Orthogonal rational functions with real poles, root asymptotics, and GMP matrices
aa r X i v : . [ m a t h . SP ] A ug ORTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOTASYMPTOTICS, AND GMP MATRICES
BENJAMIN EICHINGER, MILIVOJE LUKIĆ, GIORGIO YOUNG
Abstract.
There is a vast theory of the asymptotic behavior of orthogonal polynomials withrespect to a measure on R and its applications to Jacobi matrices. That theory has an obviousaffine invariance and a very special role for ∞ . We extend aspects of this theory in the settingof rational functions with poles on R = R ∪ {∞} , obtaining a formulation which allows multiplepoles and proving an invariance with respect to R -preserving Möbius transformations. We obtaina characterization of Stahl–Totik regularity of a GMP matrix in terms of its matrix elements; asan application, we provide an alternative proof of a theorem about a Cesàro–Nevai property ofregular Jacobi matrices on finite gap sets. Introduction
There is a vast theory of orthogonal polynomials with respect to measures on C and their rootasymptotics, exemplified by the Ullman–Stahl–Totik theory of regularity. Let µ be a compactlysupported probability measure and { p n } ∞ n =0 the corresponding orthonormal polynomials, obtainedby the Gram–Schmidt process from { z n } ∞ n =0 in L ( dµ ). Thenlim inf n →∞ | p n ( z ) | /n ≥ e G E ( z, ∞ ) (1.1)for z outside the convex hull of µ , where E is the essential support of µ and G E denotes thepotential theoretic Green function for the domain C \ E ; if that domain is not Greenian, one takes G E = + ∞ instead. For measures compactly supported in R , this theory can be interpreted in termsof self-adjoint operators. In particular, for any bounded half-line Jacobi matrix J = b a a b a a . . . . . .. . . with a ℓ > b ℓ ∈ R , lim sup n →∞ n Y ℓ =1 a ℓ ! /n ≤ Cap σ ess ( J ) , (1.2)where Cap denotes logarithmic capacity. For both of these universal inequalities, the case of equality(and existence of limit) is called Stahl–Totik regularity [21]; the theory originated with the case E = [ − , B.E. was supported by Austrian Science Fund FWF, project no: J 4138-N32.M.L. was supported in part by NSF grant DMS–1700179.G.Y. was supported in part by NSF grant DMS–1745670.
We extend aspects of this theory to the setting of rational functions with poles in R = R ∪ {∞} .One motivation for this is the search for a more conformally invariant theory. Statements such as(1.1), (1.2) rescale in obvious ways with respect to affine transformations (automorphisms of C )which preserve R , so it is obvious that an affine pushforward of a Stahl–Totik regular measure isStahl–Totik regular. However, the point ∞ has a very special role throughout the theory: for aMöbius transformation f which does not preserve ∞ , p n ◦ f are rational functions with a pole at f − ( ∞ ), and f ( J ) as defined by the functional calculus is not a finite band matrix. Thus, it is anontrivial question whether a Möbius pushforward of a Stahl–Totik regular measure is Stahl–Totikregular.The set of Möbius transformations which preserve R is the semidirect group product PSL(2 , R ) ⋊ { id , z
7→ − z } , whose normal subgroup PSL(2 , R ) corresponds to the orientation preserving case.Denote by f ∗ µ the pushforward of µ , defined by ( f ∗ µ )( A ) = µ ( f − ( A )) for Borel sets A . As anexample of our techniques, we obtain the following: Theorem 1.1.
Let f ∈ PSL(2 , R ) ⋊ { id , z
7→ − z } . If µ is a Stahl–Totik regular measure on R and ∞ / ∈ supp( f ∗ µ ) , then the pushforward measure f ∗ µ is also Stahl–Totik regular. However, we will mostly work in the more general setting when multiple poles on R are al-lowed, which arises naturally in the spectral theory of self-adjoint operators. Denote by T f,dµ themultiplication operator by f in L ( dµ ). The matrix representation for T x,dµ ( x ) in the basis oforthogonal polynomials is a Jacobi matrix, and through this classical connection, the theory of or-thogonal polynomials is inextricably linked to the spectral theory of Jacobi matrices. In this matrixrepresentation, resolvents T ( c − x ) − ,dµ ( x ) are not finite-diagonal matrices. However, in a basis of or-thogonal rational functions with poles at c , . . . , c g , ∞ , the multiplication operators T ( c − x ) − ,dµ ( x ) ,. . . , T ( c g − x ) − ,dµ ( x ) , T x,dµ ( x ) all have precisely 2 g +1 nontrivial diagonals. The corresponding matrixrepresentations are called GMP matrices; they were introduced by Yuditskii [25].We should also compare this to the case of CMV matrices: for a measure supported on the unitcircle, Stahl–Totik regularity is still defined in terms of orthogonal polynomials, but the CMV basis[3, 18] is given in terms of positive and negative powers of z , i.e., orthonormal rational functionswith poles at ∞ and 0. The symmetries in that setting lead to explicit formulas for the CMV basisin terms of the orthogonal polynomials; it is then a matter of calculation to relate the exponentialgrowth rate of the CMV basis to that of the orthogonal polynomials, and to interpret regularity interms of the CMV basis. In our setting, there is no such symmetry and no formula for orthonormalrational functions in terms of orthonormal polynomials.In order to state our results in a conformally invariant way, we will use the following notationsand conventions throughout the paper. The measure µ will be a probability measure on R . Wedenote by supp µ its support in R , and we consider its essential support (the support with isolatedpoints removed), denoted E = ess supp µ. We will always assume that µ is nontrivial; equivalently, E = ∅ .Fix a finite sequence with no repetitions, C = ( c , . . . , c g +1 ) with c k ∈ R \ supp µ for all k .Consider the sequence { r n } ∞ n =0 where r = 1 and for n = j ( g + 1) + k , 1 ≤ k ≤ g + 1, r n ( z ) = ( c k − z ) j +1 c k ∈ R z j +1 c k = ∞ (1.3)Applying the Gram–Schmidt process to this sequence in L ( dµ ) gives the sequence of orthonormalrational functions { τ n } ∞ n =0 whose behavior we will study. We note that the special case supp µ ⊂ R , RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 3 g = 0, C = ( ∞ ) corresponds to the standard construction of orthonormal polynomials associatedto the measure µ (note that, since we denote by supp µ the support in R , the statement supp µ ⊂ R implies that µ is compactly supported in R ), and our first results are an extension of the sametechniques.The first result is a universal lower bound on the growth of { τ n } ∞ n =0 in terms of a potentialtheoretic quantity. If E is not a polar set, we use the (potential theoretic) Green function for thedomain C \ E , denoted G E , and we define G E ( z, C ) = ( g +1 P g +1 k =1 G E ( z, c k ) E is not polar+ ∞ E is polar (1.4) Theorem 1.2.
For all z ∈ C \ R , lim inf n →∞ | τ n ( z ) | /n ≥ e G E ( z, C ) . This is a good place to point out that our current setup is not related to the recent paper [11], inwhich the behavior was compared to a Martin function at a boundary point of the domain. Here,the behavior is compared to a combination of Green functions (1.4), all the poles are in the interiorof the domain C \ E , and the difficulty comes instead from the multiple poles.Another universal inequality for orthonormal polynomials comes from comparing their leadingcoefficients to the capacity of E . In our setting, the analog of the leading coefficient must beconsidered in a pole-dependent way. Denote L n = span { r ℓ | ≤ ℓ ≤ n } . By the nature of the Gram–Schmidt process, there is a κ n > τ n − κ n r n ∈ L n − . The Gram–Schmidt process can be reformulated as the L ( dµ )-extremal problem κ n = max (cid:8) Re κ : f = κr n + h, h ∈ L n − , k f k L ( dµ ) ≤ (cid:9) . (1.5)By strict convexity of the L -norm, these L -extremal problems have unique extremizers given by f = τ n , and κ n is explicitly characterized as a kind of leading coefficient for τ n with respect to thepole at c k where n = j ( g + 1) + k , 1 ≤ k ≤ g + 1. Below, we will also relate the constants κ n tooff-diagonal coefficients of certain matrix representations.The growth of the leading coefficients κ n will be studied along sequences n = j ( g + 1) + k fora fixed k , and bounded by quantities related to the pole c k . If E is not a polar set, it is a basicproperty of the Green function that the limits γ k E = ( lim z → c k ( G E ( z, c k ) + log | z − c k | ) , c k = ∞ lim z → c k ( G E ( z, c k ) − log | z | ) , c k = ∞ exist. Note that, if c k = ∞ , γ k E is precisely the Robin constant for the set E . We further defineconstants λ k by log λ k = γ k E + P ≤ ℓ ≤ g +1 ℓ = k G E ( c k , c ℓ ) E is not polar+ ∞ E is polar (1.6) Theorem 1.3.
For all ≤ k ≤ g + 1 , for the subsequence n ( j ) = j ( g + 1) + k , lim inf j →∞ κ /n ( j ) n ( j ) ≥ λ / ( g +1) k . (1.7) RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 4
Theorem 1.4.
The following are equivalent:(i) For some ≤ k ≤ g + 1 , for the subsequence n ( j ) = j ( g + 1) + k , lim j →∞ κ /n ( j ) n ( j ) = λ / ( g +1) k ; (ii) For all ≤ k ≤ g + 1 , for the subsequence n ( j ) = j ( g + 1) + k , lim j →∞ κ /n ( j ) n ( j ) = λ / ( g +1) k ; (iii) lim n →∞ g +1 Y ℓ =1 κ n + ℓ ! /n = g +1 Y k =1 λ k ! / ( g +1) (iv) For q.e. z ∈ E , we have lim sup n →∞ | τ n ( z ) | /n ≤ ;(v) For some z ∈ C + , lim sup n →∞ | τ n ( z ) | /n ≤ e G E ( z, C ) ;(vi) For all z ∈ C , lim sup n →∞ | τ n ( z ) | /n ≤ e G E ( z, C ) ;(vii) Uniformly on compact subsets of C \ R , lim n →∞ | τ n ( z ) | /n = e G E ( z, C ) . Definition 1.5.
The measure µ is C -regular if it obeys one (and therefore all) of the assumptionsof Theorem 1.4.In this terminology, Stahl–Totik regularity is precisely ( ∞ )-regularity, i.e., C -regularity for thespecial case supp µ ⊂ R , g = 0, C = ( ∞ ). Theorems 1.2, 1.3, 1.4 are closely motivated byfoundational results for Stahl–Totik regularity. A new phenomenon appears through the periodicitywith which poles are taken in (1.3) and the resulting subsequences n ( j ) = j ( g + 1) + k : since κ n is a normalization constant for τ n , it is notable that control of κ n along a single subsequence n ( j ) = j ( g + 1) + k in Theorem 1.4.(i) provides control over the entire sequence. This phenomenondoesn’t have an exact analog for orthogonal polynomials, where g = 0. We will also see below thatthis is essential in order to characterize the regularity of a GMP matrix using only the entries ofthe matrix itself and not its resolvents.Moreover, we show that the regular behavior described by Theorem 1.4 is independent of the setof poles C : Theorem 1.6.
Let C , C be two finite sequences of elements from R \ supp µ , not necessarily ofthe same length. Then µ is C -regular if and only if it is C -regular. Corollary 1.7.
Let supp µ ⊂ R . Let C be a finite sequence of elements from R \ supp µ . Then µ is C -regular if and only if it is Stahl–Totik regular. Thus, Theorem 1.4 should not be seen as describing equivalent conditions for a new class ofmeasures, but rather a new set of regular behaviors for the familiar class of Stahl–Totik regularmeasures.We consistently work with poles on R since our main interest is tied to self-adjoint problems.Some of our results are in a sense complementary to the setting of [21, Section 6.1], where poles areallowed in the complement of the convex hull of supp µ , and the behavior of orthogonal rationalfunctions is considered with respect to a Stahl–Totik regular measure. Due to this, it is natural toexpect that these results hold more generally, for measures on C and general collections of poles andMöbius transformations. Moreover, in our setup the poles are repeated exactly periodically, butwe expect this can be generalized to a sequence of poles which has a limiting average distribution.Related questions for orthogonal rational functions were also studied by [2, 8]. RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 5
As noted in [21, Section 6.1], poles in the gaps of supp µ can cause interpolation defects inthe problem of interpolation by rational functions. In our work, these interpolation defects showup as possible reductions in the order of the poles. For example, consider C = ( ∞ , τ j +1 is allowed a pole at 0 of order at most j . However, if µ is symmetric with respectto z
7→ − z , the functions τ n will have an even/odd symmetry. Since τ j +1 contains a nontrivialmultiple of z j +1 , it follows that τ j +1 ( z ) = ( − j +1 τ j +1 ( − z ). By this symmetry, the actual orderof the pole at 0 is j + 1 − k for some even k , so it cannot be equal to j (it will follow from ourresults that in this case, the order of the pole is j − ∞ for C = (0 , ∞ ). In the polynomial case, this does not occur: p n always has a pole at ∞ of orderexactly n .We will consider at once the distribution of zeros of τ n and the possible reductions in the orderof the poles. We will prove that all zeros of τ n are real and simple, and that n − g ≤ deg τ n ≤ n .We define the normalized zero counting measure ν n = 1 n X w : τ n ( w )=0 δ w . Although we normalize by n , ν n may not be a probability measure: however 1 − g/n ≤ ν n ( R ) ≤ τ n instead of by n would not affect the limits as n → ∞ .We will now describe the weak limit behavior of the measures ν n as n → ∞ . To avoid pathologicalcases, we assume that E is not polar; in that case, denoting by ω E ( dx, w ) the harmonic measure forthe domain C \ E at the point w , we define the probability measure on E , ρ E , C = 1 g + 1 g +1 X j =1 ω E (d x, c j ) . The results below describe weak limits of measures in the topology dual to C ( R ). Theorem 1.8.
Let µ be a probability measure on R . Assume that E is not a polar set.(a) If µ is regular, then w-lim n →∞ ν n = ρ E , C .(b) If w-lim n →∞ ν n = ρ E , C , then µ is regular or there exists a polar set X ⊂ E such that µ ( R \ X ) =0 . We now turn to matrix representations of self-adjoint operators. Fix a sequence C = ( c , . . . , c g +1 )such that c k = ∞ for some k . A half-line GMP matrix [25] is the matrix representation for multi-plication by x in the basis { τ n } ∞ n =0 ; its matrix elements are A mn = Z τ m ( x ) xτ n ( x ) dµ ( x ) . The condition that c k = ∞ for some k guarantees that A mn = 0 for | m − n | > g + 1, so these matrixelements generate a bounded operator A on ℓ ( N ) such that A mn = h e m , Ae n i , where ( e n ) ∞ n =0 denotes the standard basis of ℓ ( N ). We say that A ∈ A ( C ).GMP matrices have the property that some of their resolvents are also GMP matrices; namely,for any ℓ = k , ( c ℓ − A ) − ∈ A ( f ( C )) where f is the Möbius transform f : z ( c ℓ − z ) − and f ( C ) = ( f ( c ) , . . . , f ( c g +1 )).Note that the special case g = 0, C = ( ∞ ) gives precisely a Jacobi matrix. A Jacobi matrix issaid to be regular if it is obtained by this construction from a regular measure; analogously, we willcall a GMP matrix regular if it is obtained from a regular measure. Just as regularity of a Jacobimatrix can be characterized in terms of its off-diagonal entries, we will show that regularity of a RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 6
GMP matrix can be characterized in terms of its entries in the outermost nontrivial diagonal. Wewill also obtain a GMP matrix analog of the inequality (1.2).The GMP matrix has an additional block matrix structure; in particular, for a GMP matrixwith c k = ∞ , on the outermost nonzero diagonal m = n − g −
1, the only nonzero terms appear for n = j ( g + 1) + k , and those are strictly positive. Thus, we denote β j = h e j ( g +1)+ k , Ae ( j +1)( g +1)+ k i (1.8) Theorem 1.9.
Fix a probability measure µ with supp µ ⊂ R and a sequence C = ( c , . . . , c g +1 ) with c k = ∞ . Then lim sup j →∞ j Y ℓ =1 β ℓ ! /j ≤ λ − k . (1.9) Moreover, the measure µ is Stahl–Totik regular if and only if lim j →∞ j Y ℓ =1 β ℓ ! /j = λ − k . (1.10)The proof will use a relation between the sequence { β j } ∞ j =1 and the constants { κ j ( g +1)+ k } ∞ j =1 ,with k still fixed so that c k = ∞ . In particular, the characterization of regularity in Theorem 1.9is made possible by the characterization of regularity in terms of the subsequence { κ j ( g +1)+ k } ∞ j =1 for a single k . Theorem 1.9 also corroborates the perspective that regularity of the measure is thefundamental notion which manifests itself equally well in many different matrix representations.Since the resolvents ( c ℓ − A ) − are also GMP matrices and their measures are pushforwards ofthe original measure, they are also regular GMP matrices; in this sense, Theorem 1.9 provides g + 1criteria for regularity, one corresponding to each subsequence n ( j ) = j ( g + 1) + ℓ .As an application of this theory, we show that it provides an alternative proof of a theorem forJacobi matrices originally conjectured by Simon [19]. Let E ⊂ R be a compact finite gap set, E = [ b , a ] \ g [ k =1 ( a k , b k ) , (1.11)and denote by T + E the set of almost periodic half-line Jacobi matrices with σ ess ( J ) = σ ac ( J ) = E [4, 12]. Through algebro-geometric techniques and the reflectionless property, this class of Jacobimatrices has been widely studied for their spectral properties and quasiperiodicity. They alsoprovide natural reference points for perturbations, which is our current interest. On boundedhalf-line Jacobi matrices J , we consider the metric d ( J, ˜ J ) = ∞ X k =1 e − k ( | a k − ˜ a k | + | b k − ˜ b k | ) . (1.12)On norm-bounded sets of Jacobi matrices, convergence in this metric corresponds to strong operatorconvergence. However, instead of distance to a fixed Jacobi matrix ˜ J , we will consider the distanceto T + E , d ( J, T + E ) = inf ˜ J ∈T + E d ( J, ˜ J ) = min ˜ J ∈T + E d ( J, ˜ J ) . Denote by S + the right shift operator on ℓ ( N ), S + e n = e n +1 . The condition d (( S ∗ + ) m JS m + , T + E ) → m → ∞ is called the Nevai condition. For E = [ − , a n → b n → n → ∞ [15]. In general, as a consequence of [17], the Nevaicondition implies regularity. The converse is false; however: RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 7
Theorem 1.10. If E ⊂ R is a compact finite gap set and J is a regular Jacobi matrix with σ ess ( J ) = E , then lim N →∞ N N X m =1 d (( S ∗ + ) m JS m + , T + E ) = 0 . (1.13)The condition (1.13) is described as the Cesàro–Nevai condition; it was first studied by Golinskii–Khrushchev [13] in the OPUC setting with essential spectrum equal to ∂ D . Theorem 1.10 wasconjectured by Simon [19] and proved in the special case when E is the spectrum of a periodicJacobi matrix with all gaps open by using the periodic discriminant and techniques from Damanik–Killip–Simon [6] to reduce to a block Jacobi setting. It was then proved in general by Krüger [14]by very different methods. We show that Simon’s strategy can be developed into a general proof,if instead of the periodic discriminant and techniques from [6] we use the Ahlfors function, GMPmatrices, and techniques of Yuditskii [25].For the compact finite gap set E ⊂ R , among all analytic functions C \ E → D which vanish at ∞ ,the Ahlfors function Ψ takes the largest value of Re( zΨ ( z )) | z = ∞ . The Ahlfors function has preciselyone zero in each gap, denoted c k ∈ ( a k , b k ) for 1 ≤ k ≤ g , a zero at c g +1 = ∞ , and no other zeros;see also [20, Chapter 8]. In particular, for the finite gap set E , this generates a particularly naturalsequence of poles C E = ( c , . . . , c g , ∞ ).The Ahlfors function was used by Yuditskii [25] to define a discriminant for finite gap sets, ∆ E ( z ) = Ψ ( z ) + 1 Ψ ( z ) . (1.14)This function is not equal to the periodic discriminant, but it has some similar properties and itis available more generally (even when E is not a periodic spectrum). Namely, ∆ E extends to ameromorphic function on C and ( ∆ E ) − ([ − , E . It was introduced by Yuditskii to solve theKillip–Simon problem for finite gap essential spectra. In fact, the discriminant is a rational functionof the form ∆ E ( z ) = λ g +1 z + d + g X k =1 λ k c k − z (1.15)for some d ∈ R ; in particular, we will explain that the constants λ j > E , by a change of oneJacobi coefficient, which does not affect regularity, we can assume that c k / ∈ supp µ (Lemma 6.1).Under this assumption, regularity of the Jacobi matrix implies regularity of the correspondingGMP matrix A and the resolvents ( c k − A ) − , k = 1 , . . . , g , which can be characterized in terms oftheir coefficients by Theorem 1.9. By properties of the Yuditskii discriminant, this further impliesregularity of the block Jacobi matrix ∆ E ( A ). Let us briefly recall that a block Jacobi matrix is ofthe form J = w v v ∗ w v v ∗ w v v ∗ . . . . . .. . . (1.16) RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 8 where v j and w j are d × d matrices, w j = w ∗ j , and det v j = 0 for each j . Type 3 block Jacobimatrices have each v j lower triangular and positive on the diagonal. An extension of regularity toblock Jacobi matrices was developed by Damanik–Pushnitski–Simon [7]; in particular, J is regularfor the set [ − ,
2] if σ ess ( J ) = [ − ,
2] andlim n →∞ n Y j =1 | det v j | /n = 1 . (1.17)This chain of arguments will result in the following lemma: Lemma 1.11.
Let J be a regular Jacobi matrix, E = σ ess ( J ) a finite gap set, and C E the corre-sponding sequence of zeros of the Ahlfors function. Assuming c k / ∈ σ ( J ) for ≤ k ≤ g , denote by A the GMP matrix corresponding to J with respect to the sequence C E . Then ∆ E ( A ) is a regulartype 3 block Jacobi matrix with essential spectrum [ − , . With this lemma, it will follow that J = ∆ E ( A ) obeys a Cesàro–Nevai condition. That Cesàro–Nevai condition will imply (1.13) by a modification of arguments of [25]. The strategy is clear:just as [25] uses a certain square-summability in terms of v j , w j to prove finiteness of ℓ -norm of { d (( S ∗ + ) m JS m + , T + E ) } ∞ m =0 , we will use Cesàro decay in terms of v j , w j to conclude the Cesàro decay(1.13). This can be expected due to a certain locality in the dependence between the terms of theseries considered; this idea first appeared in [19] in the setting of periodic spectra with all gapsopen. However, some care is needed, since the locality is only approximate in some steps; this isalready visible in (1.12). Also, substantial modifications are needed throughout the proof due to thepossibility of lim inf k v j k = 0 (this cannot happen in the Killip–Simon class), which locally breakssome of the estimates. The fix is that this can only happen along a sparse subsequence, but thecombination of a bad sparse subsequence and approximate locality means that we cannot simplyignore a bad subsequence once from the start; we must maintain it throughout the proof. Finally, itwill be natural to use a Hilbert–Schmidt type functional instead of a Killip–Simon type functionalused in [25]. We will describe these modifications and otherwise freely rely on the detailed analysisof [25].The rest of the paper will not exactly follow the order given in this introduction. In Section 2,we describe the behavior of our problem with respect to Möbius transformations, and we describethe distribution of zeros of the rational function τ n . In Section 3, we recall the structure of GMPmatrices and relate their matrix coefficients to the quantities κ n , and use this to provide a firststatement about exponential growth of orthonormal rational functions on C \ R . In Section 4,we combine this with potential theoretic techniques to characterize limits of n log | τ n | as n → ∞ and prove the universal lower bounds. In Section 5, we prove the results for C -regularity andStahl–Totik regularity. In Section 6 we describe a proof of Theorem 1.10.2. Orthonormal rational functions and Möbius transformations
In the introduction, starting from the measure µ and sequence of poles C , we defined a sequence { r n } ∞ n =0 and the orthonormal rational functions { τ n } ∞ n =0 . In the next statement, we will denotethese by r n ( z ; C ) and τ n ( z ; µ, C ), in order to state precisely the invariance of the setup with respectto Möbius transformations. Lemma 2.1. If f is a Möbius transformation which preserves R , then τ n ( z ; µ, C ) = ρ n τ n ( f ( z ); f ∗ µ, f ( C )) , (2.1) RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 9 where f ( C ) = ( f ( c ) , . . . , f ( c g +1 )) and ρ = ( +1 f ∈ PSL(2 , R ) − f ∈ (PSL(2 , R ) ⋊ { id , z
7→ − z } ) \ PSL(2 , R ) Proof.
Note that the sequence { r n } ∞ n =0 does not have this property: r n ( z ; C ) is not equal to ρ n r n ( f ( z ); f ( C )). However, if we denote L n ( C ) = span { r ℓ ( · ; C ) | ≤ ℓ ≤ n } , then it suffices to have r n ( f ( z ); f ( C )) − c n ρr n ( z ; C ) ∈ L n − ( C ) (2.2)for some constants c n >
0. If (2.2) holds, then applying the Gram–Schmidt process to the sequences { r n ( f ( z ); f ( C )) } ∞ n =0 and { r n ( z ; C ) } ∞ n =0 will give the same sequence of orthonormal functions, upto the sign change ρ n , which is precisely (2.1).Note that, if (2.1) holds for f , f , it holds for their composition, so it suffices to verify (2.2) fora set of generators of PSL(2 , R ) ⋊ { id , z
7→ − z } . In particular, (2.2) is checked by straightforwardcalculations for affine transformations and for the inversion f ( z ) = − /z , which implies the generalstatement since affine maps and inversion generate PSL(2 , R ) ⋊ { id , z
7→ − z } . (cid:3) Let us emphasize what this lemma does and what it doesn’t do. Since the Möbius transformationacts on both the measure and the sequence of poles, Lemma 2.1 does not by itself prove Theorem 1.1.Lemma 2.1 can only say that if µ is Stahl–Totik regular, then f ∗ µ is ( f ( ∞ ))-regular, which is notsufficient unless f is affine. The proof of Theorem 1.1 will be more involved.However, Lemma 2.1 provides a very useful conformal invariance for many of our proofs. Thiscan be compared to choosing a convenient reference frame. Since potential theoretic notions suchas Green functions are conformally invariant, our results will be invariant with respect to Möbiustransformations. We will often use this invariance in the proofs to fix a convenient point at ∞ ; notethat technical ingredients of the proof, such as polynomial factorizations, give a preferred role to ∞ so they break symmetry. For instance, we will often use the observation that the subspace L n can be represented as L n = (cid:26) PR n | P ∈ P n (cid:27) (2.3)for some suitable polynomial R n with factors which account for finite poles c k = ∞ . We will usethe representation (2.3) after placing a convenient point at ∞ . This idea is already seen in the nextproof. Lemma 2.2.
All zeros of the rational function τ n are simple and lie in R . Moreover, n − g ≤ deg τ n ≤ n .Let n = j ( g + 1) + k , ≤ k ≤ g + 1 , and denote by I the connected component of c k in R \ supp µ .Then τ n has no zeros in I and at most one zero in any other connected component of R \ supp µ .Proof. Fix 1 ≤ k ≤ g +1 and without loss of generality, assume c k = ∞ . Then, in the representations(2.3), we can notice that R n − = R n . In particular, then τ n ∈ L n \ L n − implies the representation τ n ( j ) = P n R n for some polynomial P n of degree n .Recall that τ n , n = k + ( j − g + 1) is the unique minimizer for the extremal problem (1.5).By complex conjugation symmetry, the minimizer is real. To proceed further, we study zeros of P n by using Markov correction terms. RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 10
We say that a rational function M is an admissible Markov correction term if M > E and M ( z ) P n ( z ) ∈ P n − . In this case, using h M τ n , τ n i >
0, we see that the function g ( ǫ ) = k τ n − ǫM τ n k obeys g ′ (0) = − h M τ n , τ n i < . Thus, for small enough ǫ >
0, the function˜ τ n = τ n − ǫM τ n obeys k ˜ τ n k L ( dµ ) < k τ n k L ( dµ ) . Since ˜ τ n is of the form ˜ τ n = κ n z j +1 + h ( z ) for some h ( z ) ∈ L n − and in particular has the same leading coefficient as τ n , the function ˜ τ n / k ˜ τ n k L ( dµ ) ∈ L n contradictsextremality of τ n . In other words, for the extremizer τ n , there cannot be any admissible Markovcorrection terms.Assume that P n has a non-real zero w ∈ C \ R . Then, since τ n is real, P n ( w ) = 0, so the Markovcorrection term M ( z ; w ) = z − w )( z − ¯ w ) would be admissible, leading to contradiction.Assume that P n has two zeros x , x in the same connected component of R \ supp µ ; then, theMarkov correction term M ( z ; x , x ) = 1( z − x )( z − x ) , would be admissible, leading to contradiction.There are no zeros of P n in I . Otherwise, if x ∈ I was a zero, the Markov term M ( z, x ) = ( z − x , x < inf E x − z , x > sup E would be admissible.Finally, all zeros of P n are simple: otherwise, if x ∈ R was a double zero, the Markov term M ( z, x ) = 1( z − x ) would be admissible.The properties of zeros of τ n follow from those of P n . There may be cancellations in the repre-sentation τ n = P n R n , but since P n has at most a simple zero at c ℓ , the only possible cancellations aresimple factors ( z − c ℓ ), ℓ = k . Thus, n − g ≤ deg τ n ≤ n . (cid:3) The use of Markov correction factors is standard in the Chebyshev polynomial literature and isapplied here with a modification for the L -extremal problem (in the L ∞ -setting, singularities in M are treated with a separate argument near the singularity, which would not work here). Corollary 2.3.
The measures ν n are a precompact family with respect to weak convergence on C ( R ) . Any accummulation point ν = lim ℓ →∞ ν n ℓ is a probability measure and supp ν ⊂ E .Proof. By Lemma 2.2, ν n ( R ) ≤
1, so precompactness follows by the Banach-Alaoglu theorem. If ν = lim ℓ →∞ ν n ℓ , then since 1 − gn ℓ ≤ ν n ℓ ( R ) ≤ ν ( R ) = 1.Let ( a , b ) be a connected component of R \ E . Let us prove that ν (( a , b )) = 0. By Möbiusinvariance, it suffices to assume that ( a, b ) is a bounded subset of R .Fix r ∈ N . As supp µ \ E is a discrete set, we have { x ∈ supp µ : a + 1 /r < x < b − /r } = M < ∞ . So, by Lemma 2.2, ν n ℓ (( a + 1 /r, b − /r )) ≤ M +1 n ℓ and by the Portmanteau theorem and sending r → ∞ , ν (( a , b )) = 0 and supp ν ⊂ E . (cid:3) RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 11 GMP matrices and exponential growth of orthonormal rational functions
In this section, we consider orthonormal rational functions through the framework of GMPmatrices. We begin by recalling the structure of GMP matrices [25]. The GMP matrix has atridiagonal block matrix structure, with the beginnings of new blocks corresponding to occurrencesof c k = ∞ . Explicitly, A = B A A ∗ B A A ∗ B A A ∗ . . . . . .. . . where B is a k × k matrix, A is a k × ( g + 1) matrix, and A j , B j for j ≥ g + 1) × ( g + 1)matrices. We will also index the entries of these matrices from 0 to g . Let X − denote the uppertriangular part of a matrix X (excluding the diagonal) and X + the lower triangular part (includingthe diagonal). Then for j ≥ A j , B j are of the form A j = ~p j ~δ ⊺ , B j = ˜ C + ( ~q j ~p ⊺ j ) + + ( ~p j ~q ⊺ j ) − , (3.1)where ~p j , ~q j ∈ R g +1 , with ( ~p j ) > C = diag { , c k +1 , . . . , c g +1 , c , . . . , c k − } (with theobvious modification of k = 1 or k = g + 1) and ~δ denotes the standard first basis vector of R g +1 .We will refer to { ~p j , ~q j } as the GMP coefficients of A . While the precise structure will not beessential throughout the paper, we point out two things. First on the outermost diagonal of A ineach block there is only one non-vanishing entry, given by ( ~p j ) , which is positive and which is ata different position depending on the position of ∞ in the sequence C . And secondly, in general asa self-adjoint matrix B j could depend on ( g + 1)( g + 2) / g + 1). This is not that surprising due to their close relation to three-diagonalJacobi matrices. A similar phenomena also appears for their unitary analogs [5].Now the various notations for the off-diagonal blocks A j , the vectors ~p j which determine them,and the coefficients β j defined in (1.8) are related as β j = h e j ( g +1)+ k , Ae ( j +1)( g +1)+ k i = ( A j ) = ( ~p j ) . Recall that k is fixed here so that c k = ∞ . The coefficients β j are a special case of the coefficients Λ n = ( h e j ( g +1)+ k , ( c k − A ) − e ( j +1)( g +1)+ k i c k = ∞h e j ( g +1)+ k , Ae ( j +1)( g +1)+ k i c k = ∞ (3.2)where β j = Λ j ( g +1)+ k , and the coefficients Λ j ( g +1)+ ℓ for ℓ = k instead occur as outermost diagonalcoefficients for the GMP matrix ( c ℓ − A ) − . In our later applications to the discriminant to A , boththe coefficients of A and of its resolvents will appear, so we will work with Λ n throughout.Next, we connect the coefficients (3.2) to the solutions of the L -extremal problem (1.5). Lemma 3.1.
For all n , κ n κ n + g +1 = Λ n . (3.3) Proof.
Let n = j ( g + 1) + k . By self-adjointness, Λ n = h e n , r k ( A ) e n + g +1 i = h r k τ n , τ n + g +1 i = h κ n r n + g +1 + h, τ n + g +1 i RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 12 for some h ∈ L n + g . By orthogonality, h τ n + g +1 , h i = 0, so h τ n + g +1 , r n + g +1 i = κ n + g +1 implies that Λ n = h τ n + g +1 , κ n r n + g +1 + h i = κ n κ n + g +1 . (cid:3) We now adapt to GMP matrices ideas from the theory of regularity for Jacobi matrices [18].
Lemma 3.2.
Let A ∈ A ( C ) . For all j ≥ , k ~p j k ≤ k A k .Proof. Fix k such that c k = ∞ . Fix j ≥ n = j ( g + 1) + k . For any ℓ = 0 , . . . , g ,( p j ) ℓ = h e n − g − ℓ , Ae n i = Z τ n − g − ℓ ( x ) xτ n ( x ) dµ ( x ) . Since the vectors τ n − g − ℓ are orthonormal, by the Bessel inequality, k ~p j k ≤ Z | xτ n ( x ) | dµ ( x ) ≤ k A k Z | τ n ( x ) | dµ ( x ) = k A k since k A k = sup x ∈ supp µ | x | . (cid:3) Lemma 3.3.
For z ∈ C \ R , lim inf n →∞ n log | τ n ( z ) | > . (3.4) Proof.
We adapt the proof of [18, Proposition 2.2]. It suffices to prove (3.4) along the subsequences n ( j ) = j ( g + 1) + k , j → ∞ , for 1 ≤ k ≤ g + 1. Moreover, due to R -preserving conformal invariance,it suffices to fix k and prove lim inf j →∞ n ( j ) log | τ n ( j ) ( z ) | > c k = ∞ . This allows us to use the associated GMP matrix A ∈ A ( C ).Note that for any m , since { τ ℓ } ∞ ℓ =0 is an orthonormal basis of L ( dµ ), X ℓ A mℓ τ ℓ ( z ) = X ℓ h zτ m ( z ) , τ ℓ ( z ) i τ ℓ ( z ) = zτ m ( z ) . This equality holds in L ( dµ ), but since all functions are rational, it also holds pointwise. Thus,if we fix z ∈ C \ R , the sequence ~ϕ = { τ ℓ ( z ) } ∞ ℓ =0 is a formal eigensolution for A at energy z , i.e.( A − z ) ~ϕ = 0 componentwise. Since A is represented as a block tridiagonal matrix, let us also write ~ϕ in a matching block form, as ~ϕ ⊤ = (cid:2) ~u ⊤ ~u ⊤ ~u ⊤ . . . (cid:3) where ~u ⊤ = (cid:2) τ ( z ) . . . τ k − ( z ) (cid:3) , ~u ⊤ j = (cid:2) τ n ( j − − ( z ) . . . τ n ( j ) − ( z ) (cid:3) , j ≥ . We also consider the projection of ~ϕ onto the first j + 1 blocks, ~ϕ ⊤ j = (cid:2) ~u ⊤ . . . ~u ⊤ j . . . (cid:3) , and compute ( A − z ) ~ϕ j . By the block tridiagonal structure of A , for m < n ( j −
1) we have h e m , ( A − z ) ~ϕ j i = 0. For 0 ≤ ℓ ≤ g , we have h e n ( j − ℓ , ( A − z ) ~ϕ i − h e n ( j − ℓ , ( A − z ) ~ϕ j i = ( p j ) ℓ τ n ( j ) ( z )so that h e n ( j − ℓ , ( A − z ) ~ϕ j i = − ( p j ) ℓ τ n ( j ) ( z ). Moreover, h e n ( j ) , ( A − z ) ~ϕ j i = h e n ( j ) , A~ϕ j i = ( ~p j ) ∗ u j ( z ) . For m > n ( j ), we again have h e m , ( A − z ) ~ϕ j i = 0. In conclusion, ( A − z ) ~ϕ j has only two nontrivialblocks, (( A − z ) ~ϕ j ) ⊤ = (cid:2) . . . − ( ~p j τ n ( j ) ( z )) ⊤ (( ~p j ) ∗ u j ) ⊤ . . . (cid:3) . RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 13
In particular, we can compute h ~ϕ j , ( A − z ) ~ϕ j i = − ~u ∗ j τ n ( j ) ( z ) ~p j . (3.6)Since A is self-adjoint and ~ϕ j ∈ ℓ ( N ), by a standard consequence of the spectral theorem [23,Lemma 2.7.], | Im z |k ~ϕ j k ≤ |h ~ϕ j , ( A − z ) ~ϕ j i| . Using (3.6) and the Cauchy–Schwarz inequality gives | Im z | j X m =0 k ~u m k ≤ | τ n ( j ) ( z ) |k ~p j kk ~u j k . By Lemma 3.2, with C = | Im z | / k A k , C j X m =0 k ~u m k ≤ | τ n ( j ) ( z ) |k ~u j k . (3.7)Applying the AM-GM inequality to the right-hand side of (3.7) gives | τ n ( j ) ( z ) |k ~u j ( z ) k ≤ (cid:0) C k ~u j ( z ) k + C − | τ n ( j ) ( z ) | (cid:1) which together with (3.7) implies | τ n ( j ) ( z ) | ≥ C j X m =0 k ~u m k . (3.8)Since | τ n ( j ) ( z ) | ≤ k ~u j +1 k , this implies that j +1 X m =0 k ~u m k ≥ (cid:0) C (cid:1) j X m =0 k ~u m k . Since k ~u k ≥ | τ ( z ) | = 1, this implies by induction that j X m =0 k ~u m k ≥ (cid:0) C (cid:1) j . Combining this with (3.8) gives a lower bound on | τ n ( j ) ( z ) | which implies (3.5). (cid:3) The estimates in the previous proof also lead to the following:
Corollary 3.4.
For any z ∈ C \ R , the quantities lim inf j →∞ j ( g + 1) + k log | τ j ( g +1)+ k ( z ) | , lim sup j →∞ j ( g + 1) + k log | τ j ( g +1)+ k ( z ) | are independent of k ∈ { , . . . , g + 1 } .Proof. Assume j ≥
1. For k − g − ≤ ℓ ≤ k −
1, the estimate (3.8) gives | τ j ( g +1)+ k ( z ) | ≥ C k ~u j k ≥ C | τ j ( g +1)+ ℓ ( z ) | which implieslim inf j →∞ j ( g + 1) + k log | τ j ( g +1)+ k ( z ) | ≥ lim inf j →∞ j ( g + 1) + ℓ log | τ j ( g +1)+ ℓ ( z ) | (3.9) RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 14 and lim sup j →∞ j ( g + 1) + k log | τ j ( g +1)+ k ( z ) | ≥ lim sup j →∞ j ( g + 1) + ℓ log | τ j ( g +1)+ ℓ ( z ) | . (3.10)Clearly, the right-hand sides don’t change if ℓ is shifted by g + 1, so (3.9), (3.10) hold for all k, ℓ ∈ { , . . . , g + 1 } with k = ℓ . By symmetry, since the roles of k, ℓ can be switched, we concludethat equality holds in (3.9), (3.10). (cid:3) Growth rates of orthonormal rational functions
In this section, we will combine the positivity (3.4) with potential theory techniques in order tostudy exponential growth rates of orthonormal rational functions. Our main conclusions will beconformally invariant, but our proofs will use potential theory arguments and objects such as thelogarithmic potential of a finite measure ν , Φ ν ( z ) = Z log | z − x | dν ( x ) , which is well defined when supp ν does not contain ∞ . Theorem 4.1.
Fix ≤ k ≤ g + 1 and denote by I the connected component of R \ supp µ containing c k . Suppose there is a subsequence n ℓ = j ℓ ( g + 1) + k such that w-lim ℓ →∞ ν n ℓ = ν and n ℓ log κ n ℓ → α ∈ R ∪ {−∞ , + ∞} as ℓ → ∞ . Then uniformly on compact subsets of ( C \ R ) ∪ ( I \ { c k } ) , we have h ( z ) := lim ℓ →∞ n ℓ log | τ n ℓ ( z ) | . The function h is determined by ν and α ; in particular, if c k = ∞ , h ( z ) = α + Φ ν ( z ) − g + 1 g +1 X m =1 m = k log | c m − z | . (4.1) Moreover,(a) α = −∞ is impossible;(b) If α = + ∞ , the limit is h = + ∞ ;(c) If α ∈ R , the limit h extends to a positive harmonic function on C \ ( E ∪ { c , . . . , c g +1 } ) suchthat h ( z ) = − g + 1 log | c m − z | + O (1) , z → c m = ∞ (4.2) h ( z ) = 1 g + 1 log | z | + O (1) , z → c m = ∞ . (4.3) Proof.
By using R -preserving conformal invariance, we can assume without loss of generality that c k = ∞ . We will use the representation (2.3) of the subspace L n . For n = j ( g + 1) + k , countingdegrees of the poles leads to τ n = P n R n , R n ( z ) = k − Y m =1 ( c m − z ) g +1 Y m =1 m = k ( c m − z ) j , RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 15 with deg P n = n . This may not be the minimal representation of τ n , but by the proof of Lemma 2.2,the only possible cancellations are simple factors ( c m − z ) for each m = k , so we get the minimalrepresentation τ n ( z ) = P ( z ) /Q ( z ) with P ( z ) = κ n Y w : τ n ( w )=0 ( z − w ) , Q ( z ) = g +1 Y m =1 m = k ( c m − z ) j + δ m,j where | δ m,j | ≤ j . All that matters is that δ m,j /j → j → ∞ . It will be useful to turnthis rational function representation into a kind of Riesz representation,log | τ n ( z ) | = log κ n + n Z log | x − z | dν n ( x ) − X ≤ m ≤ g +1 m = k ( j + δ m,j ) log | c m − z | . (4.4)Since c k = ∞ , note that K = R \ I is a compact subset of R . Denote Ω = C \ K . For any z ∈ Ω ,the map x log | x − z | is continuous on K , so Φ ν nℓ ( z ) → Φ ν ( z ) as ℓ → ∞ . In fact, convergence isuniform on compact subsets of Ω : since supp( ν n ℓ ) ⊂ K and ν n ℓ ( K ) ≤ ℓ , the estimatelog (cid:12)(cid:12)(cid:12)(cid:12) x − z x − z (cid:12)(cid:12)(cid:12)(cid:12) ≤ log (cid:18) | z − z | dist( z , K ) (cid:19) ≤ | z − z | dist( z , K ) , z , z ∈ Ω implies uniform equicontinuity of the potentials Φ n ℓ on compact subsets of Ω , and the Arzelà–Ascolitheorem implies uniform convergence on compacts.Note that (b) follows from (4.1). By Corollary 2.3, supp ν ⊂ E and Φ ν ( z ) is harmonic on C \ E ,so the right hand side extends to a harmonic function on C \ ( E ∪ { c , . . . , c g +1 } ) and we denotethis extension also by h . By Lemma 3.3, h is positive on C + ∪ C − , so α = −∞ ; moreover, by themean value property, h is positive on C \ ( E ∪ { c , . . . , c g +1 } ).The remaining asymptotic properties follow from (4.1). Under the assumption c k = ∞ , supp ν is a compact subset of R , and Φ ν ( z ) = log | z | + O (1), z → ∞ . It then follows that h ( z ) = g +1 log | z | + O (1) as z → ∞ . Of course, h ( z ) = − g +1 log | z − c m | + O (1) near each c m = c k . (cid:3) The previous theorem motivates interest in positive harmonic functions on C \ ( E ∪{ c , . . . , c g +1 } ).If E is polar, by Myrberg’s theorem [1, Theorem 5.3.8], any such function is constant. If E is notpolar, knowing the asymptotic behavior of h at the poles, positivity of h improves to the followinglower bound on h . The following Lemma reflects a standard minimality property of the Greenfunction [9, Section VII.10]. Lemma 4.2.
Assume that E is a nonpolar closed subset of R . Let h be a positive superharmonicfunction on C \ ( E ∪ { c , . . . , c g +1 } ) . Suppose h ( z ) + g +1 log | z − c k | has an existent limit at c k foreach finite c k , and h ( z ) − g +1 log | z | has an existent limit at ∞ if one of the c k = ∞ . Then h ( z ) ≥ G E ( z, C ) (4.5) for z ∈ C \ E . For ≤ k ≤ g + 1 , define α k = lim z → c k ( h ( z ) + g +1 log | z − c k | ) , c k = ∞ lim z →∞ ( h ( z ) − g +1 log | z | ) , c k = ∞ Then α k ≥ log λ k g + 1 (4.6) RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 16
Proof.
We will use a stronger, q.e. version of the maximum principle [16, Thm 3.6.9]. Define˜ h ( z ) := G E ( z, C ) − h ( z ) , which is bounded at c k for 1 ≤ k ≤ g + 1 and so extends to a subharmonic function on C \ E . Since G E vanishes q.e. on E , we have for q.e. t ∈ E ,lim sup z → t ˜ h ( z ) = − lim inf z → t h ( z ) ≤ . Now we show ˜ h is bounded above on C \ E . Let U be a union of small neighborhoods containingthe points c k in C \ E . By the definition of the Green function, G E ( z, C ) defines a harmonic andbounded function on C \ ( E ∪ U ). That is, there exists M such that for all z ∈ C \ ( U ∪ E ) we have G E ( z, C ) ≤ M. Since h ≥
0, it follows on C \ ( U ∪ E ) that˜ h ( z ) = G E ( z, C ) − h ( z ) ≤ G E ( z, C ) ≤ M. On the other hand, by properties of the Green functions we havelog λ k g + 1 = lim z → c k ( G E ( z, C ) + g +1 log | z − c k | ) , c k = ∞ lim z →∞ ( G E ( z, C ) − g +1 log | z | ) , c k = ∞ Then, by assumption, for 1 ≤ k ≤ g + 1, ˜ h ( z ) = log λ k g +1 − α k + o (1) as z → c k and, in particular, thedifference is bounded in a small neighborhood of c k . Thus, ˜ h is bounded above on C \ E .So, by the maximum principle ˜ h ≤ ⇒ G E ( z, C ) ≤ h ( z ) on C \ E . Since 0 ≥ lim z → c k ˜ h ( z ) = log λ k g +1 − α k , we have (4.6). (cid:3) Lemma 4.3.
Under the same assumptions as Lemma 4.2, the following are equivalent:(i) Equality in (4.6) for all k with ≤ k ≤ g + 1 (ii) Equality in (4.6) for a single k with ≤ k ≤ g + 1 (iii) Equality holds in (4.5) Proof. (i) = ⇒ (ii) is trivial. Suppose then (ii); with the notation of the previous lemma, by as-sumption, ˜ h ( c k ) = 0 and ˜ h achieves a global maximum. By the maximum principle for subharmonicfunctions [16, Theorem 2.3.1], ˜ h ≡ C \ E . Finally, if (iii) holds, then evaluating ˜ h ( c k ) for each1 ≤ k ≤ g + 1 yields (i). (cid:3) We will now prove Theorems 1.2 and 1.3.
Proof of Theorem 1.2.
Using conformal invariance, we take c k = ∞ . Fix z ∈ C \ R and select asequence ( n ℓ ) ∞ ℓ =1 such that lim inf n →∞ n log | τ n ( z ) | = lim ℓ →∞ n ℓ log | τ n ℓ ( z ) | . By precompactness of the ( ν n ), we may pass to a further subsequence, which we denote againby ( n ℓ ) ∞ ℓ =1 , so that w-lim ℓ →∞ ν n ℓ = ν and n ℓ log κ n ℓ → α for some ν and α . Then for h as inTheorem 4.1, lim ℓ →∞ n ℓ log | τ n ℓ ( z ) | = h ( z ) . RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 17 on C \ R . If α = + ∞ , then there is nothing to show. Suppose α < ∞ . If E is not polar we apply (a)of Theorem 4.1 to find α ∈ R , and we may use (c) of the same theorem and Lemma 4.2 to conclude.If instead E is polar, by Myrberg’s theorem, h is constant on C \ ( E ∪ { c , . . . , c g +1 } ). Computingthe limit at c k we see h ≡ + ∞ . In particular, lim inf n →∞ n log | τ n ( z ) | = + ∞ for z ∈ C \ R . (cid:3) Proof of Theorem 1.3.
We fix 1 ≤ k ≤ g +1 and assume again by conformal invariance that c k = ∞ .Using precompactness of the measures ( ν n ), we find a subsequence n ℓ = j ℓ ( g + 1) + k withlim ℓ →∞ n ℓ log κ n ℓ = lim inf j →∞ n ( j ) log κ n ( j ) =: α and w-lim ℓ →∞ ν n ℓ = ν . If α = + ∞ , we are done. Suppose then α < ∞ , then we have byTheorem 4.1 (a), α ∈ R . Furthermore, if E is nonpolar, by (c) and Lemma 4.2, h ( z ) ≥ G E ( z, C ) on C \ E . In particular, by the representation (4.1) we see that α = lim z →∞ ( h ( z ) − g +1 log | z | ), andso (4.6) yields the desired inequality.If instead E is polar, by Theorem 1.2, for each z ∈ C \ R , h ( z ) = lim ℓ →∞ n ℓ log | τ n ℓ | ≥ lim inf n →∞ n log | τ n ( z ) | = + ∞ . and so by Theorem 4.1 (b), α = + ∞ . (cid:3) Regularity
We will begin by proving a version of Theorem 1.4 for a fixed k . Lemma 5.1.
Fix k ∈ { , . . . , g + 1 } . Along the subsequence n ( j ) = j ( g + 1) + k , the following areequivalent:(i) lim j →∞ κ /n ( j ) n ( j ) = λ / ( g +1) k ;(ii) For q.e. z ∈ E , we have lim sup j →∞ | τ n ( j ) ( z ) | /n ( j ) ≤ ;(iii) For some z ∈ C + , lim sup j →∞ | τ n ( j ) ( z ) | /n ( j ) ≤ e G E ( z, C ) ;(iv) For all z ∈ C , lim sup j →∞ | τ n ( j ) ( z ) | /n ( j ) ≤ e G E ( z, C ) ;(v) Uniformly on compact subsets of C \ R , lim j →∞ | τ n ( j ) ( z ) | /n ( j ) = e G E ( z, C ) .Proof. Using conformal invariance, we will assume throughout the proof that c k = ∞ . First,suppose that E is polar. In this case (ii) is vacuous, and since G E ≡ + ∞ , (iii) and (iv) are triviallytrue. Since λ k = + ∞ , (i) follows from Theorem 1.3. As in the proof of Theorem 4.1, weakconvergence of measures implies uniform on compacts convergence of their potentials. Thus, since ν n are a precompact family, so are Φ ν n . Thus, the convergence lim j →∞ n ( j ) log κ n ( j ) = + ∞ impliesthat lim j →∞ n ( j ) log | τ n ( j ) ( z ) | = + ∞ uniformly on compact subsets of C \ R , so (v) holds.For the remainder of the proof, we will assume E is not polar. Moreover, we will repeatedlyuse the fact that if any subsequence of a sequence in a topological space has a further subsequencewhich converges to a limit, then the sequence itself converges to this limit. In particular, whenconcluding (v), we apply this fact in the Fréchet space of harmonic functions on C \ R with thetopology of uniform convergence on compact sets.(iii) = ⇒ (v): Given a subsequence of n ( j ) = j ( g +1)+ k , using precompactness of the measures ν n ,we pass to a further subsequence n ℓ = j ℓ ( g +1)+ k with w-lim ℓ →∞ ν n ℓ = ν and lim ℓ →∞ n ℓ log κ n ℓ =: α , with α real or infinite. By Theorem 4.1, uniformly on compact subsets of C \ R , h ( z ) = lim ℓ →∞ n ℓ log | τ n ℓ ( z ) | RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 18 with h given by (4.1). Using the assumption, for some z ∈ C + , we have h ( z ) ≤ lim sup j →∞ n ( j ) log | τ n ( j ) ( z ) | < ∞ . So, by Theorem 4.1, α ∈ R and h has a harmonic extension to C \ ( E ∪{ c , . . . , c g +1 } ). Furthermore,by Lemma 4.2, h ≥ G E . By assumption, we have the opposite inequality at z ∈ C + , and so, by themaximum principle for harmonic functions, h = G E on C \ ( E ∪ { c , . . . , c g +1 } ), and in particularon C \ R . Thus, we have (v).(v) = ⇒ (iv): For z ∈ { c , . . . , c g +1 } , G E ( z, C ) = + ∞ and there is nothing to show. Fix z ∈ C \ { c , . . . , c g +1 } and let n ℓ = j ℓ ( g + 1) + k be a subsequence with lim ℓ →∞ n ℓ log | τ n ℓ ( z ) | =lim sup j →∞ n ( j ) log | τ n ( j ) ( z ) | . By passing to a further subsequence, we may assume w-lim ℓ →∞ ν n ℓ = ν , and lim ℓ →∞ n ℓ log κ n ℓ =: α where α is real or infinite. By the assumption, we have h =lim ℓ →∞ n ℓ log | τ n ℓ | = G E on C \ R . So, by (a) and (b), α ∈ R and h extends to a harmonic functionon C \ ( E ∪ { c , . . . , c g +1 } ). By the representation (4.1), we may extend h subharmonically to C \ { c , . . . , c g +1 } . On this set, G E is also subharmonic, so, by the weak identity principle [16,Theorem 2.7.5], h = G E on C \ { c , . . . , c g +1 } . Thus, by the principle of descent [21, A.III], we havelim ℓ →∞ n ℓ log | τ n ℓ ( z ) | ≤ h ( z ) = G E ( z, C ) (5.1)and (iv) follows.(v) = ⇒ (i): Given a subsequence of n ( j ) = j ( g + 1) + k , we use precompactness of the ν n topass to a further subsequence n ℓ = j ℓ ( g + 1) + k with lim ℓ →∞ n ℓ log κ n ℓ =: α ∈ R ∪ {−∞ , + ∞} andw-lim ℓ →∞ ν n ℓ = ν . Then in the notation of Theorem 4.1 and by assumption, for a z ∈ C \ R lim ℓ →∞ log | τ n ℓ ( z ) | = h ( z ) = G E ( z, C ) . So by, Lemma 4.3, α = log λ k g +1 . Thus, λ / ( g +1) k is the only accummulation point of κ /n ( j ) n ( j ) in R ∪ {−∞ , + ∞} and we have (i).(i) = ⇒ (v): As before, we fix a subsequence of n ( j ) = j ( g + 1) + k and use precompactness topass to a further subsequence n ℓ = j ℓ ( g + 1) + k with w-lim ℓ →∞ ν n ℓ = ν . Then, by Theorem 4.1and in the notation introduced there, uniformly on compact subsets of C \ R ,lim ℓ →∞ n ℓ log | τ n ℓ ( z ) | = h ( z )where h is given by (4.1) with α = log λ k g +1 . Thus, by Lemma 4.3 (ii), h ( z ) = G E ( z, C ) on C \ R . Sincethe initial subsequence was arbitrary, we have (v).(iv) = ⇒ (ii): Recalling that the Green function vanishes q.e. on E , the claim follows.(ii) = ⇒ (v): Fixing a subsequence of n ( j ), we again use precompactness to select a furthersubsequence n ℓ = j ℓ ( g + 1) + k such that w-lim ℓ →∞ ν n ℓ = ν and lim ℓ →∞ n ℓ log κ n ℓ =: α , α ∈ R ∪ {−∞ , + ∞} . By the upper envelope theorem, there is a polar set X ⊂ C such that on C \ X ,lim sup ℓ →∞ Φ ν nℓ = Φ ν . Now, we let X := { t ∈ E : lim sup n →∞ n log | τ n ( t ) | > } , which is polar byassumption, and X := { z ∈ C : Φ ∞ ( z ) = −∞} , which is polar by [16, Theorem 3.5.1]. Then, for a t ∈ E \ ( X ∪ X ∪ X ), we have α ≤ lim sup n →∞ n log | τ n ( t ) | − Φ ν ( t ) + 1 g + 1 g +1 X m =1 m = k log | c m − t | < ∞ . RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 19 So α ∈ R by Theorem 4.1 (a). Thus, by (c) of the same theorem, uniformly on compact subsets of C \ R h ( z ) = lim ℓ →∞ n ℓ log | τ n ℓ ( z ) | and h extends to a positive harmonic function on C \ ( E ∪ { c , . . . , c g +1 } ) with logarithmic polesat each of the c m . So, h − G E extends to a harmonic function on C \ E , and h − G E ≥ h = G E using the stronger, q.e. maximum principle.We use the equality in (4.1) to extend h to a subharmonic function on C \ { c , . . . , c g +1 } . Bythe upper envelope theorem and the assumption again, for t ∈ E \ ( X ∪ X ) h ( t ) = lim sup ℓ →∞ n ℓ log | τ n ℓ ( t ) | ≤ . Then, for these t , since G E is positive, we havelim sup z → tz ∈ C \ E ( h ( z ) − G E ( z, C )) ≤ lim sup z → tz ∈ C \ E h ( z ) ≤ h ( t ) ≤ z → tz ∈ C \ E ( h ( z ) − G E ( z, C )) ≤ t ∈ E .Since h is upper semicontinuous on the compact set E , there is an M so that sup t ∈ E h ( t ) ≤ M .As in the above, now for any t ∈ E , we havelim sup z → tz ∈ C \ E ( h ( z ) − G E ( z, C )) ≤ lim sup z → tz ∈ C \ E h ( z ) ≤ h ( t ) ≤ M. So, there is a neighborhood U of E with sup z ∈U∩ ( C \ E ) ( h − G E ) ≤ M + 1. Since the difference isharmonic on C \ U , we conclude that sup z ∈ C \ E ( h ( z ) − G E ( z, C )) < ∞ . Thus, by the maximumprinciple and the reverse inequality, h = G E on C \ E . Since the first sequence was arbitrary, wehave (v).Since the implication (iv) = ⇒ (iii) is clear, we may conclude. (cid:3) We now put the subsequences together and use Corollary 3.4 to show that regular behavioroccurs for one k if and only if it happens for all. Proof of Theorem 1.4.
Applying Lemma 5.1 for all k implies equivalence of conditions (ii), (iv), (v),(vi), (vii) from Theorem 1.4. By Corollary 3.4, for some z ∈ C + , the conditionlim sup j →∞ j ( g + 1) + k log | τ j ( g +1)+ k ( z ) | ≤ G E ( z, C )holds for one value of k if and only if it holds for all. Due to Lemma 5.1, this immediately impliesequivalence of conditions (i) and (iii) from Theorem 1.4. It remains to prove equivalence of (ii),(iii).(ii) = ⇒ (iii): For n ∈ N and 1 ≤ k ≤ g + 1, denote by N ( n, k ) the integer such that n + 1 ≤ N ( n, k ) ≤ n + g + 1 and N ( n, k ) − k is divisible by g + 1. Then N ( n, k ) /n → n → ∞ so (ii)implies lim n →∞ κ /nN ( n,k ) = λ / ( g +1) k . Taking the product over k = 1 , . . . , g + 1 gives (iii).(iii) = ⇒ (ii): Similarly to the above, Theorem 1.3 shows that for all k ,lim inf n →∞ κ /nN ( n,k ) ≥ λ / ( g +1) k . (5.2) RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 20
Thus, if (ii) was false, this would mean that for some k = m , lim sup n →∞ κ /nN ( n,m ) > λ / ( g +1) m .Taking products over k , we would havelim sup n →∞ g +1 Y k =1 κ N ( n,k ) ! /n ≥ lim sup n →∞ κ /nN ( n,m ) lim inf n →∞ Y ≤ k ≤ g +1 k = m κ N ( n,k ) ! /n > g +1 Y k =1 λ k ! / ( g +1) (the last step again uses (5.2) for all k = m ). This would contradict (iii), so the proof is complete. (cid:3) We now prove a seemingly special case of Corollary 1.7.
Proposition 5.2.
Assume that the sequence C contains ∞ . Then µ is Stahl–Totik regular if andonly if it is C -regular.Proof. Assume that µ is regular. To prove that µ is C -regular, we will use general results for theregular behavior of polynomials with respect to varying weights [21, Theorem 3.2.1(vi)]. Use thepolynomials R n defined by (2.3) and the representation τ n = P n R n where deg P n ≤ n . Consider themeasures dµ n ( x ) = 1 | R n ( x ) | dµ ( x ) (5.3)Since lim n →∞ | R n ( z ) | /n = Y ≤ m ≤ g +1 c m = ∞ | z − c m | uniformly on E and the limit is strictly positive and continuous on E , by [21, Theorem 3.2.1],regularity implies that lim sup n →∞ n log | R n ( z ) | − | P n ( z ) |k P n k L ( dµ n ) ≤ z ∈ E . Since k P n k L ( dµ n ) = k τ n k L ( dµ ) = 1, this implies thatlim sup n n log | τ n ( z ) | ≤ z ∈ E . Thus, µ is C -regular.Let us now assume that µ is C -regular and let p n denote the orthonormal polynomial withrespect to µ . Fix z ∈ C . Since ∞ is in C , p n ∈ L n ( g +1) , so the orthonormal polynomials can beexpressed in the basis of orthonormal rational functions as p n ( z ) = n ( g +1) X m =0 c m τ m ( z ) , n ( g +1) X m =0 | c m | = 1 . Thus, in particular, | c ℓ | ≤ | p n ( z ) | ≤ (1 + n ( g + 1)) sup ≤ m ≤ n ( g +1) | τ m ( z ) | . (5.4)By Theorem 1.4, for q.e. z ∈ E , lim sup ℓ →∞ ℓ log | τ ℓ ( z ) | ≤
0. Thus, for q.e. z ∈ E , (5.4) implieslim sup n →∞ n log | p n ( z ) | ≤ . Thus, µ is Stahl–Totik regular. (cid:3) RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 21
From this seemingly special case, Theorem 1.6, and Corollary 1.7 follow easily:
Proof of Theorem 1.6.
By applying a conformal transformation, the special case shows that µ is C -regular if and only if it is ( c k )-regular for any single c k in C . By applying this twice, weconclude that if C , C have a common element, then µ is C -regular if and only if it is C -regular.By applying that conclusion twice, we will finish the proof. Namely, for arbitrary C , C , choosea sequence C which has common elements with both C and C . Then µ is C -regular if and onlyif it is C -regular if and only if it is C -regular. (cid:3) Proof of Corollary 1.7.
The result follows by taking C = ( ∞ ) in Theorem 1.6. (cid:3) Proof of Theorem 1.1.
By Lemma 2.1, f ∗ µ is Stahl–Totik regular if and only if µ is ( f − ( ∞ ))-regular, and by Corollary 1.7, this is equivalent to Stahl–Totik regularity of µ . (cid:3) Proof of Theorem 1.8. (a) We note that by Corollary 1.7 we may use Theorem 1.4. Fix 1 ≤ k ≤ g + 1, and use conformal invariance to assume c k = ∞ . Given a subsequence of n ( j ) = j ( g + 1) + k ,we use precompactness to pass to a further subsequence n ℓ = j ℓ ( g + 1) + k with w-lim ℓ →∞ ν n ℓ = ν .We write G E ( z, C ) = Φ ρ E , C ( z ) + 1 g + 1 log λ k − g + 1 g +1 X m =1 m = k log | z − c m | (5.5)which we will use to show Φ ν = Φ ρ E , C . By (ii), we may apply Theorem 4.1 with α = g +1 log λ k .Then, (vii) yields h = G E off the real line, and thus the equality between the representations (4.1)and (5.5) gives Φ ν ( z ) = Φ ρ E , C ( z ) on C \ R . By the weak identity principle, this equality extends to C . Applying the distributional Laplacian to both sides gives ν = ρ E , C . Thus, w-lim n →∞ ν n = ρ E , C .(b) The main ingredient is a variant of Schnol’s theorem; for any n , R | τ n | dµ = 1, so ∞ X n =1 n − Z | τ n | dµ < ∞ . By Tonelli’s theorem, it follows that P ∞ n =1 n − | τ n | < ∞ µ -a.e., so there exists a Borel set B ⊂ C with µ ( C \ B ) = 0 such that lim sup n →∞ n log | τ n ( z ) | ≤ , ∀ z ∈ B. (5.6)Suppose µ is not regular. Then, by Theorem 1.4 (ii), there is a 1 ≤ k ≤ g + 1 withlim sup j →∞ n ( j ) log κ n ( j ) > g + 1 log λ k . Using conformal invariance, we may assume c k = ∞ , and we can pass to a subsequence n ℓ = j ℓ ( g + 1) + k such that α := lim ℓ →∞ n ℓ log κ n ℓ > g +1 log λ k , where α ∈ R ∪ { + ∞} by Theorem 4.1(a). Since w-lim n →∞ ν n = ρ E , C , by comparing (4.1) and (5.5), we have for z ∈ C \ R ,lim ℓ →∞ n ℓ log | τ n ℓ ( z ) | = G E ( z, C ) + d (5.7)where d = α − log λ k g +1 >
0. By the upper envelope theorem applied to the sequence { ν n ℓ } ℓ ∈ N , thereexists a polar set X such that (5.7) also holds for all z ∈ C \ X . Moreover, since G E ( z, C ) ≥ RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 22 all z ∈ C , we conclude thatlim sup n →∞ n log | τ n ( z ) | ≥ lim ℓ →∞ n ℓ log | τ n ℓ ( z ) | ≥ d, ∀ z ∈ C \ X. Comparing with (5.6) shows that B ⊂ X , so µ is supported on the polar set X . (cid:3) Proof of Theorem 1.9.
Defining n ( j ) = j ( g + 1) + k and using Lemma 3.1 to compute a telescopingproduct, j Y ℓ =1 β ℓ ! /j = j Y ℓ =1 κ n ( ℓ ) κ n ( ℓ +1) ! /j = κ /jn (1) κ − /jn ( j +1) . (5.8)The first term on the right-hand side is independent of j , so κ /jn (1) → j → ∞ . For the secondfactor, using Theorem 1.3 we compute lim inf j →∞ κ /jn ( j +1) ≥ λ k and we have the upper bound (1.9) for the lim sup of (5.8). Similarly, using the criterion Theorem 1.4(ii), it follows from (5.8) that µ is C -regular if and only if (1.10) holds. (cid:3) Cesàro-Nevai condition for finite gap sets
We begin by mentioning that in Theorem 1.10 we can assume without loss of generality that c k / ∈ σ ( J ). This follows from the following Lemma: Lemma 6.1.
Let J be a Jacobi matrix with σ ess ( J ) = E and let ˜ J = J + t h· , e i e . For all butfinitely many values of t ∈ R , ˜ J is a Jacobi matrix with σ ess ( ˜ J ) = E and c k / ∈ σ ( ˜ J ) for ≤ k ≤ g .Proof. For any t ∈ R , σ ess ( ˜ J ) = σ ess ( J ), so it suffices to ensure that c k / ∈ σ d ( ˜ J ). By coefficientstripping, if m ( z ) is the m function corresponding to ˜ J , then m ( z ) = 1 b + t − z − a m ( z )where m ( z ) is the m function for S ∗ + JS + . In particular, eigenvalues of ˜ J correspond to zeros of b + t − z − a m ( z ). Thus, for any t such that b + t − c k − a m ( c k ) = 0 for 1 ≤ k ≤ g , we have c k / ∈ σ ( ˜ J ) for 1 ≤ j ≤ g . (cid:3) It was noted in [25, Section 2.2] that − log | Ψ ( z ) | = g +1 X k =1 G E ( z, c k ) (6.1)and that the Yuditskii discriminant has the form (1.15) for some λ k > d ∈ R . Note that theconstants λ k can be found by computing the residue of ∆ E at the poles c k . By using (1.14) and(6.1), we find the residues to be the same constants λ k defined in a more general setting in (1.6). Proof of Lemma 1.11.
Denote by µ the canonical spectral measure for J . Note that σ ess ( A ) = ess supp µ = E = ∆ − E ([ − , . Since ∆ E maps R \ { c , . . . , c g } to R and is piecewise strictly monotone, by a spectral mappingtheorem, this implies that for J = ∆ E ( A ), σ ess ( J ) = [ − , J implies C E -regularity by Corol-lary 1.7, and this can be characterized in terms of GMP matrix coefficients by Theorem 1.9. The RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 23
GMP matrix structure together with (1.15) implies that J = ∆ E ( A ) is a type 3 block Jacobi matrix(1.16); the diagonal entries of the off-diagonal blocks v j are given by λ k Λ j ( g +1)+ k for k = 0 , . . . , g ,with the convention λ = λ g +1 . Thus,det v j = g Y k =0 λ k Λ j ( g +1)+ k . By applying the criterion for regularity in Theorem 1.9 to the GMP matrix A and to its resolvents( c k − A ) − , we conclude that J obeys (1.17). It follows that J is regular with σ ess ( J ) = [ − , (cid:3) In this section, in order to make precise statements, it will be necessary to also introduce full-lineGMP matrices, that is, operators acting on ℓ ( Z ). For this reason, we will denote half-line GMPmatrices by A + . If A + is such that σ ess ( A + ) = E and the corresponding measure is regular on E ,then ∆ E ( A + ) is a block Jacobi matrix which due to Lemma 1.11 is regular for [ − , { v ℓ , w ℓ } denote the block Jacobi coefficients of ∆ E ( A + ) , by [19, Theorem 3.1] we havelim N →∞ N N X ℓ =1 k v ℓ − I k + k w ℓ k = 0 . (6.2)We note that since C = sup ℓ ( k v ℓ ( A ) − I k + k w ℓ k ) < ∞ , it follows from Cauchy-Schwarz and theAM-GM inequality that N N X ℓ =1 k v ℓ − I k + k w ℓ k ! ≤ N N X ℓ =1 k v ℓ − I k + k w ℓ k ≤ C N N X ℓ =1 k v ℓ − I k + k w ℓ k and thus lim N →∞ N N X ℓ =1 k w ℓ k + k v ℓ − I k = 0 ⇐⇒ lim N →∞ N N X ℓ =1 k w ℓ k + k v ℓ − I k = 0 . (6.3)We will use this equivalence freely in the following.In the setting of periodic Jacobi matrices and polynomial discriminants (i.e., ∆ is a polynomialand { v ℓ , w ℓ } are the coefficients of the block Jacobi matrix ∆ ( J + )) it is shown in [6] that ∞ X ℓ =1 k w ℓ k + k v ℓ − I k < ∞ ⇐⇒ ∞ X m =1 d (( S ∗ + ) m JS m + , T + E ) < ∞ . (6.4)It was then stated in [19] that since all the arguments in [6] are local, in this setting (6.3) yields(1.13). Let us emphasize that finite gap sets whose isospectral torus consists of periodic Jacobimatrices are very special and the arguments in [19] only apply to this setting. Yuditskii [25] hasextended the work of [6] and one has the same localness, but since the construction is quite involved,we will sketch the main ideas of proof. In this case, the condition on the right-hand side of (6.4) isstill the same, i.e., a condition for a Jacobi matrix J + , but on the left-hand side { v ℓ , w ℓ } are thecoefficients of the block Jacobi matrix ∆ E ( A ), where A is an associated GMP matrix and ∆ E is therational function as defined in (1.15).We will start with the main ingredients of the proof that the left-hand side in (6.4) implies theright-hand side and mention certain modifications to our setting. After this preparatory work willshow how this can be applied to our setting. Recall that in the beginning of Section 3 we mentionedthat except certain initial conditions affecting the blocks A , B , all blocks of half-line GMP matricesare of the same structure (3.1). This structure can clearly be extended to an operator on ℓ ( Z ). We RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 24 denote this class of operators by A ( C , Z ). We then say an operator A ∈ A ( C , Z ) is a full-line GMPmatrix, if all the resolvents ( c ℓ − A ) − exist for any ℓ = k and ( c ℓ − A ) − ∈ A ( f ( C ) , Z ) where f isthe Möbius transform f : z ( c ℓ − z ) − and f ( C ) = ( f ( c ) , . . . , f ( c g +1 )). In this case we write A ∈ GMP( C , Z ) . Again we call the generating coefficients { ~p j , ~q j } j ∈ Z the GMP coefficients of A .We will also need the notion of the isospectral torus of periodic GMP matrices [10]. Let E bea finite gap set and C E be the collection of zeros of the Ahlfors function. We call a GMP matrix1-periodic or simply periodic if S g +1 AS − ( g +1) = A where S denotes the shift operator on ℓ ( Z ).We then define the isospectral torus of periodic GMP matrices by T E ( C E ) = { ˚ A ∈ A ( C E ) , A is periodic and σ ( A ) = E } . As for Jacobi matrices, one can show that the spectrum is purely absolutely continuous and ofmultiplicity 2. However, we point out that for arbitrary finite gap sets, the isospectral torus ofJacobi matrices usually consists of almost periodic operators, whereas for GMP matrices we canalways work with periodic operators. This also makes it possible to characterize the isospectraltorus by a magic formula for GMP matrices: let A ∈ GMP( C E , Z ), then A ∈ T E ( C E ) ⇐⇒ ∆ E ( A ) = S g +1 + S − ( g +1) , (6.5)where S denotes the right shift operator on ℓ ( Z ). We point out that if as in (1.16) J = ∆ E ( A )denotes the ( g + 1) × ( g + 1)-block Jacobi matrix with coefficients { v ℓ , w ℓ } ℓ ∈ Z , then (6.5) meansthat v ℓ ≡ I and w ℓ ≡ Λ k ( A ), for k = 0 , . . . g , denotes the outermost positive entry of the resolvent( c k − A ) − and is an algebraic expression in terms of the GMP coefficients { ~p j , ~q j } of A . Therefore,having in mind 1.15 and taking a look at the outermost diagonal of (6.5), we get that A ∈ T E ( C E )implies that Λ k ( A ) λ k = 1 . (6.6)Together with an additional equation related to the value of d in (1.15), see [10, Theorem 1.10]for details, this gives an description of T E ( C E ) as an algebraic manifold. That is, there existsan algebraic polynomial F E : R g +1) → R g +2 , so that the periodic GMP matrix A with GMPparameters { ~p, ~q } ∈ R g +1) lies in T E ( C E ) if and only if F E ( ~p, ~q ) = 0 . (6.7)We will denote this manifold by IS E . Considering the dimensions in the definition of F E it will notcome as a surprise that T E ∼ = R g / Z g , which can be rigorously seen from the parametrization (6.14).For a GMP matrix A ∈ GMP( C E , Z ) define the functional H + ( A ) = ∞ X ℓ =0 k v ℓ − − I k + k w ℓ k + k v ℓ − I k . (6.8)If P + denotes the orthogonal projection onto ℓ ( N ) and k · k HS the Hilbert-Schmidt norm, then wehave H + ( A ) = k P + ( ∆ E ( A ) − ( S g +1 + S − ( g +1) )) k . (6.9)A key observation is that the functional H + ( A ) is related to the shift action of S g +1 on the GMPmatrix A . But finally we want to conclude something about S J = S ∗ JS.
RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 25
This leads to the introduction of the Jacobi flow on GMP matrices. Due to [25, Proposition 5.5.]there is, up to a certain identification, a one-to-one correspondence between Jacobi matrices, J ,satisfying c k / ∈ σ ( J ) and GMP matrices A ∈ GMP( C E , Z ). Let us denote this mapping by F . Thenthe Jacobi flow on GMP( C E , Z ) is defined by the following commutative diagram:GMP J −→ GMP F y F y Jacobi S −→ Jacobi (6.10)This is one of the reasons why it is natural to work with full-line operators. If we would considerin this construction the shift action on ℓ ( N ), which is not unitary, then it may happen that forsome m , c k ∈ σ (( S ∗ + ) m J + S m + ) and thus the corresponding half-line GMP matrix would not bewell defined. It is shown in [25, Equation (4.8) and Lemma 4.4] that there exists a block-diagonalunitary mapping U A , such that J A = S − U ∗ A AU A S. (6.11)The following lemma, which follows essentially from (6.11), allows to compute the “derivative” inJacobi flow direction and is essential in order to extract from finiteness of H + ( A ) properties of theassociated Jacobi matrix J . Lemma 6.2.
Let δ J H + ( A ) = k ( ∆ E ( J A ) − ( S g +1 + S − ( g +1) )) e − k . Then H + ( A ) = H + ( J A ) + δ J H + ( A ) . (6.12)This is an analog of [25, Lemma 6.1], but note that there H + ( A ) was defined slightly differently.This formulation was natural in this setting, since it comes from spectral theoretical sum rules.Since we will start already with the condition (6.2), defining H + ( A ) by (6.8) seems more natural.Let us define ˜ H + ( A ) = ∞ X m =0 δ J H + ( A ( m )) , where A ( m ) = J ◦ m ( A ) . Since all terms are positive, iterating (6.12) yields˜ H + ( A ) ≤ H + ( A ) . One can now use ˜ H + ( A ) < ∞ to show that A ( m ) is ℓ -close to be periodic and that the periodicoperator is ℓ -close to IS E . That is, if { ~p j ( m ) , ~q j ( m ) } m ∈ N denote the GMP parameters of A ( m ),then [25, Theorem 1.20] { ~p ( m ) − ~p − ( m ) } m ∈ N ∈ ℓ ( N , R g +1 ) , { ~q ( m ) − ~q − ( m ) } m ∈ N ∈ ℓ ( N , R g +1 ) , (6.13) { F E ( ~p ( m ) , ~q ( m )) } m ∈ N ∈ ℓ ( N , R g +2 ) . To show how one obtains from (6.13) convergence of ( S ∗ + ) m JS m + to T + E in the sense of (6.4), weneed one more ingredient: it is well known that there are continuous functions, A , B , on R g / Z g , RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 26 which can be expressed explicitly in terms of the theta function associated to E [22, Theorem 9.4.],and a fixed element χ ∈ R g / Z g , such that T E = { J ( α ) : α ∈ R g / Z g } (6.14)and J ( α ) is the Jacobi matrix built from the coefficients a m ( α ) = A ( α − mχ ) , b m ( α ) = B ( α − mχ ) . (6.15)Recall that by the definition of the Jacobi flow, if J + is the Jacobi matrix associated to A + , then( S ∗ + ) m J + S m + is the Jacobi matrix associated to P + A ( m ) P + . Using a certain “open-gap” conditionfor ∆ E , one can now conclude from (6.13) that there is an element J ( α m ) ∈ T + E , such that { a m − A ( α m − mχ ) } m ∈ N ∈ ℓ ( N ) , { b m − B ( α m − mχ ) } m ∈ N ∈ ℓ ( N ) , where { a m , b m } m ∈ N denote the coefficients of J + . This is used to prove (6.4).Before we start with our construction, we have to mention a certain technical issue. In proving(6.4), it is constantly used that for all 0 ≤ k ≤ g { λ k Λ k ( A ( m )) − } m ∈ N ∈ ℓ ( N ) = ⇒ inf m Λ k ( A ( m )) > . Let us introduce the notation { f m } ∈ CS for sequences { f m } satisfyinglim N →∞ N N X m =1 | f m | = 0 , and we call a set T ⊂ N sparse if lim N →∞ | T ∩ { , , . . . , N }| N = 0 . An elementary observation, which will be used repeatedly, is that for f ∈ CS, the set { m ∈ N || f m | ≥ δ } is sparse for any δ >
0. This follows immediately from Markov’s inequality. Note thatwe will only have { λ k Λ k ( A ( m )) − } m ∈ N ∈ CS and thereforelim inf m →∞ λ k Λ k ( A ( m )) = 0is possible. But the set where λ k Λ k ( A ( m )) < / N →∞ |{ ≤ m ≤ N : λ k Λ k ( A ( m )) < / }| N = 0 . (6.16)This allows us to apply all estimates assuming λ k Λ k ( A ( m )) ≥ / m using(6.16). For a given GMP matrix A , let us define I N = { m : ∃ k : λ k Λ k ( A ( m )) < / } ∩ [1 , N ] (6.17)and we note that H + ( A ) ∈ CS implies thatlim N →∞ | I N | N = 0 . (6.18)We are finally ready to adapt Yuditskii’s construction [25] to our setting. Let µ be a regularmeasure with ess supp µ = E and c k / ∈ supp µ and let J + and A + be the associated Jacobi and GMPmatrix, respectively. Moreover let { v ℓ , w ℓ } denote the block Jacobi coefficients of ∆ E ( A + ). First ofall, by [25, Lemma 5.1] we can extend J + respectively A + by an element from the isospectral torusto full-line operators such that still c k / ∈ σ ( A ) and therefore A ∈ A ( C E , Z ). Furthermore, let A N bethe GMP matrix obtained by truncating A after the N -th block and extending it by some element RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 27 ˚ A ∈ T E ( C E ). Again we can do this and still ensure that c k / ∈ σ ( A N ). Having in mind that elementsfrom T E ( C E ) satisfy the magic formula (6.5) and since computing the coefficients of ( c k − A ) − inthe ℓ th block only requires the entries from the consecutive blocks of A N , cf. [25, Lemma 3.2.], itfollows from compactness of T E ( C E ) that H + ( A N ) = N X ℓ =1 ( k v ℓ − − I k + k w ℓ k + k v ℓ − I k ) + O (1) , where the constant in O (1) depends on E and A . Therefore,˜ H + ( A N ) ≤ H + ( A N ) ≤ N X ℓ =1 ( k v ℓ − − I k + k w ℓ k + k v ℓ − I k ) + O (1) . (6.19)Using that also the Jacobi flow can be computed locally and analyzing the proof of [25, Theorem1.20] one can get a more quantitative statement instead of (6.13). That is, there is an N independentconstant C such that N X m =1 k ~p ( m ) − ~p − ( m ) k ≤ C ( ˜ H + ( A N ) + | I N | ) + O (1) , N X m =1 k ~q ( m ) − ~q − ( m ) k ≤ C ( ˜ H + ( A N ) + | I N | ) + O (1) , N X m =1 k F E ( ~p ( m ) , ~q ( m ) k ≤ C ( ˜ H + ( A N ) + | I N | ) + O (1) , where the O (1) comes from the fact that in order to compute the GMP coefficients for m = N , onewould need also the ( N + 1)st block of A , which was truncated in the definition of A N . However,this suffices to show that there is an element J ( α m ) ∈ T + E , so that for some N independent constant C we have N X m =1 ( a m − A ( α m − mχ )) ≤ C ( ˜ H + ( A N ) + | I N | ) + O (1) , N X m =1 ( b m − B ( α m − mχ )) ≤ C ( ˜ H + ( A N ) + | I N | ) + O (1) , where again { a m , b m } m ∈ N denote the coefficients of J + . Thus, dividing by N and sending N → ∞ ,we obtain by (6.3), (6.18) and (6.19) that { a m − A ( α m − mχ ) } m ∈ N , { b m − B ( α m − mχ ) } m ∈ N ∈ CS . (6.20)Moreover, similarly to the proof of [25, Theorem 1.5], a Lipschitz estimate on the difference ofcharacters with respect to the corresponding ~p vectors shows that { α m +1 − α m } m ∈ N ∈ CS.
Lemma 6.3.
For fixed L ∈ N and δ > , the set B L,δ = { m : | α m + ℓ − α m | ≤ δ for all ℓ = 0 , . . . , L − } has a sparse complement, i.e., | B L,δ ∩{ ,...,N }| N → as N → ∞ . RTHOGONAL RATIONAL FUNCTIONS WITH REAL POLES, ROOT ASYMPTOTICS, GMP MATRICES 28
Proof.
Since shifts and linear combinations of CS sequences are in CS, { α m + ℓ − α m } ∞ m =0 ∈ CS forany ℓ . Thus, for any ℓ , the set { m : | α m + ℓ − α m | > δ } is sparse; the complement of B L,δ is a unionof finitely many sparse sets, so it is sparse. (cid:3)
To prove Theorem 1.10, it remains to prove that, for every ǫ > N →∞ N N X m =1 dist( T + E , ( S ∗ + ) m J + S m + ) ≤ ǫ. (6.21)Fix L so that P ∞ ℓ = L e − ℓ k J + k ≤ ǫ/
16. Choose δ > |A ( β ) − A ( β ) | ≤ ǫ L , |B ( β ) − B ( β ) | ≤ ǫ L (6.22)whenever | β − β | ≤ δ .Since dist( T + E , ( S ∗ + ) m J + S m + ) is uniformly bounded in m and the complement of B L,δ is sparse,lim sup N →∞ N X ≤ m ≤ Nm/ ∈ B L,δ dist( T + E , ( S ∗ + ) m J + S m + ) = 0 . For m ∈ B L,δ , estimating the distance to T + E by the distance to J ( α m − mχ ) givesdist( T + E , ( S ∗ + ) m J + S m + ) ≤ ∞ X ℓ =0 e − ℓ ( | a m + ℓ − A ( α m − ( m + ℓ ) χ ) | + | b m + ℓ − B ( α m − ( m + ℓ ) χ ) | ) . Using (6.22) for ℓ < L and using our choice of L to bound the tail of the series, we obtaindist( T + E , ( S ∗ + ) m J + S m + ) ≤ ǫ L − X ℓ =0 e − ℓ ( | a m + ℓ − A ( α m + ℓ − ( m + ℓ ) χ ) | + | b m + ℓ − B ( α m + ℓ − ( m + ℓ ) χ ) | ) . Thus, to prove (6.21), it remains to provelim sup N →∞ N X ≤ m ≤ Nm ∈ B L,δ L − X ℓ =0 e − ℓ g m + ℓ ≤ ǫ , (6.23)where g p = | a p − A ( α p − pχ ) | + | b p − B ( α p − pχ ) | . Note g ∈ CS by (6.20). Enlarging the range ofsummation, we obtainlim sup N →∞ N X ≤ m ≤ Nm ∈ B L,δ L − X ℓ =0 e − ℓ g m + ℓ ≤ lim sup N →∞ N N + L X p =1 L − X ℓ =0 e − ℓ g p . Now the sum in ℓ can be separated as an explicit constant, so this lim sup is zero since g ∈ CS.Then (6.23) follows, and the proof of (6.21) is complete.
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Institute of Analysis, Johannes Kepler University of Linz, 4040 Linz, Austria.
E-mail address : [email protected] Department of Mathematics, Rice University MS-136, Box 1892, Houston, TX 77251-1892, USA.
E-mail address : [email protected] Department of Mathematics, Rice University MS-136, Box 1892, Houston, TX 77251-1892, USA.
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