a j , and υ is defined in (7.5) . Proof.
Let K be a compact interval with non-empty interior contained in Λ and let L > . We select acompact interval ˜ K ⊂ Λ 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 nN + i , x + uN α i ρ i ; n ∈ ˜ K . Given x ∈ K and 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 . In view of Remark 1, ( X j N + i : j ∈ N ) , ( R j N + i : j ∈ N ) ∈ D (cid:0) K , GL ( , R ) (cid:1) . Hence, by Theorem 4, there are c > and M ∈ N such that for all x , y ∈ K , and k ≥ M , a ( k + ) N + i − p k N + i ( x ) p k N + i ( y ) = π s |[R i ( x )] , ]| µ ′ ( x ) p − discr R i ( x ) s |[R i ( y )] , | µ ′ ( y ) p − discr R i ( y )× sin (cid:16) k Õ j = M + θ j N + i ( x ) + η i ( x ) (cid:17) sin (cid:16) k Õ j = M + θ j N + i ( y ) + η i ( y ) (cid:17) + E k N + i ( x , y ) where sup x , y ∈ K | E k N + i ( x , y )| ≤ c ∞ Õ j = k sup K k X ( j + ) N + i − X j N + i k + sup K k R ( j + ) N + i − R j N + i k . Therefore, we obtain n Õ k = M p k N + i ( x ) p k N + i ( y ) = π s |[R i ( x )] , ]| µ ′ ( x ) p − discr R i ( x ) s |[R i ( y )] , | µ ′ ( y ) p − discr R i ( y )× n Õ k = M a ( k + ) N + i − sin (cid:16) k Õ j = M + θ j N + i ( x ) + η i ( x ) (cid:17) sin (cid:16) k Õ j = M + θ j N + i ( y ) + η i ( y ) (cid:17) + n Õ k = M a ( k + ) N + i − E k N + i ( x , y ) . Observe that by the Stolz–Cesáro theorem, lim n →∞ ρ i − n n Õ k = M + a ( k + ) N + i − E k N + i ( x , y ) = lim n →∞ a nN + i − a ( n + ) N + i − E nN + i ( x , y ) = . In view of Proposition 6, we can apply Theorem 9 with ξ j ( x ) = θ j N + i ( x ) , γ j = N α i − a ( j + ) N + i − , and ψ ( x ) = πυ ( x ) . Therefore, for any i ′ ∈ { , , . . . , N − } , as n tends to infinity 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 nN + i ′ ) + η i ( x nN + i ′ ) (cid:17) × sin (cid:16) k Õ j = M + θ j N + i ( y nN + i ′ ) + η i ( y nN + i ′ ) (cid:17) SYMPTOTIC BEHAVIOUR OF CHRISTOFFEL–DARBOUX KERNEL 39 approaches to sin (cid:0) ( v − u ) πυ ( x ) (cid:1) ( v − u ) πυ ( x ) uniformly with respect to x ∈ K and u , v ∈ [− L , L ] . Moreover, lim n →∞ |[R i ( x nN + i ′ )] , ]| µ ′ ( x nN + i ′ ) p − discr R i ( x nN + i ′ ) = lim n →∞ |[R i ( y nN + i ′ )] , ]| µ ′ ( y nN + i ′ ) p − discr R i ( y nN + i ′ ) = |[R i ( x )] , ]| µ ′ ( x ) p − discr R i ( x ) . Hence,(9.10) lim n →∞ ρ i − n K i ; n ( x nN + i ′ , y nN + i ′ ) = sin (cid:0) ( v − u ) πυ ( x ) (cid:1) ( v − u ) πυ ( x ) · |[R i ( x )] , ]| π µ ′ ( x ) p − discr R i ( x ) . Finally, we write 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) . Observe that sup x , y ∈ K (cid:12)(cid:12) K i − n ( x , y ) − K i ; n ( x , y ) (cid:12)(cid:12) sup x , y ∈ K p p nN + i ( x ) p nN + i ( y ) ≤ c , thus, by (9.10) and (8.4), lim n →∞ ρ nN + i ′ K nN + i ′ ( x nN + i ′ , y nN + i ′ ) = lim n →∞ N − Õ i = ρ i − n K nN + i ( x nN + i ′ , y nN + i ′ ) · ρ i − n ρ nN + i ′ = µ ′ ( x ) sin (cid:0) ( v − u ) πυ ( x ) (cid:1) ( v − u ) πυ ( x ) N π N − Õ i = |[R i ( x )] , ]| p − discr R i ( x ) · α i − . Hence, by (7.5), lim n →∞ ρ nN + i ′ K nN + i ′ ( x nN + i ′ , y nN + i ′ ) = υ ( x ) µ ′ ( x ) · sin (cid:0) ( v − u ) πυ ( x ) (cid:1) ( v − u ) πυ ( x ) , and the theorem follows. (cid:3) References [1] P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Venakides, and X. Zhou,
Strong asymptotics of orthogonal polynomials withrespect to exponential weights , Comm. Pure Appl. Math. (1999), no. 12, 1491–1552.[2] P.A. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach , Courant Lecture Notes in Mathe-matics, vol. 3, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society,Providence, RI, 1999.[3] J. Dombrowski,
Eigenvalues and spectral gaps related to periodic perturbations of Jacobi matrices , Spectral Methods forOperators of Mathematical Physics, Operator Theory: Advances and Applications, vol. 154, Birkhäuser Basel, 2004, pp. 91–100.[4] J.S. Geronimo and W. Van Assche,
Approximating the weight function for orthogonal polynomials on several intervals , J.Approx. Theory (1991), 341–371.[5] A. Ignjatović, Asymptotic behaviour of some families of orthonormal polynomials and an associated Hilbert space , J. Approx.Theory (2016), 41–79.[6] A. Ignjatovic and D.S. Lubinsky,
On an asymptotic equality for reproducing kernels and sums of squares of orthonormalpolynomials , Progress in approximation theory and applicable complex analysis, Springer Optim. Appl., vol. 117, Springer,Cham, 2017, pp. 129–144.[7] J. Janas and S. Naboko,
Spectral analysis of selfadjoint Jacobi matrices with periodically modulated entries , J. Funct. Anal. (2002), no. 2, 318–342.[8] S. Khan and D.B. Pearson,
Subordinacy and spectral theory for infinite matrices , Helv. Phys. Acta (1992), no. 4, 505–527.[9] A.B. J. Kuijlaars, K.T.-R. McLaughlin, W. Van Assche, and M. Vanlessen, The Riemann-Hilbert approach to strong asymptoticsfor orthogonal polynomials on [− , ] , Adv. Math. (2004), no. 2, 337–398. [10] E. Levin and D.S. Lubinsky, Orthogonal polynomials for exponential weights , CMS Books in Mathematics/Ouvrages deMathématiques de la SMC, vol. 4, Springer-Verlag, New York, 2001.[11] D.S. Lubinsky,
An update on local universality limits for correlation functions generated by unitary ensembles , SIGMASymmetry Integrability Geom. Methods Appl. (2016), Paper No. 078, 36.[12] A. Máté and P. Nevai, Orthogonal polynomials and absolutely continuous measures , Approximation theory, IV (CollegeStation, Tex., 1983), Academic Press, New York, 1983, pp. 611–617.[13] E.B. Saff and V. Totik,
Logarithmic Potentials with External Fields , vol. 316, Springer-Verlag, 1997.[14] K. Schmüdgen,
The moment problem , Graduate Texts in Mathematics, vol. 277, Springer, Cham, 2017.[15] B. Simon,
The Christoffel-Darboux kernel , Perspectives in partial differential equations, harmonic analysis and applications,Proc. Sympos. Pure Math., vol. 79, Amer. Math. Soc., Providence, RI, 2008, pp. 295–335.[16] ,
Szegő’s theorem and its descendants: Spectral theory for L perturbations of orthogonal polynomials , PrincetonUniversity Press, 2010.[17] H. Stahl and V. Totik, General orthogonal polynomials , Encyclopedia of Mathematics and its Applications, vol. 43, CambridgeUniversity Press, Cambridge, 1992.[18] G. Świderski,
Periodic perturbations of unbounded Jacobi matrices II: Formulas for density , J. Approx. Theory (2017),67–85.[19] ,
Periodic perturbations of unbounded Jacobi matrices III: The soft edge regime , J. Approx. Theory (2018), 1–36.[20] G. Świderski and B. Trojan,
Periodic perturbations of unbounded Jacobi matrices I: Asymptotics of generalized eigenvectors ,J. Approx. Theory (2017), 38–66.[21] ,
Asymptotic behaviour of Christoffel–Darboux kernel via three-term recurrence relation I , arXiv:1909.09107, 2019.[22] ,
About discrete spectra of unbounded Jacobi matrices , preprint, 2020.[23] ,
Asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients , J. Funct. Anal. (2020),no. 3, 108326, 55.[24] V. Totik,
Universality and fine zero spacing on general sets , Ark. Mat. (2009), no. 2, 361–391. Grzegorz Świderski, Department of Mathematics, KU Leuven, Celestijnenlaan 200B box 2400, BE-3001 Leuven,Belgium & Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
E-mail address : [email protected] Bartosz Trojan, The Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-696 Warszawa,Poland
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