Strong asymptotic of Cauchy biorthogonal polynomials and orthogonal polynomials with varying measure
aa r X i v : . [ m a t h . C A ] F e b STRONG ASYMPTOTIC OF CAUCHY BIORTHOGONAL POLYNOMIALSAND ORTHOGONAL POLYNOMIALS WITH VARYING MEASURE
L. G. GONZ ´ALEZ RICARDO AND G. L ´OPEZ LAGOMASINO
Abstract.
We give the strong asymptotic of Cauchy biorthogonal polynomials under theassumption that the defining measures are supported on non intersecting intervals of the realline and satisfy Szeg˝o’s condition. The biorthogonal polynomials are connected with certainmixed type Hermite-Pad´e polynomials, which verify full orthogonality relations with respect tosome varying measures. Thus, the strong asymptotic of orthogonal polynomials with respectto varying measures plays a key role in the study.
Keywords: biorthogonal polynomials, Hermite-Pad´e approximation, varying measures, strongasymptotic
AMS classification:
Primary: 42C05; 30E10, Secondary: 41A211.
Introduction
Cauchy biorthogonal polynomials.
Let ∆ = (∆ , ∆ ) be a pair of intervals, containedin the real line R , which have at most one common point. By M ( ∆ ) we denote the cone of allpairs ( σ , σ ) of positive Borel measures with finite moments whose supports verify supp σ k ⊂ ∆ k and Z Z d σ ( x ) d σ ( y ) | x − y | < ∞ . Fix ( σ , σ ) ∈ M ( ∆ ). For each pair of non negative integers ( m, n ) ∈ Z ≥ there exists a pair( P m , Q n ) of monic polynomials whose degrees verify deg P m ≤ m, deg Q n ≤ n, and(1.1) Z ∆ Z ∆ P m ( x ) Q n ( y ) d σ ( x ) d σ ( y ) x − y = C n δ m,n , C n = 0 . (As usual, δ m,n = 0 , m = n, δ n,n = 1.) These polynomials were introduced in [7] and calledCauchy biorthogonal polynomials. Some interesting properties were revealed. In particular, itwas shown that deg P n = n , its zeros are simple, interlace for consecutive values of n , and lie in ◦ ∆ (the interior of ∆ with the Euclidean topology of R ). The same goes for the Q n on ∆ . Date : February 9, 2021.The first author was supported in part by a research fellowship from Department of Mathematics of UniversidadCarlos III de Madrid, Spain.The second author was supported in part by the research grant PGC2018-096504-B-C33 of Ministerio deCiencia, Innovaci´on y Universidades, Spain.
Cauchy biorthogonal polynomials appear in the analysis of the two matrix model [6, 8] andwere used to find discrete solutions of the Degasperis-Procesi equation [7] through a Hermite-Pad´e approximation problem for two discrete measures. In [8], the authors apply the nonlinearsteepest descent method to a class of 3 × , ∆ are bounded non intersecting intervals and the measures σ , σ verify Szeg˝o’s condition(1.2) Z ∆ k ln σ ′ k ( x ) d x p ( b k − x )( x − a k ) > −∞ , k = 1 , , where σ ′ denotes the Radon-Nikodym derivative of σ with respect to the Lebesgue measure. Inthis case we write ( σ , σ ) ∈ S ( ∆ ). Theorem 1.1.
Let ( σ , σ ) ∈ S ( ∆ ) and ( P n ) n ≥ , ( Q n ) n ≥ are the sequences of monic polyno-mials determined by (1.1) . Then (1.3) lim n P n ( z )Φ n ( z ) = G ∗ ( z ) G ∗ ( ∞ ) , lim n Q n ( z )Φ n ( z ) = G ( z ) G ( ∞ ) , uniformly on each compact subset of Ω = C \ ∆ and Ω = C \ ∆ , respectively, where Φ k ∈H (Ω k ) , k = 1 , , (holomorphic in Ω k ) is defined in (3.22) , G ∗ is the function which appears in (3.53) and G appears in (3.45) . The logarithmic and ratio asymptotic of biorthogonal polynomials were obtained in [15] formore general Cauchy type kernels involving m ≥ ii Hermite-Pad´e poly-nomials of Angelesco and Nikishin systems. We give more details in Section 3 where Theorem1.1 is obtained as a consequence of a more general result (Theorem 3.8) connected with thestrong asymptotic of polynomials arising from an associated mixed type Hermite-Pad´e approxi-mation problem. Here we limit ourselves to saying that the proof heavily depends on the strongasymptotic of sequences of orthogonal polynomials with respect to varying measures.1.2.
Orthogonal polynomials with varying measures.
Let ∆ = [ a, b ] ⊂ R . Consider asequence (d µ n /w n ) n ≥ where µ n is a finite positive Borel measure supported on ∆ and w n is a polynomial with real coefficients, deg w n = i n ≤ n , whose zeros ( x n,i ) ni =2 n − i n +1 lie in C \ ∆. This is called a sequence of varying measures. Let Q n ( x ) = x n + · · · be the n -th monicorthogonal polynomial satisfying Z x ν Q n ( x ) d µ n ( x ) | w n ( x ) | = 0 , ν = 0 , , . . . , n − . TRONG ASYMPTOTIC OF POLYNOMIALS 3
The sequence ( Q n ) n ≥ is called the sequence of monic polynomials with respect to the givenvarying measures. A common normalization is to take τ n := (cid:18)Z Q n ( x ) d µ n ( x ) | w n ( x ) | (cid:19) − / , and define q n ( x ) := τ n Q n ( x ) as the orthonormal polynomial of degree n .In the context of multipoint Pad´e and Hermite-Pad´e approximation, orthogonal polynomialswith respect to varying measures arise naturally (see, for example, [2, 4, 9, 17, 19]). Dependingon the type of asymptotic one wishes to obtain for the sequence ( Q n ) n ≥ (or ( q n ) n ≥ ), someconditions must be imposed on the varying measures. Combinations of the following ones areappropriate in our case.i) There exists a finite positive Borel measure supported on ∆ such that lim n µ n = µ inthe weak star topology of measures, µ ′ > n Z | µ ′ n − µ ′ | d x = 0 . ii) The measure µ satisfies Szeg˝o’s condition on ∆; that is, Z ∆ ln µ ′ ( x ) d x p ( x − a )( b − x ) > −∞ . and lim inf n Z ln µ ′ n ( x ) d x p ( x − a )( b − x ) ≥ Z ln µ ′ ( x ) d x p ( x − a )( b − x ) . iii) Let Ψ be the conformal map from Ω = C \ ∆ onto the exterior of the unit circle suchthat Ψ( ∞ ) = ∞ and Ψ ′ ( ∞ ) >
0. The sequence of zeros of the w n verifylim n →∞ n X i =2 n − i n +1 (cid:18) − | Ψ( x n,i ) | (cid:19) = ∞ . (iv) There exist positive non-negative continuous functions ϕ and ψ on ∆ such thatlim n ϕ n ( x ) | w n ( x ) | = 1 /ψ ( x )uniformly on compact subsets of ( a, b ) and(1.4) lim n →∞ Z ba ln( ϕ n ( x ) | w n ( x ) | ) p ( b − x )( x − a ) d x = − Z ba ln ψ ( x ) p ( b − x )( x − a ) d x < + ∞ . In many applications, d µ n = h n d µ, µ ′ > h n ) n ≥ is a sequence of positivecontinuous functions which converges uniformly on ∆ to a positive continuous function h , andthe zeros of the polynomials w n are uniformly bounded away from ∆ in which case i ) and iii )are immediate, and ii ) holds if µ verifies Szeg˝o’s condition.Conditions i ) − iii ) are sufficient to prove strong asymptotic for ( Q n ) n ≥ . The first result inthis direction appeared in [18] and was later improved in [11] and [5]. An alternative proof of themain result in [18] may be found in [25]. The answer in [18] is given in terms of a Szeg˝o functionassociated with µ and a Blaschke product in which the zeros of w n intervene (see (2.2) below). L. G. GONZ´ALEZ RICARDO AND G. L ´OPEZ LAGOMASINO
This is too general for our purpose. In order to give the answer in a closer form, some knowledgeof the asymptotic behavior of the polynomials w n is required and condition iv ) comes in.Let ϕ be a positive continuous function on ∆. Let λ ϕ be the (unitary) equilibrium measuresupported on ∆ which solves the equilibrium problem for the logarithmic potential with externalfield − ln ϕ . It is well known that λ ϕ is uniquely determined by the equilibrium conditions on∆ [24, Theorem i .3.3]:(1.5) V λ ϕ ( x ) −
12 ln ϕ ( x ) ( = γ, x ∈ supp λ ϕ ,> γ, x ∈ ∆ \ supp λ ϕ , where γ is a constant and V λ ϕ ( u ) = Z ln 1 | u − x | d λ ϕ ( x )is the logarithmic potential of λ ϕ . We will assume that ϕ is such that supp λ ϕ = ∆. (This istrue, for example, if ln ϕ = V τ is the logarithmic potential of a measure τ , of total mass c ≤ λ ϕ is the balayage of τ on ∆ plus (1 − c )times the Chebyshev measure on ∆.)Set(1.6) Φ( u ) := e − v ϕ ( u ) , v ϕ := V λ ϕ + i e V λ ϕ , C := e γ , where e V λ ϕ denotes the harmonic conjugate of V λ ϕ on C \ ∆. Though e V λ ϕ is multi-valued, it hasan increment of 2 π if we surround once the interval ∆ in the positive direction; consequently, Φ isa single-valued analytic function in C \ ∆ with a simple pole at ∞ since Φ( u ) = u + O (1) , u → ∞ .We write µ ∈ S (∆) when µ verifies Szeg˝o’s condition on ∆. Then, the Szeg˝o function of µ isdefined as(1.7) G ( µ, u ) := exp " p ( u − b )( u − a )2 π Z ∆ ln( p ( b − x )( x − a ) µ ′ ( x )) x − u d x p ( b − x )( x − a ) , The square root outside the integral is taken to be positive for u > b and those inside the integralare positive when x ∈ ( a, b ). Theorem 1.2.
Assume that ( µ n , w n ) n ≥ verifies i ) − iv ) , µ ∈ S (∆) , and supp λ ϕ = ∆ (see (1.5) ). Then, (1.8) lim n q n ( u ) C n Φ n ( u ) = 1 √ π G ( ψµ, u ) , uniformly on compact subsets of Ω , and ψµ is the measure with differential expression ψ d µ .Moreover, (1.9) lim n τ n C n = 1 √ π G ( ψµ, ∞ ) , and (1.10) lim n Q n ( u )Φ n ( u ) = G ( ψµ, u ) G ( ψµ, ∞ ) . uniformly on compact subsets of Ω . TRONG ASYMPTOTIC OF POLYNOMIALS 5
The following result is obtained from Theorem 1.2. It is in the spirit of [26, Theorem 14.3]. Theassumptions have points in common but they are not the same. In some regards the conditionsin [26] are more general, in others our assumptions are weaker.
Theorem 1.3.
Let ( µ n ) n ≥ be a sequence of measures verifying i ) − ii ) . Let τ, d τ = v d x, be aprobability measure on ∆ such that supp τ = ∆ , v is continuous on ∆ , and let there be constants A, β > − , and β such that (1.11) A − (( b − x )( x − a )) β ≤ v ( x ) ≤ A (( b − x )( x − a )) β , x ∈ ( a, b ) . Set Φ τ ( u ) = exp ( − V τ ( u ) − i e V τ ( u )) , where e V τ is the harmonic conjugate on C \ ∆ of the logarithmic potential V τ . Then (1.12) lim n p n ( u )Φ nτ ( u ) = 1 √ π G ( µ, u ) , uniformly on compact subsets of Ω , where p n is the n -th orthonormal polynomials verifying Z p m ( x ) p n ( x ) d µ n ( x ) | Φ nτ ( x ) | = ( , m < n, , m = n. The paper is organized as follows. Section 2 is dedicated to the study of the strong asymptoticof polynomials orthogonal with respect to varying measures; in particular, Theorems 1.2 and 1.3are proved. Section 3 is devoted to the study of the strong asymptotic of a sequence of Hermite-Pad´e polynomials intimately connected with the Cauchy biorthogonal polynomials defined above.Here, the results of Section 2 are used to prove Theorem 3.8 from which Theorem 1.1 follows.2.
Strong asymptotic of orthogonal polynomials with varying measures
As mentioned above our goal here is to prove Theorems 1.2 and 1.3. They are essential in theproof of Theorem 1.1, but have independent interest and may find other applications. We beginexplainig our choice of Szeg˝o function for measures supported on an interval of the real line.2.1.
The Szeg˝o function.
Let µ ∈ S ([ − , Z − ln µ ′ ( x ) d x √ − x > −∞ . On the unit circle T one can define a symmetric measure σ with the property that σ ( B ) = µ ( B ∗ )whenever B is a Borel set contained either in the upper or lower half of the unit circle and B ∗ is its orthogonal projection on [ − , σ ′ ( e it ) = | sin t | µ ′ (cos t ) , t ∈ [0 , π ]where σ ′ and µ ′ denote the Radon-Nikodym derivatives of σ and µ with respect to the Lebesguemeasure on T and [ − , ζ = e it and x = Re( ζ ) = cos t , we can also write σ ′ ( ζ ) = p − x µ ′ ( x ) , ζ ∈ T , x = Re( ζ ) . L. G. GONZ´ALEZ RICARDO AND G. L ´OPEZ LAGOMASINO
Let S ( σ, z ) = exp (cid:20) π Z T ζ + zζ − z ln σ ′ ( ζ ) | d ζ | (cid:21) , be the (usual) Szeg˝o function associated with the measure σ . Notice that if µ satisfies the Szeg˝ocondition on [ − ,
1] then R T ln σ ′ ( ζ ) | d ζ | > −∞ ; that is, σ verifies Szeg˝o’s condition on T .In general, when supp µ = ∆ = [ a, b ] (not necessarily [ − , σ as it was done beforeout of the measure e µ supported on [ − ,
1] such that e µ ( B ) = µ ( { x ∈ [ a, b ] : b − a ( x − b + a ) ∈ B } ),for every Borel set B ⊂ [ − , σ ′ ( e it ) = p ( b − x )( x − a ) µ ′ ( x ) , x = b − a t + b + a . We wish to define a Szeg˝o function with respect to the measure µ so that G ( µ, u ) = S ( σ, Ψ( u )) , u ∈ C \ ∆ . Then, from known properties of the Szeg˝o function for measures on the unit circle, we have(2.1) lim u → x | G ( µ, u ) | = lim u → x | S ( σ, Ψ( u ) | = 1 /σ ′ ( ζ ) = ( p ( b − x )( x − a ) µ ′ ( x )) − , a.e. on ∆ , where ζ = Ψ( x ) (Ψ can be extended continuous to ∆ as usual assuming that the interval has twosides and since σ is symmetric with respect to the real line we can take ζ either on the upper halfor the lower half of T ). Straightforward calculations show that the explicit expression of G ( µ, u )is (1.7).When h is a function on ∆ such that ln h is integrable with respect to d x/ p ( b − x )( x − a )we also write G ( h, u ) = exp " p ( u − b )( u − a )2 π Z ∆ ln h ( x ) x − u d x p ( b − x )( x − a ) , u ∈ C \ ∆ , Notice that in the definition of the Szeg˝o function with respect to a measure µ we take not theRadon-Nikodym derivative µ ′ of µ with respect to the Lebesgue measure d x but with respect tod x/ p ( b − x )( x − a ) which is precisely p ( b − x )( x − a ) µ ′ ( x ).2.2. A starting point.
Let x n,i , n − i n + 1 ≤ i ≤ n, denote the zeros of w n . If i n < n wedefine x n,i = ∞ , ≤ i ≤ n − i n . Set B n ( u ) := n Y i =1 Ψ( u ) − Ψ( x n,i )1 − Ψ( x n,i )Ψ( u ) . When x n,i = ∞ the corresponding factor in the Blaschke product is replaced by 1 / Ψ( u ).In [11, Theorem 4] a strong asymptotic result is given. We state it as a lemma for convenienceof the reader and further reference in the paper. Lemma 2.1.
Assume that ( µ n , w n ) n ≥ verifies i ) − iii ) . Then (2.2) lim n q n ( u ) w n ( u ) B n ( u ) = 12 π G ( µ, u ) , uniformly on compact subsets of Ω . TRONG ASYMPTOTIC OF POLYNOMIALS 7
We wish to point out that in [11, Theorem 4] there is a typo when writing the condition ii ). There, it appears in terms of the Lebesgue measure d x instead of the Chebyshev measured x/ p ( b − x )( x − a ). Except for that, the proof given is correct.When µ n = µ is fixed and w n ≡ B n ≡ / Ψ n ) we retrieve the standard resultfor strong asymptotic of orthogonal polynomials with respect to measures supported on ∆. Thedrawback of Lemma 2.1 is the appearance of the Blaschke product on the left hand side of (2.2),but nothing can be done to simplify the expression unless some restriction is imposed on theasymptotic behavior of the sequence of polynomials ( w n ) n ≥ .If d µ n = h n d µ , where µ is a fixed measure satisfying Szeg˝o’s condition on ∆, ( h n ) n ≥ is asequence of positive continuous functions such that lim n h n = h , and lim n →∞ | w n ( x ) | ϕ n ( x ) =1 /ψ ( x ) > √ π G ( ψhµ, u ).2.3. Proof of Theorem 1.2.
We begin with an auxiliary lemma.
Lemma 2.2.
Assume that the sequence of polynomials ( w n ) n ≥ verifies iv ) . Then (2.3) lim n →∞ C n Φ n ( u ) B n ( u ) w n ( u ) = G − ( ψ, u ) uniformly on compact subsets of C \ ∆ , where Φ and C are defined as in (1.6) .Proof. Notice that(2.4) C n Φ n ( u ) B n ( u ) w n ( u ) = (cid:18) C n Φ n ( u )Ψ n ( u ) (cid:19) (cid:18) Ψ n ( u ) B n ( u ) w n ( u ) (cid:19) and consider each factor in parenthesis on the right hand side separately.Define the function f n,i ( u ) := Ψ( u ) u − x n,i Ψ( u ) − Ψ( x n,i )1 − Ψ( x n,i )Ψ( u ) . It is easy to verify that this function is holomorphic and never vanishes in C \ ∆. Also, | f n,i | canbe extended continuously to ∆ with boundary values | f n,i ( x ) | = | x − x n,i | − , x ∈ ∆. Moreover, f n,i ( u ) = Ψ( u ) u − x n,i u − − Ψ( x n,i )Ψ − ( u )Ψ − ( u ) − Ψ( x n,i ) , thus f n,i ( ∞ ) = − Ψ ′ ( ∞ ) / Ψ( x n,i ). As | f n,i | is continuous and different from zero in C , it followsthat f n,i and f − n,i are in H ( C \ ∆) with respect to the Lebesgue measure on ∆; consequently, f n,i is an outer function (see [23, Chap. 17]). Then, f n,i ( u ) = c i exp " p ( u − a )( u − b ) π Z ∆ ln | x − x n,i | x − u d x p ( b − x )( x − a ) , (see (2.1)) where c i is a constant, | c i | = 1. Should w n be monic, an easy consequence of thisrepresentation is(2.5) (Ψ n B n )( u ) w n ( u ) = n Y i =1 f n,i ( u ) = exp " p ( u − a )( u − b ) π Z ∆ ln | w n ( x ) | x − u d η ( x ) , L. G. GONZ´ALEZ RICARDO AND G. L ´OPEZ LAGOMASINO where, for simplicity in the notation, hereafterd η ( x ) := d x p ( b − x )( x − a ) . (The product of all the constants c i gives 1.) If w n is not monic then the same representationholds due to the fact that for any positive constant κ exp " p ( u − a )( u − b ) π Z ∆ ln κx − u d η ( x ) = 1 κ . On the other hand, ( C Φ) / Ψ is analytic and different from zero in C \ ∆. Moreover, | C Φ( x ) / Ψ( x ) | = exp(2 γ − V λ ϕ ( x )), x ∈ ∆ , and using the equilibrium condition | C Φ( x ) / Ψ( x ) | =exp ( − ln ϕ ( x )) = 1 /ϕ ( x ), x ∈ ∆. Consequently, C Φ / Ψ is an outer function and we have(2.6) C n Φ n ( u )Ψ n ( u ) = exp p ( u − a )( u − b ) π Z ∆ n ln ϕ ( x ) x − u d η ( x ) ! . Putting together (2.4), (2.5), and (2.6), we have C n Φ n ( u ) B n ( u ) w n ( u ) = exp p ( u − a )( u − b ) π Z ∆ ln( | w n ( x ) | ϕ n ( x )) x − u d η ( x ) ! . To deduce (2.3) it remains to use (1.4) and the definition of G ( ψ, u ). (cid:3) Now, Theorem 1.2 is easy to derive. Indeed q n ( u ) C n Φ n ( u ) = q n ( u ) B n ( u ) w n ( u ) w n ( u ) C n Φ n ( u ) B n ( u ) . As n → ∞ , the limit of the first factor on the right is given by Lemma 2.1 and that of the secondone by Lemma 2.2. The proof of (1.8) has been concluded.Next, we deduce the asymptotic behavior of the monic orthogonal polynomials Q n and theleading coefficients τ n of q n . It is easy to see thatΦ( u ) = e − v λϕ ( u ) = u + O (1) , u → ∞ . Using (1.8) at u = ∞ , we obtain (1.9). Then, (1.10) follows directly from (1.8) and (1.9). (cid:3) Proof of Theorem 1.3.
We wish to express the orthogonality relations of the polynomials p n in such a way that we can apply Theorem 1.2.Notice that | Φ τ ( x ) | − = exp V τ ( x ). According to [26, Theorem 10.2] (see also [26, Lemma9.1]), there exists a sequence of polynomials ( H n − ) n ≥ , deg H n − ≤ n − , which do not vanishon ∆ whose zeros verify condition iii ) (see assertion on page 94 in [26]) such that(2.7) | H n − ( x ) / Φ nτ ( x ) | ≤ , x ∈ ( a, b ) , (2.8) lim n | H n − ( x ) / Φ nτ ( x ) | = 1 , TRONG ASYMPTOTIC OF POLYNOMIALS 9 uniformly on compact subsets of ( a, b ), and(2.9) lim n Z ba ln( | H n − ( x ) / Φ nτ ( x ) | ) p ( b − x )( x − a ) d x = 0 . Now, the orthogonality relations satisfied by the polynomials p n can be rewritten as Z p m ( x ) p n ( x ) | H n − ( x ) || Φ nτ ( x ) | d µ n ( x ) | H n − ( x ) | = ( , m < n, , m = n. Let us check that the sequence (cid:18) | H n − ( x ) | d µ n | Φ nτ ( x ) | , H n − ( x ) (cid:19) n ≥ verifies i ) − iv ). Indeed, the zeros of the polynomials H n − , and thus of the polynomials H n − , deg H n − ≤ n , verify condition iii ). On the other hand, (2.7), (2.8), and condi-tion i ) for the sequence of measures ( µ n ) n ≥ imply condition i ) for the sequence of measures( | H n − | d µ n / | Φ nτ | ) n ≥ andlim inf n Z ln (cid:18) | H n − ( x ) || Φ nτ ( x ) | µ ′ n ( x ) (cid:19) d x p ( b − x )( x − a ) ≥ Z ln µ ′ ( x ) d x p ( b − x )( x − a ) ;therefore ii ) takes place. Take w n = H n − and ϕ = e V τ . Using (2.9) we obtain iv ) with ψ ≡ V τ ( x ) − V τ ( x ) ≡ γ = 0; therefore C = 1. Applying Theorem 1.2 the thesis ofTheorem 1.3 readily follows. (cid:3) Applications to rational approximation.
Let µ be a positive measure with supp µ = ∆that satisfies Szeg˝o’s condition. (In this section we take h n ≡ , n ≥ µ ( z ) = Z ∆ d µ ( x ) z − x . Consider a sequence of polynomials ( w n ) n ≥ as above, positive on ∆.It is well known that for each n ≥
1, there exists a rational function R n = P n − Q n , deg P n − ≤ n − Q n ≤ n such that( Q n ˆ µ − P n − )( z ) w n ( z ) = A n z n +1 + · · · , z → ∞ where the function on the left hand side is analytic in C \ ∆. The fraction R n is called the n -thmulti-point Pad´e approximant of ˆ µ with respect to w n . It is well known and easy to prove that Q n is an n -th orthogonal polynomial with respect to the varying measure µ/w n and it can betaken to be monic. The remainder of ˆ µ − R n has the integral expression (see [16])(ˆ µ − R n )( z ) = w n ( z ) Q n ( z ) Z ∆ Q n ( x ) d µ ( x ) w n ( x )( z − x ) = w n ( z ) q n ( z ) Z ∆ q n ( x ) d µ ( x ) w n ( x )( z − x ) , where q n denotes the corresponding orthonormal polynomial. Taking into account [10, Theorem 8], we know thatlim n Z ∆ q n ( x ) d µ ( x ) | w n ( x ) | ( z − x ) = 1 p ( z − b )( z − a ) , uniformly on compact subsets of C \ ∆, where the square root is chosen to be positive when z > b . So, a direct consequence of Lemma 2.1 and Theorem 1.2 is the next result. Corollary 2.3.
Assume that i ) − iii ) take place where h n ≡ , n ≥ . We have lim n (ˆ µ − R n )( z ) B n ( z ) = 2 π G − ( µ, z ) p ( z − a )( z − b ) , If, additionally, iv ) holds and supp λ ϕ = ∆ , then lim n ( C Φ) n ( z )(ˆ µ − R n )( z ) w n ( z ) = 2 π G − ( ψµ, z ) p ( z − a )( z − b ) . The limits are uniform on compact subsets of Ω . Biorthogonal polynomials and multi level Hermite Pad´e polynomials
Let us briefly outline its contents. In subsection 3.1, we establish a connection betweenbiorthogonal polynomials and an associated Hermite-Pad´e approximation problem. In subsec-tions 3.2 and 3.3 we prove some useful formulas verified by these approximants and their associ-ated polynomials. In subsection 3.4 we introduce the functions Φ , Φ which are used to comparethe asymptotic behavior of the Hermite-Pad´e polynomials. These functions are expressed interms of the solution of a vector valued equilibrium problem. An equivalent expression is givenin subsection 3.11 using the branches of a conformal map defined on a Riemann surface of genuszero. The implementation, in our setting, of Aptekarev’s method is carried out in subsections3.5-3.8 leading to the proof of Theorem 3.8 from which Theorem 1.1 is derived in subsection 3.9.In subsection 3.10 some corollaries of Theorem 3.8 are obtained.Our method differs from Aptekarev’s in two aspects. Proposition 3.3, which plays a key role, isderived using arguments from complex function theory. The corresponding result in [2, Theorem2] uses a quite intricate approximative construction on a Riemann surface. Secondly, in [2],Widom’s approach introduced in [27] is followed closely to obtain L estimates, on segments ofthe real line, of the asymptotic behavior of the multiple orthogonal polynomials. Thus, the resultsare obtained for measures in the Szeg˝o class which are absolutely continuous with respect to theLebesgue measure. We use instead the results obtained in Section 2 on orthogonal polynomialswith respect to varying measures and do not need to restrict to absolutely continuous measures.In consequence, we only give the asymptotic in the complement of the intervals.3.1. Multilevel HP polynomials.
Let ∆ , ∆ be non-intersecting closed intervals of the realline. Let ( σ , σ ) ∈ M ( ∆ ) where ∆ = (∆ , ∆ ). Using the differential notation, we define athird measure s , by d s , ( x ) := b σ ( x ) d σ ( x ) , b σ ( x ) = Z d σ ( t ) x − t . TRONG ASYMPTOTIC OF POLYNOMIALS 11
Inverting the role of the measures we define similarly s , , d s , ( x ) = b σ ( x ) d σ ( x ). The pair N ( σ , σ ) := ( s , , s , ), where s , = σ , is called the Nikishin system generated by ( σ , σ ).Notice that the order in which the measures are taken is important and N ( σ , σ ) = N ( σ , σ ).General Nikishin systems of m ≥ m ≥ n ∈ N , there exists a vector polynomial ( a n, , a n, , a n, ) , not identically equal to zero,with deg a n, ≤ n −
1, deg a n, ≤ n − , and deg a n, ≤ n , that satisfies A n, ( z ) := ( a n, − a n, b s , + a n, b s , ) ( z ) = O (1 /z n +1 ) , (3.1) A n, ( z ) := ( − a n, + a n, b s , ) ( z ) = O (1 /z ) . (3.2)Here and below, the symbol O ( ∗ ) is taken as z → ∞ . By extension we take A n, ≡ a n, . Thepolynomials a n, , a n, , a n, are called multilevel Hermite-Pad´e polynomials.It can be shown that deg a n, = n and the vector polynomial can be normalized taking a n, monic. With this normalization ( a n, , a n, , a n, ) is unique. Moreover, all the zeros of a n, aresimple and lie in the interior ◦ ∆ (with the Euclidean topology of R ) of the interval ∆ . For moredetails, see [20, Theorem 1.4] and Lemma 3.1 below.Combining Cauchy’s theorem, Fubini’s theorem, and Cauchy’s integral formula, from (3.1) itfollows that Z x ν A n, ( x ) d σ ( x ) = 0 , ν = 0 , . . . , n − , and from (3.2) we get the integral representation A n, ( x ) = Z a n, ( y ) d σ ( y ) x − y . Therefore,
Z Z x ν a n, ( y ) x − y d σ ( x ) d σ ( y ) = 0 , ν = 0 , . . . , n − . Consequently a n, , normalized to be monic, verifies the same orthogonality relations as thebiorthogonal polynomial Q n (see (1.1)) and coincides with it.Analogously, for each n ∈ N , there exists a vector polynomial ( b n, , b n, , b n, ) , not identicallyequal to zero, with deg b n, ≤ n −
1, deg b n, ≤ n − , and deg b n, ≤ n , that satisfies B n, ( z ) := ( b n, − b n, b s , + a n, b s , ) ( z ) = O (1 /z n +1 ) , (3.3) B n, ( z ) := ( − b n, + b n, b s , ) ( z ) = O (1 /z ) . (3.4)By extension we take B n, ≡ b n, . Normalizing b n, to be monic, we have b n, = P n (the otherbirthogonal polynomial in (1.1). Therefore, in order to prove Theorem 1.1, we need to find the strong asymptotic of thesequences of polynomials ( a n, ) n ≥ and ( b n, ) n ≥ . Because of the symmetry of the problem, itsuffices to analyze the first sequence and the results for the second one are immediate.Indeed, we will give the strong asymptotic of the forms A n, , A n, and the polynomials a n, , a n, , a n, , as n → ∞ , under the assumption that the generating measures σ , σ are inthe Szeg˝o class; that is ( σ , σ ) ∈ S ( ∆ ) (see (1.2)). For general Nikishin systems of m ≥ Some useful properties.
The forms A n,k , k = 0 , , , are interlinked and satisfy interestingorthogonality relations which will be of great use. The following result, is a special case ( m = 2)of [15, Lemma 2,4]. It is stated here for convenience of the reader. Lemma 3.1.
Consider the Nikishin system N ( σ , σ ) . For each fixed n ∈ Z + and j = 1 , , A n,j has exactly n zeros in C \ ∆ j +1 they are all simple and lie in ◦ ∆ j (∆ = ∅ ) . A n, has no zeroin C \ ∆ . Let Q n,j , j = 1 , , denote the monic polynomial of degree n whose zeros are those of A n,j in ∆ j . For j = 0 , , (3.5) A n,j ( z ) Q n,j ( z ) = Z A n,j +1 ( x ) z − x d σ j +1 ( x ) Q n,j ( x ) , where Q n, ≡ , and (3.6) Z x ν A n,j +1 ( x ) d σ j +1 ( x ) Q n,j ( x ) = 0 , ν = 0 , . . . , n − . From (3.6) with j = 1 we get (recall that A n, = a n, = Q n, )(3.7) 0 = Z x ν Q n, ( x ) d σ ( x ) Q n, ( x ) , ν = 0 , . . . , n − . On the other hand, from (3.6) with j = 0, it follows that(3.8) 0 = Z x ν Q n, ( x ) H n, ( x ) d σ ( x ) Q n, ( x ) , ν = 0 , . . . , n − . where, using (3.5) with j = 0 and (3.7)(3.9) H n, ( z ) := Q n, ( z ) A n, ( z ) Q n, ( z ) = Q n, ( z ) Z Q n, ( x ) z − x d σ ( x ) Q n, ( x ) = Z Q n, ( x ) z − x d σ ( x ) Q n, ( x ) . Proposition 3.2.
There is a unique pair of monic polynomials with real coefficients ( Q n, , Q n, ) each one of degree n , whose zeros lie in C \ ∆ and C \ ∆ , respectively, satisfying (3.7) - (3.8) with H n, ( z ) = Z Q n, ( x ) z − x d σ ( x ) Q n, ( x ) . Proof.
The existence of such polynomials is guaranteed from Lemma 3.1. We must show thatif ( Q n, , Q n, ) is a pair of monic polynomials of degree n which satisfy (3.7)-(3.8) with H n, asindicated then we can construct forms A n, , A n, , A n, verifying (3.1)-(3.2) whose zeros are thoseof the polynomials Q n, , Q n, . TRONG ASYMPTOTIC OF POLYNOMIALS 13
So, let ( Q n, , Q n, ) be an arbitrary pair of monic polynomials of degree n which satisfy (3.7)-(3.8). Take A n, = a n, := Q n, and a n, ( z ) := Z Q n, ( z ) − Q n, ( x ) z − x d σ ( x ) . Obviously, a n, is a polynomial of degree ≤ n −
1. Rearranging this equality and using (3.7), weget A n, ( z ) := ( − a n, + a n, b s , )( z ) = Z ( Q n. Q n, )( x ) z − x d σ ( x ) Q n, ( x ) = Q n, ( z ) Z Q n, ( x ) z − x d σ ( x ) Q n, ( x ) . The first equality tells us that A n, ( z ) = O (1 /z ), so that (3.2) takes place, and the last equalityimplies that the zeros of A n, in C \ ∆ coincide with the simple roots that Q n, has in the interiorof ∆ . Moreover, these relations together with (3.7)-(3.8) imply that for each ν = 0 , , . . . , n − Z x ν A n, ( x ) d σ ( x ) = Z x ν Q n, ( x ) Z Q n, ( t ) x − t d σ ( t ) Q n, ( t ) d σ ( x ) = Z x ν Q n, ( x ) Z Q n, ( t ) x − t d σ ( t ) Q n, ( t ) d σ ( x ) Q n, ( x ) = Z x ν Q n, ( x ) H n, ( x ) d σ ( x ) Q n, ( x ) = 0 . These orthogonality relations verified by A n, in turn imply that(3.10) Z A n, ( x ) z − x d σ ( x ) = 1 z n Z x n A n, ( x ) z − x d σ ( x ) = O (1 /z n +1 )Using the definition of A n, ( x ), we get a n, ( z ) := a n, ( z ) b s , ( z ) − a n, ( z ) b s , ( z ) + Z A n. ( x ) z − x d σ ( x ) = Z a n, ( z ) − a n, ( x ) z − x d σ ( x ) − Z a n, ( z ) − a n, ( x ) z − x d s , ( x ) , which is obviously a polynomial of degree ≤ n −
1. Rearranging this equality and taking accountof (3.10), it follows that A n, ( z ) := a n, ( z ) − a n, ( z ) b s , ( z ) + a n, ( z ) b s , ( z ) = Z A n. ( x ) z − x d σ ( x ) = O (1 /z n +1 ) . Thus, A n, verifies (3.1).From our findings, we deduce that the vector polynomials ( a n, , a n, , a n, ) defined previouslyis the unique solution of (3.1)-(3.2). In particular, a n, = Q n, is uniquely determined and by(3.8) so is Q n, since the measure ( H n, d σ ) /Q n, has constant sign on ∆ . We are done. (cid:3) A normalization.
Set(3.11) κ − n, := Z Q n, ( x ) d σ ( x ) | Q n, ( x ) | , ( κ n, κ n, ) − := Z Q n, ( x ) |H n, ( x ) | d σ ( x ) | Q n, ( x ) | . Take(3.12) q n, := κ n, Q n, , q n, := κ n, Q n, , h n, := κ n, H n, . Notice that κ − n, := Z Q n, ( x ) | h n, ( x ) | d σ ( x ) | Q n, ( x ) | . We can rewrite (3.7)-(3.8) as(3.13) 0 = Z x ν q n, ( x ) d σ ( x ) | Q n, ( x ) | = 0 , ν = 0 , . . . , n − , and(3.14) 0 = Z x ν q n, ( x ) | h n, ( x ) | d σ ( x ) | Q n, ( x | ) , ν = 0 , . . . , n − . We also have(3.15) Z q n, ( x ) d σ ( x ) | Q n, ( x ) | = 1 , and(3.16) Z q n, ( x ) | h n, ( x ) | d σ ( x ) | Q n, ( x ) | = 1 . Consequently, q n, and q n, are the n -th orthonormal polynomials with respect to the varyingmeasures | h n, | d σ | Q n, | and d σ | Q n, | , respectively. Recall that the zeros of Q n,j lie in ◦ ∆ j = ( a j , b j ) , j =1 , σ ′ > , then for any bounded measurablefunctions g on ∆ , (3.17) lim n Z g ( x ) q n, ( x )d σ ( x ) | Q n, ( x ) | = 1 π Z b a g ( x ) d x p ( b − x )( x − a ) . Taking into account (3.9) and using (3.17) with g ( x ) = | t − x | − , t ∈ ∆ , and (3.9), it followsthat lim n | h n, ( t ) | = lim n Z q n, ( x )d σ ( x ) | t − x || Q n, ( x ) | =(3.18) = 1 π Z b a d x | t − x | p ( b − x )( x − a ) = 1 p | t − a || t − b | =: h ( t ) , uniformly for t ∈ ∆ . Then (3.16), (3.18), and [10, Theorem 8] imply that if σ ′ > ,then for any bounded Borel measurable function g on ∆ we have(3.19) lim n Z g ( x ) q n, ( x ) | h n, ( x ) | d σ ( x ) | Q n, ( x ) | = 1 π Z b a g ( x ) d x p ( b − x )( x − a ) | x − b || x − a | . The comparison functions.
The logarithmic asymptotic of general ML Hermite-Pad´epolynomials was studied in [15, Section 3]. In particular, it was proved that this asymptoticbehavior can be described in terms of the solution of a vector equilibrium problem which, in thecase we are dealing with, reduces to finding a pair of probability measures ( λ , λ ) and a pair ofconstants ( γ , γ ) such that(3.20) ( V λ ( x ) − V λ ( x ) ≡ γ , x ∈ ∆ V λ ( x ) − V λ ( x ) ≡ γ , x ∈ ∆ . TRONG ASYMPTOTIC OF POLYNOMIALS 15
It is well known that this problem has a unique solution. From [15, Theorem 3.4] it follows thatif the measures σ , σ are regular then for k = 1 , n | Q n,k | /n = exp( − V λ k ) , lim n κ /nn,k = γ k , where the first limit is uniform on compact subsets of C \ ∆ k .Since strong asymptotic implies weak asymptotic, (3.21) reveals that the functions with whichone must compare the polynomials q n, , q n, in order to have strong asymptotic (should it exist)are tightly connected with the potentials V λ , V λ and the constants γ , γ . With this in mind(see (1.6)), we define(3.22) Φ k ( z ) := e − v k ( z ) , v k := V λ k + i e V λ k , C k := e γ k , k = 1 , , where e V λ k denotes the harmonic conjugate of V λ k in C \ ∆ k .For a different expression of the comparison functions see (3.63).In (3.7)-(3.8) we see that the orthogonality relations verified by the polynomials Q n, , Q n, areinterconnected. This prevents the direct use of Theorem 1.2 to obtain their asymptotic becauseto give the asymptotic of one of the sequences one must know that of the second, and viceversa.So, we will follow an indirect approach implemented by A.I. Aptekarev in [1] and [2] for thestudy of the strong asymptotic of type ii Hermite-Pad´e polynomials.3.5.
A prescribed asymptotic behavior.
An important ingredient of the method consists inbeing capable of producing a sequence of functions of the form P n,k / Φ nk , k = 1 ,
2, where P n,k isa polynomial of degree n , whose limit is a predetermined Szeg˝o function.Let ( λ , λ ) be the solution of the vector equilibrium problem (3.20). From [12, Theorem1.34] it follows that d λ k = v k d x, d λ = v d x on ∆ k , k = 1 , v , v verify theassumptions relative to v in Corollary 1.3 on the intervals ∆ , ∆ , respectively. In the sequelΩ k := C \ ∆ k , k = 1 , . Proposition 3.3.
Assume that ( µ , µ ) ∈ S ( ∆ ) and for each n ≥ , (˜ q n, , ˜ q n, ) is the pair ofpolynomials of degree n such that (3.23) Z ˜ q n,k ( x )˜ q m,k ( x ) d µ k ( x ) | Φ nk ( x ) | = ( , ≤ m < n, , m = n, k = 1 , . Then (3.24) lim n →∞ ˜ q n,k ( z )Φ nk ( z ) = G ( µ k , z ) √ π , and (3.25) lim n →∞ ˜ Q n,k ( z )Φ nk ( z ) = G ( µ k , z ) G ( µ k , ∞ ) , uniformly on each compact subset of Ω k , k = 1 , , where the Φ k were introduced in (3.22) and ˜ Q n,k , k = 1 , is ˜ q n,k renormalized to be monic. Proof.
As was mentioned above, [12, Theorem 1.34] guarantees that the components of theequilibrium measures ( λ , λ ) are absolutely continuous with respect to the Lebesgue measure onthe corresponding intervals and their weights v , v verify (1.11) with parameters β = β = − / , ∆ , respectively. The remaining assumptions of Corollary 1.3 are easilyverified. Then, (3.24) follows directly from (1.12). If ˜ κ n,k is the leading coefficient of ˜ q n,k ,applying (3.24) at ∞ we get lim n ˜ κ n,k = G ( µ k , ∞ ) √ π and (3.25) follows at once. (cid:3) The operator ˜ T n . Let ( σ , σ ) ∈ M ( ∆ ). Let (˜ h n ) n ≥ be a sequence of positive continuousfunctions on ∆ such that(3.26) lim n ˜ h n = ˜ h > , uniformly on ∆ . Let P n,k , k = 1 , , be the set of all monic polynomials with real coefficients ofdegree n whose zeros lie in C \ ∆ when k = 1 and in C \ ∆ when k = 2. Define an operator˜ T n : P n, × P n, −→ P n, × P n, where, for every ( ˆ Q n, , ˆ Q n, ) ∈ P n, × P n, (3.27) ˜ T n ( ˆ Q n, , ˆ Q n, ) := ( Q ∗ n, , Q ∗ n, ) , with ( Q ∗ n, , Q ∗ n, ) the unique pair of monic polynomials of degree n which satisfies0 = Z x ν Q ∗ n, ( x ) d σ ( x ) | ˆ Q n, ( x ) | , ν = 0 , . . . , n − , and 0 = Z x ν Q ∗ n, ( x )˜ h n ( x ) d σ ( x ) | ˆ Q n, ( x ) | , ν = 0 , . . . , n − . Set(3.28) ( κ ∗ n, ) − := Z ( Q ∗ n, ( x )) ( x ) d σ ( x ) | ˆ Q n, ( x ) | , ( κ ∗ n, ) − := Z ( Q ∗ n, ( x )) ( x ) ˜ h n ( x )d σ ( x ) | ˆ Q n, ( x ) | . It is easy to verify that ˜ T n is continuous on P n, × P n, .From Proposition 3.2 it follows that if we take ˜ h n = | h n, | then˜ T n ( Q n, , Q n, ) = ( Q n, , Q n, ) , where ( Q n, , Q n, ) is the unique pair of polynomials of degree n verifying (3.13)-(3.16). Therefore,in this case ( Q n, , Q n, ) is a fixed point of the operator ˜ T n . In the case of arbitrary ˜ h n it is notdifficult to prove that ˜ T n also has fixed points. (In general, it may not be unique.)Indeed, given n if ( ˜ Q n, , ˜ Q n, ) is a fixed point then the n zeros of ˜ Q n, must lie in ∆ and the n zeros of ˜ Q n, must be in ∆ . Consequently, it is sufficient to restrict the operator ˜ T n to the TRONG ASYMPTOTIC OF POLYNOMIALS 17 class ˜ P n, × ˜ P n, of all pairs of monic polynomials whose first component has all its zeros on ∆ and the second has its zeros on ∆ . Suppose that˜ Q n, ( x ) = n Y j =1 ( x − x n,j ) , ˜ Q n, ( x ) = n Y j =1 ( x − y n,j ) . Assume that the zeros and indexed in such a way that a ≤ x n, ≤ · · · ≤ x n,n ≤ b , a ≤ y n, ≤ · · · ≤ y n,n ≤ b . There is a canonical homeomorphism between ˜ P n, × ˜ P n, and ˜∆ × ˜∆ , where ˜∆ k , k = 1 , , isthe subset of ∆ nk made up of all points whose coordinates are increasing, given by( ˜ Q n, , ˜ Q n, ) −→ (( x n, , . . . , x n,n ) , ( y n, , . . . , y n,n ))The operator ˜ T n induces an operator from ˜∆ × ˜∆ into itself, where the image is determinedby the zeros of ( Q ∗ n, , Q ∗ n, ) = ˜ T n ( Q ∗ n, , ˜ Q n, ). The induced operator is continuous and ˜∆ × ˜∆ is a convex compact subset of R n × R n ; therefore, by Brouwer’s fixed point theorem the inducedoperator has at least one fixed point. Consequently, so does ˜ T n .We are ready to use Theorem 1.2. Proposition 3.4.
Assume that ( µ , µ ) ∈ S ( ∆ ) and for each n ≥ , ( ˜ Q n, , ˜ Q n, ) is the pair ofmonic polynomials of degree n which satisfies (3.25) . Let ( σ , σ ) ∈ S ( ∆ ) and let ( Q ∗ n, , Q ∗ n, ) =˜ T n ( ˜ Q n, , ˜ Q n, ) where (˜ h n ) n ≥ fulfills (3.26) . Then (3.29) lim n q ∗ n, ( z ) C n Φ n ( z ) = 1 √ π G ( f − ˜ hσ , z ) , lim n q ∗ n, ( z ) C n Φ n ( z ) = 1 √ π G ( f − σ , z ) , uniformly on compact subsets of Ω and Ω , respectively, f k = G ( µ k , · ) / G ( µ k , ∞ ) , k = 1 , , and q ∗ n,k = κ ∗ n,k Q ∗ n,k is the corresponding orthonormal polynomial of degree n (see (3.28) ). Addition-ally, (3.30) lim n κ ∗ n, C n = 1 √ π G ( f − ˜ hσ , ∞ ) , lim n κ ∗ n, C n = 1 √ π G ( f − σ , ∞ ) , k = 1 , . Consequently, (3.31) lim n Q ∗ n, ( z )Φ n ( z ) = G ( f − ˜ hσ , z ) G ( f − ˜ hσ , ∞ ) , lim n Q ∗ n, ( z )Φ n ( z ) = G ( f − σ , z ) G ( f − σ , ∞ ) . Proof.
It is easy to see that the sequences (˜ h n d σ , ˜ Q n, )) n ≥ , (d σ , ˜ Q n, )) n ≥ verify i ) − iv ) onthe intervals ∆ and ∆ . Therefore, the assumptions of Theorem 1.2 are fulfilled. Consequently,(3.29)-(3.31) follow directly from Proposition 3.3 and Theorem 1.2 taking into account the equi-librium equations (3.20) verified by the equilibrium measures and the defining formulas (3.22)for the functions Φ k and constants C k , k = 1 , (cid:3) Formulas (3.29)-(3.31) describe the strong asymptotic behavior of the components of the imageof ˜ T n . In the next section, we give an operator approach to interpret the limiting functionsappearing in these relations. The operator T . The Szeg˝o functions which describe the limits (3.29)-(3.31) verify theboundary equations (see (2.1))(3.32) | G ( f − ˜ hσ , x ) | = | f ( x ) | p ( b − x )( x − a )(˜ hσ ′ )( x ) , a.e. on [ a , b ] = ∆ , and(3.33) | G ( f − σ , x ) | = | f ( x ) | p ( b − x )( x − a ) σ ′ ( x ) , a.e. on [ a , b ] = ∆ . The functions f , f themselves are expressed in terms of Szeg˝o functions and ˜ h is a positivecontinuous function on ∆ . The Szeg˝o functions above are symmetric with respect to the realline, never equal zero, and are positive at infinity. Consequently, on the real line, outside of theintervals supporting their defining measures, they are positive. Relations (3.32)-(3.33) suggestthe definition of an operator.Let ∆ = (∆ , ∆ ). We denote by C ∆ the space of all pairs g = ( g , g ) of real valued functionssuch that g is continuous on ∆ and g is continuous on ∆ . Set k g k C ∆ := max {k g k ∆ , k g k ∆ } , where k · k X denotes the sup norm on X . Obviously ( C ∆ , k · k C ∆ ) is a Banach space. Considerthe cone C + ∆ of all the vectors in C ∆ such that g is positive on ∆ and g is positive on ∆ . Theapplication ( g , g ) (ln g , ln g ) establishes a homeomorphism between C + ∆ and C ∆ . Given g (1) = ( g (1)1 , g (1)2 ) , g (2) = ( g (2)1 , g (2)2 ) ∈ C + ∆ , set d ( g (1) , g (2) ) := max {k ln( g (1)1 /g (2)1 ) k ∆ , k ln( g (1)2 /g (2)2 ) k ∆ } . It is easy to check that ( C + ∆ , d ) is a complete metric space. Certainly, on C + ∆ we can alsoconsider the norm k · k C ∆ but C + ∆ is not complete with that norm; however, given a sequence( g ( n ) ) n ≥ ⊂ C + ∆ and g ∈ C + ∆ , we have(3.34) lim n k g ( n ) − g k C ∆ = 0 ⇔ lim n d ( g ( n ) , g ) = 0 . Define T : C + ∆ −→ C + ∆ , where T ( g , g ) = ( g ∗ , g ∗ ) is the pair of Szeg˝o functions, g ∗ k ∈ H (Ω k ) , k = 1 , , verifying | g ∗ ( x ) | = g ( x ) p ( b − x )( x − a )(˜ hσ ′ )( x ) , a.e. on [ a , b ] = ∆ , and | g ∗ ( x ) | = g ( x ) p ( b − x )( x − a ) σ ′ ( x ) , a.e. on [ a , b ] = ∆ . (From the definition of the Szeg˝o function it readily follows that g ∗ is positive and continuouson R \ ∆ ⊃ ∆ and g ∗ is positive and continuous on R \ ∆ ⊃ ∆ .) Here ˜ h, σ , σ are the sameas in (3.32)-(3.33). Set w ( x ) = p ( b − x )( x − a )(˜ hσ ′ )( x ) , w ( x ) = p ( b − x )( x − a ) σ ′ ( x ) . TRONG ASYMPTOTIC OF POLYNOMIALS 19
Finding g ∗ k , k = 1 , , reduces to solving the Dirichlet problems for a harmonic function u k inΩ k , with boundary values integrable on ∆ k and equal to ln( g /w ) a.e. on ∆ in the case of u and ln( g /w ) a.e. on ∆ in the case of u , and the subsequent problem of finding theirharmonic conjugates e u k , e u k ( ∞ ) = 0 , in Ω k . Then g ∗ k = exp( u k + i e u k ) ., k = 1 , . Set Ω = (Ω , Ω ). Let h Ω be the set of pairs of harmonic functions in Ω and Ω , respectively,with integrable boundary values. Given g = ( g , g ) ∈ C + ∆ let χ = ( χ , χ ) = (ln g , ln g )) ∈ C ∆ . The map T induces the map t : C ∆ −→ h Ω ⊂ C ∆ , where t ( χ ) := 12 ( P ( χ ) + β ) , χ = ( χ , χ ) t ∈ C ∆ ,β = ( β , β ) t is the (column) vector made up of harmonic functions with boundary values β k ( x ) = − ln w k ( x ) , a.e. on ∆ k , k = 1 , P is the linear operator P := P , P , ! , such that P , ( χ ) is the harmonic function on Ω with boundary values equal to χ on ∆ , and P , ( χ ) is the harmonic function on Ω with boundary values equal to χ on ∆ .The following result is contained in [2, Proposition 1.1]. Proposition 3.5.
The map T is a contraction in C + ∆ with respect to the metric d . More precisely d ( T ( g (1) ) , T ( g (2) )) ≤ d ( g (1) , g (2) ) , g (1) , g (2) ∈ C + ∆ . Therefore, the map T has a unique fixed point in C + ∆ .Proof. The proof is simple so, for completeness, we include it. As mentioned above, ( C + ∆ , d ) isa complete metric space so the second statement follows from the first.Set χ ( k ) := (ln g ( k )1 , ln g ( k )2 ) , k = 1 ,
2. From the definitions of d, T, and t it follows that d ( T ( g (1) ) , T ( g (2) )) = k t ( χ (1) ) − t ( χ (2) ) k C ∆ = 12 k P ( χ (1) ) − P ( χ (2) ) k C ∆ ≤ k P kk χ (1) − χ (2) k C ∆ = 12 d ( g (1) , g (2) ) , where in the last inequality the maximum principle is used to establish that k P k = 1. (cid:3) Let G = ( G , G ) be the unique fixed point of the operator T . That is T ( G ) = G and the components of G are characterized by the system of boundary values | G ( x ) | = G ( x ) p ( b − x )( x − a )(˜ hσ ′ )( x ) , a.e. on [ a , b ] = ∆ , and | G ( x ) | = G ( x ) p ( b − x )( x − a ) σ ′ ( x ) , a.e. on [ a , b ] = ∆ . Obviously, the components of G are Szeg˝o functions in Ω and Ω , respectively.The method developed by Aptekarev to study the strong asymptotic of type ii Hermite-Pad´e polynomials consists in showing that any neighborhood of a fixed point of the operator T determines fixed points of the operators ˜ T n for all sufficiently large n . By Proposition 3.2,when we take ˜ h n = | h n, | as in (3.12), the operator ˜ T n has only one fixed point. To completeAptekarev’s approach, we need one last ingredient. Proposition 3.6.
Let ( ˜ Q n, , ˜ Q n, ) n ≥ be an arbitrary sequence of vector polynomials such that ( ˜ Q n, , ˜ Q n, ) ∈ P n, × P n, . Set f ( n ) = ˜ Q n, Φ n , ˜ Q n, Φ n ! . Assume that there exists f = ( f , f ) ∈ C + ∆ and a sequence of non-negative integers Λ such that (3.35) lim n ∈ Λ k f ( n ) − f k C ∆ = 0 . Let ( σ , σ ) ∈ S ( ∆ ) and let ( Q ∗ n, , Q ∗ n, ) = ˜ T n ( ˜ Q n, , ˜ Q n, ) where (˜ h n ) n ≥ fulfills (3.26) . Then (3.36) lim n ∈ Λ q ∗ n, ( z ) C n Φ n ( z ) = 1 √ π G ( f − ˜ hσ , z ) , lim n ∈ Λ q ∗ n, ( z ) C n Φ n ( z ) = 1 √ π G ( f − σ , z ) , uniformly on compact subsets of Ω and Ω , respectively. Additionally, (3.37) lim n ∈ Λ κ ∗ n, C n = 1 √ π G ( f − ˜ hσ , ∞ ) , lim n ∈ Λ κ ∗ n, C n = 1 √ π G ( f − σ , ∞ ) , k = 1 , . Consequently, (3.38) lim n ∈ Λ Q ∗ n, ( z )Φ n ( z ) = G ( f − ˜ hσ , z ) G ( f − ˜ hσ , ∞ ) , lim n ∈ Λ Q ∗ n, ( z )Φ n ( z ) = G ( f − σ , z ) G ( f − σ , ∞ ) . Proof.
The proof is identical to that of Proposition 3.4. In that proof, it is not used that the fullsequences of indices is considered and the result only depends on the asymptotic behavior of thesequences of denominators of the varying part of the measures of orthogonality on the intervals∆ and ∆ , respectively, for which assumption (3.35) was included. The details are left to thereader. (cid:3) Proof of Theorem 3.7.
Let G = ( G , G ) be the fixed point of the operator T . Thefunction G k , k = 1 , k ; therefore, the value G k ( ∞ ) is well defined. Set H + ,n := (cid:26)(cid:18) G ( ∞ ) P n, Φ n , G ( ∞ ) P n, Φ n (cid:19) : P n,k ∈ P n,k , k = 1 , (cid:27) . (Recall that P n,k is the set of all monic polynomials of degree n with real coefficients whose zeroslie in Ω j , j = k, j, k = 1 , TRONG ASYMPTOTIC OF POLYNOMIALS 21
Let ˜ T n, and ˜ T n, be the operators defined on P n, × P n, which determine the components of˜ T n ; that is, ˜ T n = ( ˜ T n, , ˜ T n, ) (see (3.27)). Define T n : H + ,n −→ H + ,n where T n (cid:18) G ( ∞ ) P n, Φ n , G ( ∞ ) P n, Φ n (cid:19) = G ( ∞ ) ˜ T n, ( P n, , P n, )Φ n , G ( ∞ ) ˜ T n, ( P n, , P n, )Φ n ! . Notice that any fixed point of T n generates a fixed point of ˜ T n . The continuity of ˜ T n implies thecontinuity of T n . Theorem 3.7.
Let ( σ , σ ) ∈ S ( ∆ ) and (˜ h n ) n ≥ fulfills (3.26) . Then, there exists a sequence ( Q n, , Q n, ) n ≥ , where ( Q n, , Q n, ) is a fixed point of ˜ T n , such that (3.39) lim n Q n,k ( z )Φ nk ( z ) = G k ( z ) G k ( ∞ ) , k = 1 , , uniformly on compact subsets of Ω k . Additionally, if ( κ n, ) − := Z ( Q n, ( x )) ( x ) d σ ( x ) | Q n, ( x ) | , ( κ n, ) − := Z ( Q n, ( x )) ( x ) ˜ h n ( x )d σ ( x ) | Q n, ( x ) | , and q n,k = κ n,k Q n,k , k = 1 , , then (3.40) lim n κ n,k C nk = 1 √ π G k ( ∞ ) , and (3.41) lim n q n,k ( z ) C nk Φ nk ( z ) = 1 √ π G k ( z ) , k = 1 , , uniformly on compact subsets of Ω k .Proof. Due to the way that in which H + ,n , T n and T were defined, the statements (3.39)-(3.41)follow directly from Proposition 3.6 if we show that there exists a sequence ( g ( n ) ) n ≥ n , where g ( n ) is a fixed point of T n , such that(3.42) lim n ≥ n k g ( n ) − G k C ∆ = 0 . The components of G are Szeg˝o functions in Ω k , k = 1 ,
2, respectively. Let K = ( K , K ) bea pair of two non intersecting closed disks, symmetric with respect to R , whose interior in theEuclidean topology of C verify ∆ ⊂ ◦ K , ∆ ⊂ ◦ K . By H + ( K ) we denote the cone of all pairs ( g , g ) of functions such that g k is holomorphic anddifferent from zero in ◦ K k , and positive on ◦ K k ∩ R . For g = ( g , g ) ∈ H + ( K ) we define k g k K := max {{ sup | g k ( z ) | : z ∈ ◦ K k } : k = 1 , } ( ≤ ∞ ) , and min ∆ g := min { min x ∈ ∆ g ( x ) , min x ∈ ∆ g ( x ) } . Fix a constant
C >
0. Define H + ( K , C ) := { g ∈ H + ( K ) : k g k K ≤ C, min ∆ g ≥ C − } . Take C sufficiently large so that G ∈ H + ( K , C ).Let g (1) , g (2) ∈ H + ( K , C ) and let 0 ≤ β ≤
1, then k β g (1) + (1 − β ) g (2) k K ≤ β k g (1) k K + (1 − β ) k g (2) k K ≤ C, and min ∆ (cid:16) β g (1) + (1 − β ) g (2) (cid:17) ≥ β min ∆ g (1) + (1 − β ) min ∆ g (2) ≥ C − , Therefore, β g (1) + (1 − β ) g (2) ∈ H + ( K , C ). This shows that H + ( K , C ) is convex.On the other hand, if ( g ( n ) ) n ≥ is an arbitrary sequence of elements in H + ( K , C ). Then thecomponents form normal families in ◦ K and ◦ K , respectively. Therefore, there exists a sequenceof indices Λ such that ( g ( n ) ) n ∈ Λ converges componentwise to some vector function g uniformlyon each compact subset of ◦ K and ◦ K , respectively. The components of g are therefore analyticon ◦ K and ◦ K , respectively. The uniform limit of holomorphic functions which never equal zeromust either be identically equal to zero or never zero. The second case is not possible becausemin ∆ g = lim n ∈ Λ min ∆ g ( n ) ≥ C − . Also k g k K = lim n ∈ Λ k g ( n ) k K ≤ C. Consequently, g ∈ H + ( K , C ). We conclude that H + ( K , C ) is compact.Fix an arbitrary θ >
0. Let ω ( θ ) = { g ∈ H + ( K , C ) : k g − G k C ∆ ≤ θ } . This is a closed subset of H + ( K , C ) and, therefore, it is compact. Obviously, it is convex.Analogously, for every ε >
0, set ω ε := { g ∈ H + ( K , C ) : d ( g , G ) ≤ ε } There exists ε such that ω ε ⊂ ω ( θ ) , < ε ≤ ε , for otherwise we could find a sequence of vector functions in H + ( K , C ) ⊂ C + ∆ which convergesto G in the d metric but not in the k · k C ∆ norm which would contradict (3.34).Take ω ε,n = ω ε ∩ H + ,n For each fixed n the set ω ε,n is a closed, bounded subset of a finite dimensional space, thereforeit is compact.Let µ k be the representating measure of G k so that G k ( z ) = G ( µ k , z ) , z ∈ Ω k . From Proposition3.3, using (3.24) and (3.34) it follows thatlim n d ( g ( n ) , G ) = 0 , TRONG ASYMPTOTIC OF POLYNOMIALS 23 where g ( n ) := G ( ∞ ) ˜ Q n, Φ n , G ( ∞ ) ˜ Q n, Φ n ! and ˜ Q n,k , k = 1 , , is given by (3.23). Consequently, for every fixed ε, < ε ≤ ε , there exists n such that ω ε,n = ∅ for all n ≥ n . Using the structure of the elements of H + n , the definition ofthe metric d , and the monotonicity of the logarithm, it is easy to verify that ω ε,n is convex.Let us show that T n ( ω ε,n ) ⊂ ω ε,n for all sufficiently large n . We claim that for every γ, <γ < / n such that for all n ≥ n and g ∈ ω ε,n . we have(3.43) d ( T n ( g ) , T ( g )) < ε/ . Should this not occur, we could find a sequence ( g ( n ) ) n ∈ Λ , g ( n ) ∈ ω ε,n , such that(3.44) d ( T n ( g ( n ) ) , T ( g ( n ) )) ≥ ε/ ω ε,n belong to H + ( K, C ); therefore, ( g ( n ) ) n ∈ Λ is uniformly bounded in the k · k K norm. Consequently, there exists g ∈ H + ( K, C ) and a subsequence of indices Λ ′ ⊂ Λ such thatlim n ∈ Λ ′ k g ( n ) − g k K = 0 . In particular, lim n ∈ Λ ′ k g ( n ) − g k C ∆ = 0 . Then, according to (3.36) in Proposition 5lim n ∈ Λ ′ k T ( g ( n ) ) − T ( g ) k C ∆ = 0which contradicts (3.44) due to (3.34).Using the triangle inequality and (3.43) for every n ≥ n and g ∈ ω ε,n d ( G , T n ( g )) ≤ d ( T ( G ) , T ( g )) + d ( T ( g ) , T n ( g )) < d ( G , g ) + 12 ε ≤ ε. Consequently, T n ( ω ε,n ) ⊂ ω ε,n as claimed. Now, using Brouwer’s fixed point Theorem we obtainthat for all n ≥ n the operator T n has a fixed point in ω ε,n .Since θ > (cid:3) If we apply Theorem 3.7 to the case of ML Hermite-Pad´e polynomials (so here ˜ h n = | h n, | and ˜ h = h (see (3.12) and (3.18)), we get Theorem 3.8.
Let ( σ , σ ) ∈ S ( ∆ ) . Let ( Q n, , Q n, ) n ≥ be the sequence of ML Hermite-Pad´epolynomials defined by (3.7) - (3.9) and let κ n, , κ n, , q n, , q n, , and h n, be defined as in (3.11) - (3.12) . Then (3.45) lim n Q n,k ( z )Φ nk ( z ) = G k ( z ) G k ( ∞ ) , k = 1 , , uniformly on compact subsets of Ω k . Additionally, (3.46) lim n κ n,k C nk = 1 √ π G k ( ∞ ) , and (3.47) lim n q n,k ( z ) C nk Φ nk ( z ) = 1 √ π G k ( z ) , k = 1 , , uniformly on compact subsets of Ω k .Proof. It is sufficient to apply Theorem 3.7 with ˜ h n = | h n, | , taking note that (3.18) takes placeand that according to Proposition 3.2 the operator ˜ T n has a unique fixed point in all of P n, ×P n, which coincides with ( Q n, , Q n, ). (cid:3) Proof of Theorem 1.1.
Recall that the biorthogonal polynomial Q n coincides with Q n, .Consequently, the second relation in (1.3) follows directly from (3.45).To obtain the asymptotic of the biorthogonal polynomials P n we need to obtain a resultsimilar to Theorem 3.8 working with the definition (3.3)-(3.4) corresponding to the Nikishinsystem N ( σ , σ ). We outline the main ingredients.From Lemma 3.1 and Proposition 3.2 it follows that there exist a unique pair ( P n, , P n, ) ofmonic polynomials of degree n , where P n, = b n, = P n , such that Z x ν P n, ( x ) d σ ( x ) P n, ( x ) = 0 , ν = 0 , , . . . , n − , (3.48) Z x ν P n, ( x ) L n, ( x ) d σ ( x ) P n, ( x ) = 0 , ν = 0 , , . . . , n − , (3.49)where L n, ( z ) := B n, ( z ) P n, ( z ) P n, ( z ) = P n, ( z ) Z P n, ( x ) z − x d σ ( x ) P n, ( x ) = Z P n, ( x ) z − x d σ ( x ) P n, ( x ) . The normalization in this case is ξ − n, = Z P n, ( x ) d σ ( x ) | P n, ( x ) | , ( ξ n, ξ n, ) − = Z P n, ( x ) |L n, ( x ) | d σ ( x ) | P n, ( x ) | . Take, p n, = ξ n, P n, , p n, = ξ n, P n, , ℓ n, = ξ n, L n, . Then, the orthogonality relation (3.48) and (3.49) can be restated as Z x ν p n, ( x ) d σ ( x ) | P n, ( x ) | = 0 , ν = 0 , , . . . , n − , Z x ν p n, ( x ) | ℓ n, ( x ) | d σ ( x ) | P n, ( x ) | = 0 , ν = 0 , , . . . , n − , and the polynomials p n, and p n, are orthonormal with respect to the corresponding varyingmeasures. Following the same arguments that led us to (3.18), we obtain(3.50) lim n | ℓ n, ( t ) | = 1 p | t − a || t − b | =: ℓ ( t )uniformly for t ∈ ∆ . TRONG ASYMPTOTIC OF POLYNOMIALS 25
Then, the operator T : C + ∆ −→ C + ∆ which is relevant to describe the strong asymptotic ofthe polynomials P n, , P n, and their orthonormal versions p n, , p n, is the one defined as follows. T ( g , g ) = ( g ∗ , g ∗ ) is the pair of Szeg˝o functions, g ∗ k ∈ H (Ω k ) , k = 1 , , verifying(3.51) | g ∗ ( x ) | = g ( x ) p ( b − x )( x − a ) σ ′ ( x ) , a.e. on [ a , b ] = ∆ , and(3.52) | g ∗ ( x ) | = g ( x ) p ( b − x )( x − a )( ℓσ ′ )( x ) , a.e. on [ a , b ] = ∆ , where ℓ is given in (3.50).Following the same reasonings as before, if ( G ∗ , G ∗ ) is the fixed point of the operator T definedthrough (3.51)-(3.52), we have(3.53) lim n P n, ( z )Φ n ( z ) = G ∗ ( z ) G ∗ ( ∞ ) , lim n P n, ( z )Φ n ( z ) = G ∗ ( z ) G ∗ ( ∞ )uniformly on compact subsets of Ω and Ω respectively. Since P n = P n, , the second limit in(3.53) gives us the strong asymptotic of the sequence ( P n ) n ≥ , We are done. (cid:3) Naturally, the asymptotic of the sequence of normalizing coefficients and of the orthonormalpolynomials p n, , p n, can also be given. We leave the details to the reader.3.10. Asymptotic of ML Hermite-Pad´e polynomials.
In this subsection, as an easy con-sequence of the strong asymptotic of the polynomials Q n, and Q n, , we obtain the strongasymptotic of A n,j , and a n,j , j = 0 ,
1. Recall that Q n, ≡ a n, ≡ A n, . Corollary 3.9.
Let ( σ , σ ) ∈ S ( ∆ ) and A n,j , j = 0 , , is defined by (3.1) – (3.2) . Then, (3.54) lim n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ n, A n, ( z )(Φ / Φ ) n ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = G ( ∞ ) G ( ∞ ) (cid:12)(cid:12)(cid:12)(cid:12) G ( z ) G ( z ) (cid:12)(cid:12)(cid:12)(cid:12) p | z − b || z − a | , and (3.55) lim n (cid:12)(cid:12)(cid:12)(cid:12) ( κ n, κ n, ) A n, ( z )Φ − n ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = | G ( z ) | π G ( ∞ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b a ( z − x ) − d x p ( b − x )( x − a ) | x − b || x − a | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where the limits are uniform in compact subsets of C \ (∆ ∪ ∆ ) and C \ ∆ , respectively.Proof. Formula (3.9) can be rewritten as A n, ( z ) = Q n, ( z ) Q n, ( z ) Z Q n, ( x ) z − x d σ ( x ) Q n, ( x ) , where the equality holds in Ω . Then,(3.56) κ n, A n, ( z )(Φ / Φ ) n ( z ) = Q n, ( z )Φ n ( z ) Φ n ( z ) Q n, ( z ) Z q n, ( x ) z − x d σ ( x ) Q n, ( x ) , z ∈ C \ (∆ ∪ ∆ ) . From [10, Th. 8], we have(3.57) lim n Z q n, ( x ) z − x d σ ( x ) | Q n, ( x ) | = 1 π Z b a d x ( z − x ) p ( b − x )( x − a ) = 1 p ( z − b )( z − a ) , uniformly in compact subsets of Ω . This, together with (3.45) and (3.56), gives us (3.54).Combining (3.5) for j = 0 with (3.9) we get A n, ( z ) = Z Q n, ( x ) z − x H n, ( x ) d σ ( x ) Q n, ( x ) . By orthogonality, we have Q n, ( z ) Z Q n, ( x ) z − x H n, ( x ) d σ ( x ) Q n, ( x ) = Z Q n, ( x ) z − x H n, ( x ) d σ ( x ) Q n, ( x ) . Therefore, A n, ( z ) = 1 Q n, ( z ) Z Q n, ( x ) z − x H n, ( x ) d σ ( x ) Q n, ( x ) , where the equality holds for z ∈ Ω . So,(3.58) ( κ n, κ n, ) A n, ( z )Φ − n ( z ) = Φ n ( z ) Q n, ( z ) Z q n, ( x ) z − x h n, ( x ) d σ ( x ) Q n, ( x ) . From [10, Th. 8] and (3.18), it follows thatlim n Z q n, ( x ) z − x | h n, ( x ) | d σ ( x ) | Q n, ( x ) | = 1 π Z b a ( z − x ) − d x p ( b − x )( x − a ) | x − b || x − a | . This formula combined with (3.58) and (3.45) gives us (3.55). We have completed the proof. (cid:3)
For the sequence of even indices n , the limits (3.54) and (3.55) hold without the need of takingabsolute value (understanding that the square root on the right hand side of (3.54) is positivewhen z > b ). For the sequence of odd indices n , in order to remove the absolute value, one musttake into consideration the relative position of the intervals ∆ and ∆ . The details are left tothe reader.Notice that b σ ( z ) − a n, ( z ) a n, ( z ) = A n, ( z ) Q n, ( z ) = Q n, ( z ) q n, ( z ) Z q n, ( x ) z − x d σ ( x ) Q n, ( x )Therefore, using (3.45), (3.47), and (3.57), we obtain(3.59) lim n (cid:12)(cid:12)(cid:12)(cid:12) C Φ ( z )Φ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) n (cid:12)(cid:12)(cid:12)(cid:12)b σ ( z ) − a n, ( z ) a n, ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = 2 π | G ( z ) | G ( ∞ ) | G ( z ) | p | z − a || z − b | , uniformly on compact subsets of C \ (∆ ∪ ∆ ). The definition of Φ , Φ , and C imply (cid:12)(cid:12)(cid:12)(cid:12) C Φ ( z )Φ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = exp ( − (2 V λ ( z ) − V λ ( z ) − γ )) . The second equilibrium equation in (3.20) and the maximum principle for subharmonic functionentail 2 V λ ( z ) − V λ ( z ) − γ < , z ∈ Ω . Consequently, (3.59) gives a precise description of the rate with which ( a n, /a n, ) n ≥ convergesto b σ . Exactly the same formula can be obtained substituting in (3.59) b σ − a n, /a n, by b s , − a n, /a n, . However, we will not dwell into this because it requires the introduction of some newtransformations which drive us off course. TRONG ASYMPTOTIC OF POLYNOMIALS 27
From [20, Th. 1.6], we know that, for j = 0 , n a n,j ( z ) a n, ( z ) = ˆ s ,j +1 ( z ) , where the limit is uniform on compact subsets of Ω . Corollary 3.10.
Let ( σ , σ ) ∈ S ( ∆ ) . Then, for j = 0 , n a n,j Φ n ( z ) = G ( z ) G ( ∞ ) ˆ s ,j +1 ( z ) , where the limit is uniform on compact subsets of Ω .Proof. Since a n, ≡ Q n, , we havelim n a n,j ( z )Φ n ( z ) Φ n ( z )( z ) Q n, ( z ) = ˆ s ,j +1 ( z ) . Taking into account (3.47) the proof readily follows. (cid:3)
Results analogous to Corollaries 3.9 and 3.10 for the forms B n, , B n, , and the polynomials b n, , b n, , follow immediately considering the Nikishin system N ( σ , σ ). The details are left tothe reader.3.11. A different expression for the functions Φ , Φ and the constants C , C . In [15,Theorem 4.2] the ratio asymptotic of general ML Hermite-Pad´e polynomials was given. The limitwas expressed in terms of the branches of a conformal map of a certain Riemann surface. Sincestrong asymptotic implies ratio asymptotic, we can use that result to interpret the comparisonfunctions Φ , Φ and the constants C , C in a different way. (Which coincides with the form inwhich they were defined in [2].)We introduce the Riemann surface which is relevant in our case of two measures. Let R denotethe compact Riemann surface R = [ k =0 R k formed by 3 consecutively “glued” copies of the extended complex plane R := C \ ∆ , R := C \ (∆ ∪ ∆ ) , R := C \ ∆ . The upper and lower banks of the slits of two neighboring sheets are identified.Let π : R −→ C be the canonical projection from R to C and denote by z ( k ) the point on R k verifying π ( z ( k ) ) = z, z ∈ C . Let ϕ : R −→ C denote a conformal mapping whose divisor consistsof one simple zero at ∞ (0) ∈ R and one simple pole at ∞ (2) ∈ R . This mapping exists and isuniquely determined up to a multiplicative constant. Denote the branches of ϕ by ϕ k ( z ) := ϕ ( z ( k ) ) , k = 0 , , , z ( k ) ∈ R k . From the properties of ϕ , we have(3.60) ϕ ( z ) = c /z + O (1 /z ) , ϕ ( z ) = c z + O (1) , z → ∞ , where c , c are non-zero constants.Let x ∈ ∆ k , k = 1 ,
2. We write z → x + when z ∈ C approaches x from above the real line.Analogously, z → x − means that z approaches x from below the real line. Let us define ϕ k ( x + ) := lim z → x + ϕ k ( z ) = lim z → x + ϕ ( z ( k ) )and ϕ k ( x − ) := lim z → x − ϕ k ( z ) = lim z → x − ϕ ( z ( k ) ) . Except when x is an end point of ∆ k , these limits are different due to the fact that lim z → x + z ( k ) =lim z → x − z ( k ) on R . However, due to the identification made of the points on the slits it is easyto verify that(3.61) ϕ k ( x + ) = ϕ k +1 ( x − ) , ϕ k ( x − ) = ϕ k +1 ( x + ) , k = 0 , , because lim z → x + z ( k ) = lim z → x − z ( k +1) , lim z → x − z ( k ) = lim z → x + z ( k +1) . Taking account of the way in which the functions ϕ k were extended to ∆ k and (3.61) itfollows that Q k =0 ϕ k is a single-valued analytic function on C without singularities; therefore, itis constant. We normalize ϕ so that p Y k =0 ϕ k = c, | c | = 1 , c > . Let us show that with this normalization c is +1.Indeed, for a point z ( k ) ∈ R k on the Riemann surface we define its conjugate z ( k ) := z ( k ) .For a points z ( k ) on the upper bank of the slit ∆ k the conjugate is the one corresponding to thelower bank. Now, we define ϕ ∗ : R −→ C as follows ϕ ∗ ( ζ ) := ϕ ( ζ ). It is easy to verify that ϕ ∗ isa conformal mapping of R onto C with the same divisor as ϕ . Therefore, there exists a constant κ such that ϕ ∗ = κϕ . The corresponding branches satisfy the relations ϕ ∗ k ( z ) = ϕ k ( z ) = κϕ k ( z ) , k = 0 , , . Comparing the Laurent expansions at ∞ of ϕ ( z ) and κϕ ( z ), using the fact that c >
0, itfollows that κ = 1. Then ϕ k ( z ) = ϕ k ( z ) , k = 0 , , . This in turn implies that for each k = 0 , , ϕ k are real numbers. Obviously, c is the product of theseleading coefficients. Since they are real numbers c is real and since it is of module 1, it has tobe either 1 or −
1. Analyzing the Laurent expansion of the branches at ∞ one easily concludesthat indeed c = 1. So, we can assume in the following that(3.62) Y k =0 ϕ k ≡ , c > . It is easy to see that conditions (3.60) and (3.62) determine ϕ uniquely.In [3, Lemma 4.2] the authors proved the following result. TRONG ASYMPTOTIC OF POLYNOMIALS 29
Lemma 3.11.
Their exists a unique pair of functions ( F , F ) such that for k = 1 , (1) F k , /F k ∈ H ( C \ ∆ k ) ,(2) F ′ k ( ∞ ) > ,(3) | F k ( x ) | | F k − ( x ) F k +1 ( x ) | = 1 , x ∈ ∆ k , ( F ≡ F ≡ . The functions may be expressed by the formulas F k := Y ν = k ϕ ν , k = 1 , . The boundary conditions for the functions F k , k = 1 , | F ( x ) | = | F ( x ) | , x ∈ ∆ , | F ( x ) | = | F ( x ) | , x ∈ ∆ . Compare with (3.20) after taking logarithm.From [15, Theorem 4.2, Corollary 4.3] we know that for k = 1 , , lim n →∞ Q n +1 ,k ( z ) Q n,k ( z ) = F k ( z ) F ′ k ( ∞ ) , uniformly on compact subsets of Ω k andlim n →∞ κ n +1 ,k κ n,k = F ′ k ( ∞ ) q F ′ k − ( ∞ ) F ′ k +1 ( ∞ ) , where by definition we take F ′ ( ∞ ) = F ′ ( ∞ ) = 1. On the other hand, (3.45) and (3.46) implythat lim n →∞ Q n +1 ,k ( z )Φ n +1 k ( z ) Φ nk ( z ) Q n,k ( z ) = 1Φ k ( z ) lim n →∞ Q n +1 ,k ( z ) Q n,k ( z ) = 1 , uniformly on compact subsets of Ω k andlim n →∞ κ n +1 ,k C n +1 k C nk κ n,k = 1 C k lim n →∞ κ n +1 ,k κ n,k = 1 . Consequently(3.63) Φ k ( z ) ≡ F k ( z ) F ′ k ( ∞ ) , C k = F ′ k ( ∞ ) q F ′ k − ( ∞ ) F ′ k +1 ( ∞ ) , k = 1 , . References [1] A.I. Aptekarev. Asymptotics of simultaneously orthogonal polynomials in the Angelesco case. Math. USSRSb. (1989), 57-84.[2] A.I. Aptekarev. Strong asymptotic of multiple orthogonal polynomials for Nikishin systems. Sb. Math. ,631-669.[3] A.I. Aptekarev, G. L´opez Lagomasino, and I.A. Rocha. Ratio asymptotic of Hermite-Pad´e orthogonal poly-nomials for Nikishin systems. Sb. Math. (2005), 1089-1107.[4] L. Baratchart, H. Stahl, and F. Wielonsky. Asymptotic error estimates for L best rational approximants toMarkov functions. J. of Approx. Theory (2001), 53-96. [5] D. Barrios, B. de la Calle Ysern, and G. L´opez Lagomasino. Ratio and relative asymptotic of polynomialsorthogonal with respect to varying Denisov-type measures. J. of Approx. Theory, 139 (2006), 223-256.[6] M. Bertola. Two matrix models and biorthogonal polynomials. The Oxford handbook of random matrixtheory, 310-328. Oxford Univ. Press, Oxford, 2011.[7] M. Bertola, M. Gekhtman, and J. Szmigielski. Cauchy biorthogonal polynomials. J. Approx. Theory (2010), 832-867.[8] M. Bertola, M. Gekhtman, and J. Szmigielski. Strong asymptotics for Cauchy biorthogonal polynomials withapplication to the Cauchy two–matrix model. J. Math. Physics (2013), 043517.[9] J. Bustamante Gonz´alez and G. L´opez Lagomasino. Hermite-Pad´e approximation for Nikishin systems ofanalytic functions. Russian Acad. Sci. Sb. Math. (1994), 367-384.[10] B. de la Calle Ysern and G. L´opez Lagomasino. Weak convergence of varying measures and Hermite-Pad´eorthogonal polynomials. Const. Approx. (1999), 553-575.[11] B. de la Calle Ysern and G. L´opez Lagomasino. Strong asymptotic of orthogonal polynomials with varyingmeasures and Hermite-Pad´e approximants. J. Comp. Appl. Math. (1998), 91-103.[12] P. Deift, T. Krieckerbauer, and K. T.-R. McLaughlin. New results on the equilibrium measure for logarithmicpotentials in the presence of an external field. J. of Approx. Theory (1998), 388-475.[13] U. Fidalgo Prieto and G. L´opez Lagomasino. Nikishin systems are perfect. Const. Approx. (2011), 297–356.[14] U. Fidalgo Prieto and G. L´opez Lagomasino. Nikishin systems are perfect. Case of unbounded and touchingsupport. J. of Approx. Theory (2011), 779-811.[15] U. Fidalgo Prieto, G. L´opez Lagomasino, and S. Medina Peralta. Asymptotic of Cauchy biorthogonal poly-nomials. Mediterr. J. Math. (2020) 17: 22.[16] A.A. Gonchar and G. L´opez Lagomasino. On Markov’s theorem for multipoint Pad´e approximants. Math.USSR Sb. (1978), 449-459.[17] A.A. Gonchar, E.A. Rakhmanov, and V.N. Sorokin. Hermite-Pad´e approximation for systems of Markov-typefunctions. Sbornik Math. (1997), 33-58.[18] G. L´opez Lagomasino. Szeg˝o’s theorem for polynomials orthogonal with respect to varying measures. In“Orthogonal Polynomials and their Applications” Eds. M. Alfaro et al. Lecture Notes in Math. Vol. 1329,Springer-Verlag, Berlin, 1988, 255-260.[19] G. L´opez Lagomasino. Convergence of Pad´e approximants of Stieltjes type meromorphic functions and com-parative asymptotics of orthogonal polynomials. Math. USSR Sb. (1989), 207-229.[20] G. L´opez Lagomasino, S. Medina Peralta, and J. Szmigielski. Mixed type Hermite-Pad´e approximationinspired by the Degasperis-Procesi equation. Adv. Math. (2019), 813-838.[21] V.G. Lysov. Mixed type Hermite-Pad´e approximants for a Nikishin System. Proc. Steklov Inst. of Math. (2020), 199–213.[22] E.M. Nikishin. On simultaneous Pad´e approximants. Math. USSR Sb. (1982), 409-425.[23] W. Rudin. Real and Complex Analysis . McGraw-Hill, Series in Higher Math., New York, 1966.[24] E.B. Saff and V. Totik.
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