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ETNA Electronic Transactions on Numerical Analysis. Volume 25, pp. 369-392, 2006. Copyright  2006, Kent State University. ISSN 1068-9613. Kent State University etna@mcs.kent.edu SZEGŐ POLYNOMIALS: A VIEW FROM THE RIEMANN-HILBERT WINDOW A. MARTÍNEZ-FINKELSHTEIN Dedicated to Ed Saff on the occasion of his 60th birthday Abstract. This is an expanded version of the talk given at the conference “Constructive Functions Tech-04”. We survey some recent results on canonical representation and asymptotic behavior of polynomials orthogonal on the unit circle with respect to an analytic weight. These results are obtained using the steepest descent method based on the Riemann-Hilbert characterization of these polynomials. Key words. zeros, asymptotics, Riemann-Hilbert problem, Szegő polynomials, Verblunsky coefficients AMS subject classification. 33C45 1. Introduction. During the Fall Semester of 2003 I was visiting the Department of Mathematics of the Vanderbilt University, where I had the opportunity to continue my collaboration with Ed Saff. I was very excited with the evolution of the Riemann-Hilbert approach to the asymptotic analysis of orthogonal polynomials, and discussed extensively with Ed the new perspectives. He was the one who posed the question: can this method tell us anything new about such a classical object as the orthogonal polynomials on the unit circle (OPUC, known also as Szegő polynomials), in particular, about their zeros? The question was more on a skeptical side. I was aware of some previous work of the founders of the method, [1], [3], but none of these papers was focused on the description of the zeros of the OPUC’s. So, we started to work and realized that we were able to find curious facts even in the simplest situations. Later on I visited Ken McLaughlin, at that time in Chapel Hill. A two-day discussion at Strong Café (a recommended place) was crucial, and Ken joined the team. This paper is a short and informal report on some of the advances we have had so far. def Let me introduce some notation and describe the setting. For , denote def , and . A positive measure on has the LebesgueRadon-Nikodym decomposition (1.1) ! .- "#%$ '&% )( $ ' & " + * ," .- where is the singular part of with respect to the Lebesgue measure on we will consider measures satisfying the Szegő condition 012,354#6 / . Throughout, ( $ & " 8 7:9<; allowing to define the Szegő function (see e.g. [19, Ch. X, = 10.2]): * 0MLAN 3P4,6 ( $RQSPT & Q SUT ",V,WYX >?$ (@A'& )BDCFEHG)J#I K O (1.2) Q SUT 7 [Z This function is piecewise analytic and non-vanishing, defined for I , and we will denote _ by >]\ and >]^ its values for I and I , respectively, given by formula (1.2). It is def easy to verify that (1.3) > \ G ( @[ I W ]> ^`$ (I @a& ; I ; b Received June 28, 2005. Accepted for publication December 22, 2005. Recommended by D. Lubinsky. University of Almerı́a and Instituto Carlos I de Fı́sica Teórica y Computacional, Granada University, Spain (andrei@ual.es). 369 ETNA Kent State University etna@mcs.kent.edu 370 A. MARTÍNEZ-FINKELSHTEIN and for the boundary values we have ( $ ' &% ]> \c$ (( @A@A''& & ( I @a& L ; d .X > ^$ > ^$ (1.4) ( The first equality in (1.4) can be regarded as a Wiener-Hopf factorization of the weight , which is a key fact for the forthcoming analysis. there exists a unique sequence of polynomials For a nontrivial positive measure on lower degree terms, , such that i f ! j 12 k e f $ '& ge l $ & ",%$ &% )m l f o ; np;Aq r ; I ;+XsX+X (1.5) We denote by t f $ '& egf $ &auvi f the corresponding monic orthogonal polynomials. They egf $ ' &h i f f * def satisfy the Szegő recurrence t vf w $ '&% t f $ '& 7 x f t f $ & ;yt O $ '&%z I ; f where we use the standard notation t f $ '& 8 t f $ I u #& . The parameters x f 7 t fvw v$R & are called Verblunsky coefficients (also reflection coefficients or Schur parameters) and satisfy x f % for q {; I ;}|F;sX+X+X . Furthermore, ~ x f is a bijection; the map ) x f def is an inverse problem, and it is known to be difficult. In [18] there is a thorough discussion of several techniques to tackle this problem. Recently, the Riemann-Hilbert approach, not described in [18], proved to be very promising in this context also (see [1], [3], [6], [13]). The main goal of this paper is a further discussion of how this method can shed new light on the study of the asymptotics of the Szegő polynomials. We are not going to provide detailed proofs that can be found elsewhere (the references are included), the aim is to show the method in action and to discuss some new results in two apparently simple situations. The structure of the paper is as follows. Section 2 is devoted to the case when is non vanishing and has an analytic extension across . The main role here is played by the scattering function1 >?$ (@+& / :$ (@a& > \ $ (@A'& > ^ $ (@A'& ; meromorphic in an annulus, containing / , and which, via its iterated Cauchy transforms, ale%f def (1.6) lows to write some canonical series representing ’s. In this situation convergence is always exponentially fast, and the Riemann-Hilbert analysis is particularly simple and transparent. Only some of the multiple corollaries of the canonical representation are discussed; a more thorough analysis is contained in [12]. In Section 3 we look at the situation when the original analytic and nonvanishing weight has been modified by a factor having a finite number of zeros on the unit circle. Now the behavior of the zeros of ’s is qualitatively different: most of them cluster at , and only a finite number stays within the disc . The method of Section 2 must be modified now in order to handle the zeros of the weight: a local analysis plays the major role. The exposition here is much more sketchy; in this sense, more than a detailed view this window gives us a glimpse of the possible techniques and results. At any rate, the main goal is to persuade the reader that the Riemann-Hilbert analysis is a powerful technique, that deserves to be in the toolbox of those interested in orthogonal polynomials on the unit circle. t f Aa 1 Function is denoted in [17] by , and in [18, Section 6.2] by . It corresponds also to the scattering matrix in [9]. I prefer to follow the notation of [12]. ETNA Kent State University etna@mcs.kent.edu 371 SZEGŐ POLYNOMIALS: A VIEW FROM THE RIEMANN-HILBERT WINDOW 2. Analytic and nonvanishing weight. Any analysis of OPUC’s can be started either . from the orthogonality measure or from the sequence of the Verblunsky coefficients Let us assume that the sequence of the Verblunsky coefficients has an exponential decay: x f 3PP f, x f f X def (2.1) Nevai-Totik [15] proved that this situation is characterized by the following conditions on : in the decomposition (1.1), , measure satisfies the Szegő condition, and PF - I ]> ^$ (@a& (2.2) is holomorphic in jX Taking into account (1.3) we see that the first identity in (1.4) can be regarded as an analytic extension of the weight . With this definition of (which we use in the sequel) we can say equivalently that ( ( PF I I u( $ & is holomorphic in I u ; and both circles and A contain singularities of I u`( . In this situation the well-known Szegő asymptotic formula 35P f, e f $ &% >]\c$ (I @A'& ; d ; can be continued analytically through the unit circle / and is valid locally uniformly in a . It shows that the number of zeros of e%f on compact subsets of r remains uniformly I (Nevai-Totik points bounded, and these zeros are attracted by the zeros of > ^ in in the terminology of B. Simon [16]). Numerical experiments show that the vast majority of zeros gather at the “critical circle” . This fact was justified theoretically by Mhaskar and Saff [14], who using potential theory arguments showed that for any subsequence qh satisfying 3P5 A f# x f, ; the zeros of ehf,w distribute asymptotically uniformly in the weak-star sense on . The behavior of the zeros inside can be intriguing. Although approaching in mass the critical circle as predicted by Mhaskar and Saff, some of them still may remain inside and follow interesting patterns. Even the convergence to the circle is different for different (2.3) measures. Can we give a full description of this behavior in terms of the weight of orthogonality? The answer is positive, and the description will involve a sequence of iterates of certain Hankel and Toeplitz operators with symbols depending on the scattering function introduced in (1.6). It is a consequence of a canonical representation of the Szegő polynomials, found by means of the Riemann-Hilbert characterization. 2.1. Steepest descent analysis and canonical representation for orthogonal polynomials. The starting point of all the analysis is the fact that under assumption (2.1) conditions (1.5) can be rewritten in terms of a non-hermitian orthogonality for and : 12 k e f $ & 12 k e f' e%f ef fF ( $ f '& " r; for {; I ;+X+XsXD;q¡7 I ; ( '& {; I ;sX+XsX;q¡7¤|r; $ & $ f " £¢ ¥ruv; i f' ;¦ q¡ 7 IX ETNA Kent State University etna@mcs.kent.edu 372 A. MARTÍNEZ-FINKELSHTEIN !§ x£¨ 7 ©w * , , are oriented counterclockwise; with Here and in what follows, all the circles this orientation we talk about the “ ” and the “ ” side of referring to its inner and outer boundary points, respectively. Analogously, and are the corresponding boundary values on for any function for which these limits exist. By standard arguments (see e.g. [1] or [7], as well as the seminal paper [8] where the Riemann-Hilbert approach to orthogonal polynomials started), § © § © ª ´+µ 12 ft $ & IK ¥ k t f f $°¯ &±( $°¯ & & "#¯ µµ | ¯ $°¯g7 $ '&% ¬«® K i f' e f' ' & i 'f 1 2 e f' ª $R¯ &³( $R¯ & "#¯ ¶ f '& ²7 | $ 7 ¥ k ¯ R$ ª ¯g7 ª ª is a unique solution of the following Riemann-Hilbert problem: is holomorphic in `· , f [ f 35P f » ; $°¯ &¸G I ( $R¯ &Au ¯ W¹; d g; and º $ &¸G (2.4) w $°¯ &% W I where » is the |]¼¸| identity matrix. This is the starting position for the steepest descent analysis as described in [7] (see also [10]), which consists in performing a series of explicit and reversible steps in order to arrive at an equivalent problem, which is solvable, at least in an asymptotic sense. Since these steps are almost standard, they will be described very schematically. We will use the I 7 I W is the Pauli matrix, and for any non-zero ¿ and integer l l & ÀsÁ $°¿ sÀ Á . [f ÀsÁ`; if I ; & ¢ (2.5) ÂÃ$ ; »; if I ' & ' & ' & and put ÄÅ$ $ ÂÃ$ 3PP . Then Ä is holomorphic in ` ; this transformation normalizes º ÄÅ$ '&· Æ» . The price we pay is the oscillatory behavior of the behavior at infinity: the new jump matrix on : f wÄ $R¯ &% Ä $R¯ &¸G ¯ ( Ç$°¯ f & WY;o¯ X ¯ ½¾ ¬G n , ¿ÀÁ G ¿ u ¿ W and ¿ I S TEP 1: Define ª following notation: def def We get rid of these oscillations in the next transformation, taking advantage of the analyticity of its entries in the annulus. S TEP 2: Choose an arbitrary , , that we fix for what follows; it determines the regions (Figure 2.1) I ' #; È ² #; È I È O ² È w Define Ê É $ '& Ä$ &Ë $ & , where »; Ñ Ë $ '& oÎ ÌÍÍ f (2.6) Í Ñ ÍÍ I u ÍÏ u #; I I I u #X5X def def d ; if d u (I $ & I+Ò fI $ ( $ '&& I+Ò ; if if È OÐ È ; È w ; d È X ETNA Kent State University etna@mcs.kent.edu SZEGŐ POLYNOMIALS: A VIEW FROM THE RIEMANN-HILBERT WINDOW 373 È È O Ô Ó È w È ÔÕ Ó F IG . 2.1. Opening lenses. É Then is holomorphic in ity, but now Öa% Ð Ð A s× . We have not modified its behavior at infin- É w $°¯ &% É $°¯ &[Ø'Ù $°¯ & o ; ¯ Ö Ð Ð A × ; where Ñ ( $°¯ & u ( & 7 I $°¯ Ò ; f u(I & ; ¯ $R¯ IDÒ uI $R¯ f I ( $°¯ &A& IDÒ ; if ¯ ; Î ÌÍÍ Ñ ÍÍ if ¯ %`; Í Ñ if ¯ A X ÍÍ ÍÍ ÍÏ The jump on and is exponentially close to the identity, which is convenient to our purposes. We have to deal now with the relevant jump on the unit circle. S TEP 3: The Szegő functions > \ and > ^ have been introduced in (1.2). Define 0MLAN 3P4,6 G I ( @ & ) D B F C E I ( $ÜQ SPT & "#V W j·X (2.7) Ú >Å\Û$ (@ & > ^ $ 9 7 J#K O Ø'Ù $°¯ %& def Hence, if we introduce the geometric mean then 0 LaN 53 4#6 Ý!Þ (/ß BDCFE?G| I K O Ö ( ÖcQ SPT × "#V × WY; def (2.8) Ú $ÜÝ!Þ (·ßà& L . ETNA Kent State University etna@mcs.kent.edu 374 A. MARTÍNEZ-FINKELSHTEIN It is straightforward to check that the piece-wise analytic matrix-valued function Ñ > ^ $ (@A'&Au Ú ; if I ; Î u ( A @ ' & Ú > ^$ Ò áÃ$ '&h áÃ$ (@A'& ÌÍÍ Ñ (2.9) > \ $ (@a&au Ú ; if ; I u ( a @ & 7/Ú > \ $ Ò ÍÍ Ï on / as Éd$ & . This motivates to make a new transforis invertible, and has the same jumps $ & . Matrix â is holomorphic in r$R Ð Ð & , mation, defining â/$ & Éd$ '& á 35P º â/$ '&% » ; def def and â w $°¯ & â R$ ¯ &[Øã $°¯ & ä ; ¯ Ð Ð A ; (2.10) where »Ñ ; f 7/¯ å$ ( @ ¯ &Au Ú I fI $R¯ :$ (@ ¯ && Ø'ã $°¯ %& á Ø Ù á w ä Î ÌÍÍ Ñ I Í ÍÍ Ú L u ÍÏ ª Our main character, , has entered the picture! (2.11) Summarizing, we have (2.12) Â Ë $ &% â/$ & áÃ$ '&±Ë ¯ ; L ; if ¯ ; Ò ; if ¯ A X I+Ò if & ' & $ Â $ X á â Here , and are explicitly defined in (2.5), (2.6) and (2.9), respectively. About we know only that it is piece-wise analytic and satisfies the jump condition (2.10)–(2.11). A feature of this situation is that we can write a formula for in terms of a series of iterates of some Cauchy operators acting on the space of holomorphic functions in with continuous boundary values. Indeed, let us denote â r$R Ð A & .L â G ââ LAæâ æâ LAL W ; and look at the equations given by the jumps on ! . According to (2.11), the first column is analytic acrossf this circle, while the second column has an additive jump equal to the first L column times 7/¯ :$ (@ ¯ &au Ú . So, if we define the operator f ç \f $c© & $ & 7 K ¥I k 1Fè © $R¯ & :$ (@ ¯ & ¯ "#¯Ç; | ÚL ¯g7 where © denotes the exterior boundary values of the function © on , then taking into def account the behavior at infinity and Sokhotsky-Plemelj’s theorem we get that (2.13) â L ç \f Ü$ â.A & ;yâ LaL I *Mç \f Ü$ â L & X ETNA Kent State University etna@mcs.kent.edu SZEGŐ POLYNOMIALS: A VIEW FROM THE RIEMANN-HILBERT WINDOW 375 Analogously, with ç ^f c$ © & $ & Ú K L ¥ k 1 c2 é è © °$ ¯ & If | : $ (@ ¯ & ¯ °$ ¯g7 & "#¯Ç; where © denotes the exterior boundary values of the function © on , we have again (2.14) â.A I *Ãç ^f $câ L & ;yâ L ç ^f $Üâ LaL & X By (2.13), (2.14), functions â satisfy the following integral equations: ë S ê ç ç $ » 7 ^fì \f & âa I ; $ » 7 ç \fì ç ^f & â L ç \f $ I & ; $ » 7 ç ^f ì ç \f & â L ç ^f $ I & ; $ » 7 ç \f ì ç ^f & â LAL I ; » is the identity operator. Straightforward bounds show that there exists a constant where í depending 1Fè on only, such that îî ç \f & '& îî{ï í fð © ð $c© $ 5 7ñ1 2c é ;è ò u ; (2.15) ç ^f $c© & $ '& ï í f 5ð © ð ; ò u ; 7 Iu í where ð ðDó is the ôAõ E -norm on ö ; in the sequel we use to denote some irrelevant constants, def different in each appearance, whose dependence or independence on the parameters will be stated explicitly. Thus, we can invert these operators using convergent Neumann series, Ñ ÷ âa Ñ ÷ ø âL ø Ñ çÖ ^f ì ç \f × $ & ; â L ÷ O Ò I Dø O ç ^f ì ç \f × ì ç ^f $ & ; â AL L O Ö Ò I Ö ç \f ì ç ^f × ì ç \f $ I & ; Ò Ñ ÷ Ö ç \f ì ç ^f × $ I & X O Ò ø In order to make this somewhat more explicit, let us define recursively two sequence of functions: O}ú .© f ù I ä ; ©.f ù û f ù O}ú I ; û f ù def def Then ÷ âAv$°q @a&% ø ÷ â L $°q @a&% ø ú ç \f & $I ; ú ç ^f & $I ; def def .© f ù L w © fù L Lú and û f ù û fù L w and ÷ ú L f '& ; â L v$Rq @A'&g O©ù $ ä ø w ú @a&h f L '& O © ù $ ;äâ LAL $°q ú ç ^f f L ú & Ü$ ©.ù ; ú ç \f f L ú & $Ü© ù ;y ; ú ç L \ f $ û fù & ; ú ç ^f û f L ú & $ ù ;y X def def def def û f ù L w ú $ & ; O ÷ û f L ú & ù $ ; ø O Z u ; I Z for X for í All these series are uniformly and absolutely convergent in their domains. Moreover, straightforward bounds on the Cauchy transform show that there exists a constant depending on only, such that for , ¥ a; | I â $ & ï 5 í ; for Z u ; I 7 Iu S and â L $ '& ï P í ; p S 7 for [ Z X ETNA Kent State University etna@mcs.kent.edu 376 A. MARTÍNEZ-FINKELSHTEIN ª â Sê have their own meaning, and We emphasize that in each of the regions above functions are not obtained in general by analytic continuation from one domain to another. Once we have computed , we may replace its expression in (2.12) in order to find . It should be performed independently in each region. For instance, in the domain , , , , so that â áÃ$ &% $Ü> ^ $ (@a&au Ú & ÀsÁ Ë $ &% )» ÂM$ '&% f $ '&h â·$ '& $ >]^$ (f @A'&Au Ú & ÀÁ üG â A $°q @a@a&& f >]^`$ ((@a@a&a&auu Ú â L $°q >]^`$ Ú ª ª [f È ÀÁ â L $°q @aa@ && Ú â LAL $°q Ú f $u f ]> ^`$ (( @a@a&A&A&& W¹X $ >]^`$ u Analogous computations are easily completed in the rest of the regions. All the information about the parameters of the orthogonal polynomials is codified in the first column of : its entry gives us , that evaluated at yields the Verblunsky coefficients, while the entry at is related to the leading coefficient of : . Hence, we have obtained the following theorem, that has been proved in [12]: T HEOREM 2.1. Let be a strictly positive analytic weight on the unit circle , the constant as defined in (2.2)–(2.3), and constant with fixed. Then with the notations introduced above, for every sufficiently large the following formulas hold: ª L R$ &· ²7 | K i L'f ( (2.16) (2.17) (2.18) ¥& t f ¥Û¥ & ¥Û¥Û¥ & xf i Lf f t $I ;I& & $c|F; I d ef' I q f Ú > ^ $ (@a& .â Av$°q A@ '& ; @a& f $ &h Î ÌÍ Ú >]^`$ (@a& â A $°q @A'& 7 ²Ú > â \ $ L ( $°q @a& ; Í Ú/â L $°q @A'& 7 > \ $ (@a& ; Í Ú L â ÍÏ L $°q * I @ & X L Ú K â LaL $Rq * I @ & X | u @ I if Iu @ ' !X if if R EMARK 2.2. It is easily seen that the method we have just described is valid also if we replace the condition of positivity of on by the requirement that its winding number on is zero. In such a case we can assure that only for sufficiently large ’s, but the rest of the argument remains the same. Before we analyze some implications of these formulas let us look more carefully at the scattering function and at the iterates of its Cauchy transforms used in the definition of . Following notation of [18, Section 6.2], let ( 3P4,6 B 6 f ý t q q â ( $ & ÷ þvÿ be the Laurent expansion (equivalently, the Fourier series) for computation shows that 3P4,6 ( . Then straightforward Ñ w÷ Ñ w÷ 7 X )BDCFE :$ (@A'& B+CFE 7 Ò Ò ø. ø Let (2.19) w. ÷ :$ (@a& Ç °$ & Dø and w ÷ G I I å$ (@a& ø W ETNA Kent State University etna@mcs.kent.edu SZEGŐ POLYNOMIALS: A VIEW FROM THE RIEMANN-HILBERT WINDOW å$ (@a& I be the Laurent expansions of into account that Iu and on , in the annulus 377 { u , respectively. Taking I w GÅI W $° & ; and ÷ $° & w l $R & ¢ I ; if n r; (2.20) {; if n '%X ø Ç L L L Â w L . Let us denote by w and the Riesz Denote by Â w the Hardy class, and Â f L L projections onto Â w and Â , respectively, and ½ f $ & :$ (@A'& . Then w w_f w_f ÷ ÷ w $R½ f & $ '& [f $à & ; $R½ f & $ &% Çf $° & X ø In particular, w. ÷ w f I & % & I & w_f Â w L ; if !; 7 R $ ½ $ 7 à $ L L Î ú Ú ø Çf © f ù $ '&% ç \f $ I &% ÌÍ Ú Í I L $Ü½ f & $ '&% I L ÷ [f $R & wf Â L ; if j!X Ú Ú Í ÍÏ side converge locally uniformly. Observe also that The series in the right hand ú (2.21) © f ù $R & 7 Ú I L $R & Çf 7 Ú I L G] I W f X If we introduce the following composition of Hankel and Toeplitz operators, having ½f as a symbol, L Â L ; given by fw $c© & 7 w $R½ f $Ü½ f © && ; f Â f $c© & $R½ f $R½ f © &A& ; w then L ú f© ù L w ú ¢ ff $c© ffù L ú && ; if ; $c© ù ; if ; ú ú L represents the values of © f ù L in · . where © fù ú For û f ù we can obtain analogous formulas: L ú û f ù L ú $ '&% ç \f $ û f ù ú & ¢ 7/ Ú L w $Üf'½ û f'û úf ù & & $ & &% 7 w f $Ü½ w f w $R½ &&f '&A& & $ '& ; if ; Ú $R½ f ù $ $R½ $Ü½ f $ ; if ; w and Lú ' û f ù L w L ú ¢ f $ ûû fù L ú && ; if !; f $ fù ; if j!X ' , For instance, taking into account (2.20), for ´ Ñ w ´ w _ w f [ f ÷ ÷ ú û f ù L $ &% 7 w $Ü½ f w $R½ f &A& $ '&% 7 w ® ® G I ¶ & « « ø Ç $à ê ê ¶ ø f W Ò ê ´ w 7 ÷ & $° & ê 7 ÷ ® ÷ $° & w l $à & ¶ l ; ° $ l ø O « [f ê [f ê def def ETNA Kent State University etna@mcs.kent.edu 378 A. MARTÍNEZ-FINKELSHTEIN and by identities in (2.20) we can rewrite last formula as (2.22) w. Ñ l ÷ * û f ù L ú $ &% ÷ & & l w ° $ R $ ; ! X I l ø O [ f Ò L R EMARK 2.3. There are some further relations and equivalent expressions that an interested reader can easily derive. For instance, if we introduce the operator on with kernel f w fÄ $ ¥ ; & ÷ R$ & - w_f,w w ú G I W _w fvw w ÷ °$ & G I W ; S ê ù ê -³ø O S Çf' ê then ç ^f ì ç \f $Ü© &% £¢ w f $Ü© & ; if I u ; 7 f $Ü© & ; if I u ; L ú and û f ù L w ú . and we can obtain expressions for © f ù def 2.2. Asymptotic behavior of OPUC. Representation (2.16) is asymptotic in nature. Using (2.15), it is immediate to show that for all sufficiently large and for , îî îî á r ; I ;}|F;+XsX+X í w f 7 u ù L L ú ; [ Z I u !; I , but neither on q nor on . Analogously, í w f 5 7p ù L ¾ ú ; [ Z !; q ÷ f L ú '& î ï îî â.Av$°q A@ '& 7 © ù $ îî 5 î ø O í where the constant depends only on and á îî w ú '& îîî ï ÷ L î a @ & f (2.23) îî â L $°q 7 O © ù $ îî ø í where has a similar meaning as above. These bounds show that (2.16) allows us to obtain approximations of t f of an arbitrarily high order. We will concentrate only on the most interesting domain including the critical circle and its interior (for a full analysis, check [12]). Let us discuss the consequences of truncating âgA and â L in (2.16) at their first terms, 1rè :$ (@ ¯ & ¯ f *! ¾ f f& * L A @ ' % & a @ % & I & â a $Rq 7 | K ¥Ú L k I $R ;yâ L $°q ¯.7 ",¯ $° ; imposing some additional conditions on the analytic continuation of our weight ( (or function ). We assume first that the critical circle contains only isolated singularities in a finite number. such that > ^ can be conT HEOREM 2.4 ([12]). Assume that there exists ï " tinued to the exterior of the circle $# , as an analytic function whose only singularities are on the circle , and these are all isolated. Denote by %;sX+X+Xs;&%' the singularities (whose number is finite) of >d^ on , % s+ % ' :X there exist constants ï md Æm $° " & and í í $° " & * 9 such Then for " I ï " and q , that for " (2.24) îî f î îî t f $ & 7 f > ^ $ (@a& 7 > \ $ (@ & ÷ ' * B ô G :$ (@ ¯ & ¯ W îîî ï í ÖÜ f m f * ¾ f × X î > ^ $ (@ 9 & > \ $ (@a& Dø.)(ø,+ ¯.7 î ETNA Kent State University etna@mcs.kent.edu SZEGŐ POLYNOMIALS: A VIEW FROM THE RIEMANN-HILBERT WINDOW í í $ñ Ë & * 9 îî îî t f $ & 7 > (2.25) î > Ë % there exist constants ï m Ym $ Ëp& I Ê¸Ë , f î \ $ (@ & ÷ ' * B ô G :$ ( @ ¯ & ¯ W îîî ï í ÖÜ f m f * ¾ f × X \ $ (@A'& ø )(ø-+ g¯ 7 î Furthermore, for every compact set such that for > ^ 379 and If can be continued as an analytic function with a finite number of isolated singularities to whole disc , then we may take in the right hand sides in (2.24)–(2.25). Otherwise the right hand side in (2.25) may be replaced by an estimate of the form . , one must be able to analyze the approximation to In order to isolate the zeros of afforded by (2.24) and (2.25). For example, zero-free regions may be determined by (i) establishing zero-free regions for the approximation, and (ii) bounding away from zero using the error estimates. Similarly, isolating the zeros can be done by first isolating the zeros of the approximation, and then using a Rouche’ type argument for . This theorem tells us that in general all the relevant information for the asymptotics of ’s in comes from the singularities of the exterior Szegő function on (that is, from the first singularities of we meet continuing it analytically inside the unit disc), and reduces the asymptotic analysis of ’s (at least, in the first approximation) to the study of the behavior of the corresponding residues. In the case when all the singular points that we met on are poles, this analysis is more or less straightforward. be a pole of a function analytic in . We denote by D EFINITION 2.5. Let its multiplicity and say that is a dominant pole of if for any other singularity of , either or , but then is also a pole and t f t f t f mH í fmf t f > ^ > ^ t f /3 . % & º 0 õ © ø-+g©g$ % 0 % 0 t f % 0 © © F I /3 . 3/. 2 & 1 º ø,õ + ©g$ º ø-õ 3 ©g$ '& X and 4 , In the sequel we use the following notation: for % 576v$8% & 8d ² 79% ' 4X (2.26) such that >]^ can be T HEOREM 2.6 ([12]). Assume that there exists ï " continued to the exterior of the circle # , as a meromorphic function whose only singularities are on the circle Ä . Denote by % ;+X+XsXD;&% ' the poles (whose number is finite) of >Ê^ on % , and assume that the dominant poles of > ^ are %r;+XsX+XD;:%<; , = ï!> , and their multiplicity is n . ï :7?4 , ò u Ð ' 5 6 $@%F & , and q , Let 4]j . Then for " ø. (2.27) ; ft $ '&% > ^ $ ((@ @a& & f * > \ $ ((@@a&& ÷ G q W % f' l w > \ $ (@ %F &B> A ^ $ (@ %F & *?C f $ '& ; >]^$ 9 >]\Û$ % 7 ø n 7 I 35P + > ^ $ (@A'& $ 7D%F & l , I ;+XsX+XD;E= . There exist a constant A ^ $ (@ %' &ò º where > íÆ* 9 independent of q and 4 , and a constant m )m $F4 & I , such that í f f * f C f $ & ï Î í ÖÜ m l ¾ f × ; if n I ; L 1 Ì 4 l q ; if n |X Ï í í $ Ëñ& 9 Furthermore, for every compact set Ë there exists a constant def ETNA Kent State University etna@mcs.kent.edu 380 A. MARTÍNEZ-FINKELSHTEIN Ê¸Ë , and q , îî (@a& ÷ ; > \ $ (@ %F & > A ^ $ (@ %F & LaN f' l w ú îîî \ · Ú > $ q î f G ' & l F f w T îî Q SRù (2.28) î î % % 7 n 7 I W t $ 7 ø £ Î í í m f ; if n I ; ï 1 Ì q ; if n |; Ï where 6 6 (2.29) V, I ; and Vv | I K $@G %Få79G %F & ; |F;sX+XsXD;H=hX ( ( In particular, on every compact set Ë ) , for all sufficiently large q polynomials t f can have at most =7 I zeros, counting their multiplicities. I $ '& â/$ & L , d ! , Observe that this result is applicable to weights of the form ( $ & I where is a rational function Z with at least one zero on (or one pole on ), and â is any function holomorphic and in any annulus, containing ï ï I u . such that for By means of (2.27) we may show also that under assumptions of Theorem 2.6 the vast majority of zeros approaching the critical circle does it in an organized way, exhibiting of can be numbered in such a way that, roughly an equidistribution pattern: if zeros speaking, ùf ú ê t f ù f ú G * I P3 4,6 G q W *! G I ·W W¹; I q S n 7 I £ q and G 6 fú 6 f ú | K * G IL W M ùw J ù 7 G S ê S q q ( ( (again, we refer the reader to [12] for details). This is also an analogue of the interlacing property of the zeros of orthogonal polynomials on the real line. Moreover, (2.28) allows us to describe the accumulation set of zeros of ’s inside . For instance, if all ’s in (2.29) are rational, this set is discrete and finite. Otherwise, as it follows from KroneckerWeyl theorem, it can be a diameter of or even fill a two dimensional domain. The situation gets much more complicated if the first singularity that we meet continuing analytically inside is more severe. Consider the simplest example of an essential singularity on : t f h V % >]^ îî îî L G I ( & D B F C E î î (2.30) $°¯ î Å 7p¯ W î ;o¯ .; . Observe that its inverse, u`( , satisfies also the conditions of Theorem 2.4. with I I However, the behavior of the zeros of the OPUC for ( and I u`( is qualitatively different, check Figure 2.2. In a few words, the explanation for this phenomenon is the following. Observe that in the case of an essential singularity of the asymptotic behavior of the Cauchy transform 1è f f© ù ú $ &% 7 K I ¥ L k ¯ :$ ( @ ¯ & ",¯ | Ú g¯ 7 ETNA Kent State University etna@mcs.kent.edu 381 SZEGŐ POLYNOMIALS: A VIEW FROM THE RIEMANN-HILBERT WINDOW 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −1 −0.5 0 F IG . 2.2. Zeros of 0.5 KMLON −1 −1 1 for weights P (left) and A:P (right), with −0.5 P 0 given in (2.30) and 0.5 1 QSRòA$T . is not as simple as when the only singular points are poles. In fact, the leading term of the asymptotics will come now from a dominant saddle point of U f $R¯ & 3P4,6 ¯ * I 3P4,6 :$ (@a& q w lying close to the singular point of , ¯ ¯ , which is the solution of the equation (2.31) I 7 * I G ¯ * I W satisfying ¯ w *WV * * GI W¹;oq? 9<; ¯ q I ¯g7 I ¯ 7ñ q I q def where we take the positive square root. It is possible to show that the zeros of the orthogonal polynomials (at least those not staying to close to ), will approach the level curve 3P4,6 G I K ¾¾ E&ZZ WY; X B $ U f $ '& 7 U f R$ ¯ w && q I |<Y q and the error decreases with q . However, for the weight I u`( we will have two dominant saddle points, and the different structure of the level curve (2.32) for weights ( and I u`( (2.32) explains the different result of the numerical experiments (see Figure (2.3)). 2.3. Verblunsky and leading coefficients. Let us finish this Section with some comments about the behavior of the notorious coefficients related to the OPUC. Evaluating the polynomial or any of its approximations at the origin we obtain information about the Verblunsky coefficients. For instance, a combination of (2.17), (2.21), and (2.23) yields the following estimate of the Verblunsky coefficients : P ROPOSITION 2.7 ([12]). Let be a strictly positive analytic weight on the unit circle . With the notation introduced above and for each , t fvw ( (2.33) where $ I u & fvw q f x f 7 G I W vf w *! $° ¾ & ; is the corresponding Laurent coefficient of Iu xf in (2.19). ETNA Kent State University etna@mcs.kent.edu 382 A. MARTÍNEZ-FINKELSHTEIN 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −1 −0.5 0 −1 −1 0.5 [H\ ^ [:] −0.5 0 0.5 é é Q R A $ T F . 2.3. Left: zeros of KM_ for P given in (2.30) with ` and aRcbed , along with the level curves fMgihkj _ hmlonqprj _ hms \ n8n R9d , fMgehtj _ hklonuprj _ hms \ n@n Rw_rv xzye{| }O~ v Á and t hkj _ hmlnupj _ h/s \ n8n R9d . _Á Right: the same, but for the weight A:P . Observe that the level curve (2.32) has now two components. IG This fact has the following reading: consider the generating function of the Verblunsky coefficients, ÷ ff ' % & $ føOx X ¾f Then the Maclaurin series of and w $7 'u :$ '&& match up to the $R & term. In consequence, we have P ROPOSITION 2.8. Function $ & * :$ (@A'& ; defined of , can be continued as a holomorphic function to the annulus Æ rin a uneighborhood ¾ . This fact has been established independently by Simon [17] and Deift and I I Östensson [6]. If the only singularities on are dominant poles, we can use formula (2.28) in order to derive the asymptotic behavior of ’s: P ROPOSITION 2.9 ([12]). Under assumptions of Theorem 2.6, the Verblunsky coefficients satisfy f x ; * f f if n I ; fx 7 ÷ G q I W % 'f l w > \ $ ( @ F% & > A ^ $ (@ %F & * ¢ $R l m L & ; f RÖ q × ; if n 1 |X ø n 7 I That is, in the situation when the first singularities of > ^ met during its analytic continuation inside are only poles, the Verblunsky coefficients are asymptotically equal to a combination of competing exponential functions with coefficients that are polynomials in q . We ( can compare it with the case of the essential singularity considered before: for the weight given in (2.30), ¾ &Z f x f 7 |<Y I K¸¯ w :$ (@ ¯ w & q G I * G q AI L W²WY;oq? 9<; ETNA Kent State University etna@mcs.kent.edu SZEGŐ POLYNOMIALS: A VIEW FROM THE RIEMANN-HILBERT WINDOW where ¯w 383 is given by the equation (2.31). A similar analysis can be carried out for the asymptotic expansion of the leading coefficients . By (2.18), if L L i Lf Ú K â LAL $°q * I @ &g Ú K I * û f,ù L w ú $Ü & * $R Z f & X | | Taking into account (2.22), we arrive at P ROPOSITION 2.10. For the leading coefficient if the following formula holds: ´ L L f ÷ ÷ $° & L * $° Z & Ú K ® 7 $à & L ¶ * $R Z f & ; L ÚK (2.34) i f I | [f' | « [f' where $° & ’s are the coefficients of the Laurent expansion of in (2.19). Observe that we can write this identity also in terms of the Riesz projections: 1{2 Li f Ú L ÖÜ½ fv w × L º ú *! $° Z f & X ù Formula (2.34) shows that 35P L Ú L f if |K ; and L i Lfvw 7 i Lf Ú K $° & Çf' L *! $° Z f & ; | in accordance with (2.33) and the well known fact that L IiLfvw 7 i I Lf 7 x i f Lf X Iu Summarizing, we see that the Laurent coefficients of (or of ) contain surprisingly good approximations of two main parameters of the OPUC: they match asymptotically the Verblunsky coefficients, and the partial sums of the squares of their absolute values represent (up to a normalizing constant) the leading coefficient of the orthonormal polynomials. 3. Weight with zeros on . Let us analyze the change of the behavior of the orthogonal polynomials if we allow zeros of the weight on the unit circle. In other words, we consider now a weight of the form l (3.1) ¹$ '& ( $ ' & 7 %F LH ; d ! ; ø. where % , 1 , I ;sX+XsXD;n , and ( is an analytic and positive weight on , such as considered in Section 2. Without loss of generality we assume that ( is analytic and u non-vanishing in the annulus I . def According to Nevai and Totik [15], the Verblunsky coefficients no longer have an exponential decay, neither the bulk of zeros accumulate on an inner circle, but how many of them stay inside? And for those approaching the unit circle, does the rate depend on the “orders” ? And how can we extend the Riemann-Hilbert method, that so nicely worked for us in the analytic situation, to the case of a weight of the form (3.1)? [ ETNA Kent State University etna@mcs.kent.edu 384 A. MARTÍNEZ-FINKELSHTEIN 3.1. Steepest descent analysis. Matrix ª +´ µ 12 f & & K| I ¥ k t ¯ f$°¯ $°¯ ¹7 $°'¯ & "#¯ µµ t f $ & $ '&% «® K i f' e 'f '& i 'f 1 2 e fF $R¯ & ¹$R¯ & ",¯ ¶ ; f ²7 | $ 7 ¥ k ¯ $R¯.7 '& ( ª solves the Riemann-Hilbert problem (2.4), with replaced by we add additional requirements at the zeros of the weight: $ &% G I I W ; I I as . It is the unique solution if %F'; d v';y I +; XsX+XD;Anpª X Â ` Nothing hinders performing Step 1: with defined by (2.5) we put becomes holomorphic in (including the infinity) and Ä w $R¯ %& Ä $R¯ &HG ¯ f Ä def Â , so that Ä ¹Ç$°f ¯ & W¹;o¯ X ¯ However, in order to get rid of the oscillatory behavior of the diagonal entries of the jump matrix the lenses we opened in Step 2 of Section 2 are no longer valid, at least because has singularities on . Since they are in a finite number, we can modify this step by opening lenses inside and outside , but “attached” to the unit circle at ’s (see Figure 3.1). This % È w È O ö^ È } ö\ v È v F IG . 3.1. Opening lenses. Ë in (2.6) consistent (after replacing ( Ë É Ä É w $°¯ & É $°¯ &[Ø Ù $°¯ & ;y¯ ö \ Ð ! Ð ö ^ ; deformation of the contours makes the definition of def with ), in such a way that has the jumps ETNA Kent State University etna@mcs.kent.edu 385 SZEGŐ POLYNOMIALS: A VIEW FROM THE RIEMANN-HILBERT WINDOW with Ñ ¹$°¯ & ; 7 u ¹$R¯ & Ò ÌÎ ÍÍ Ñ I Ø'Ù $°¯ & Í f u I & ; (3.2) ÍÍ Ñ ¯ ¹$°¯ I+Ò ÍÍ I u $R¯ f I¹$°¯ && IDÒ ; ÍÍ S TEP 3: Our next goal is toÍÏ handle the jump on / in Section 2 using the Szegő function. However, for if ¯ ; if ¯ ö'\Ü; if ¯ ö^X á via the global parametrix built as in (3.1) the Szegő function is, in general, no longer single-valued in the neighborhood of ’s, and a short digression is convenient in order to discuss briefly the form of this function and its multivaluedness. def For the sake of brevity we define the set of singularities of the weight, . We fix for what follows , such that additionally , so that all neighborhoods (see definition (2.26)) are disjoint. Denote also % l m 7¡ dP ø % %r7;% sX+X+ X+;&% m ¾ S ê S ê I 5v$8%' & ] ² 79% m ;ä I ;sX+XsXD;n; (3.3) l and a value % we will as well as 5 Ð ø. 5 $@% & . Furthermore, given a subset l ¿ ; consistently, Ð ø. $@% & . use the standard notation % 8 %#¿ In order to construct explicitly the Szegő function for the modified weight we introduce the generalized polynomial l & (3.4) ¡F$ $ 7%F & L ø l * and select its single-valued analytic branch in $ Ð ø. %F Þ I ; 9 && by fixing the value of ¡F$R & . With this convention we can write the Szegő functions for the modified weight : L '& (@A'& (3.5) > \ $ @A'&h ¡¡ L $$R & > \ $ (@A'& ;ä> ^ $ @A'&% L > ^ & $ uq,¢ & L X ¡ $R ¡F$ I l In particular, > \²$ @A'& is holomorphic in $ Ð Dø. %F Þ I ; I u &A& , > ^ $ @A'& is holomorl d Ð j{v$ Dø. % $R; I ß°& , and phic in I I @ & > ^ $ (@ 9 & Ú¡/; (3.6) >]\c$ @ & >]\Ü$ (@ & > ^ $8 9 def def def def Ú where has been defined in (2.7). Furthermore, with the orientation of the cuts toward infinity we have for : (3.7) I +; XsX+XD;An Þ > \ $8 @A'&Ûß w Q LaN S Þ > \ $8 a@ &³ß ; d F% $ I ; I u & ; Þ > ^ $8 @A'&Ûß w Q aL N S Þ > ^ $ A@ '&Ûß ; d %F °$ [; I & X By definition (1.6) and formulas (3.5) we have :$ @A'&% ]> \c$8 A@ '& ]> ^$8 @a&% Ñ L ¡F$ '& (@a& ; ¡ L $Ü & ¡F$ I u£,¢ & Ò : $ ETNA Kent State University etna@mcs.kent.edu 386 A. MARTÍNEZ-FINKELSHTEIN g u { v$t $R; u && ; ¨ I I 6 6 if d 5¤,$8%F & and G 6 $ & cG 6 $8%F & ; ; if d 5¤,$8%F & and G ( $ & G ( $8%F & ; ( ( that is also analytic and single-valued in the cut annulus furthermore, with our assumptions on , function m N S :$ @A'& ; Q A @ ' & A ¢ $8 (3.8) Q N S å$8 @A'& is holomorphic in 5¤v$8%' & , I ;+XsX+XD;An . So, we can define ¥ .A F$8 @ %' &! ¸;y ;+XsX+XD;AnÆX I ' & Now let us get back to the global parametrix áÃ$ áÃ$ @A'& , given by formula (2.9), ' & that is well defined, has the same jumps it exhibits the same as ÉÊ» $ , and '& á $ on& tends by9 (3.6), behavior at infinity. Hence, to as , and is holomorphic in Ê É $ F$°ö ^ Ð ö \ & . The jump on these curves is again exponentially close to identity, except in a neighborhood of the zeros % of the weight . This is a new feature, and we have to deal def def with this problem separately. %{ w Ê 0 % 6 '%F& 6 Ç& 0 57#$@%F & +6 0 G $ ¦G $@% ÈA È 0 0 ¡ 0 G $ & G s6 $@% & M( ; B ;A{;}9( ê2§ ê ( ( !A 0 0 r$R/ Ð ö \ Ð ö ^ & § ØÙ A Ð Aö \ Ð öA ^ w ØÙ p % É á . For the sake of brevity S TEP 4: local analysis. Let us pick a singular point def def along this subsection we use the following shortcuts for the notation: , , def def def , (where were defined in (3.3)), and , def def . We also write , where , and def , etc. analogous notation for curves: The goal is to build a matrix such that it is holomorphic in , satisfies across the jump relation , with given in (3.2), with the same local behavior as close to , and matching on . This analysis is very technical, and we refer the reader to [11] for details, and describe here the main ideas in a very informal fashion. È ^ È v È \ ö\ È O ¨\ ¨^ È ö^ È O ö'^ È ^ ö\ È \ F IG . 3.2. Local analysis in . we reduce the problem to the one with constant jumps. Let us denote * ¨ \ As% a $ first7 step ¨ m & ; I I and ^ % $ I ; I mv& , oriented both from ò % to infinity (see Fig. def def ETNA Kent State University etna@mcs.kent.edu SZEGŐ POLYNOMIALS: A VIEW FROM THE RIEMANN-HILBERT WINDOW 0 387 ( A L $ & and L denote the principal holomorphic branches of these functions in L & '& ¡F$ u #&'( A L $ & , with ¡ defined in (3.4). Then L is holomorphic I 0 ©% $ 7 $ m ; * ¡Fmv$ & , and according to (3.7), I I w A L $ & Q N S on ¨ \#; and w A L $ & Q N S on ¨ ^ X A L $ & A L $ & 3.2). Let , and in def Thus, if we define ª $F a@ & Î ÍÌÍ def ÍÍÏ and set Q N S N Q N S QN S Q S A L $ '&± f L ; f A LL $ & Ç f L ; L AL $ &[ f L ; A $ '&± ; d d d d $ ÈÈA $ ÈA $ ÈA $A ^ Ð ÈÈA ^ Ð ÈA \ Ð ÈA \Ð A & 0w ; & § 0 ; O & § 0 w ; ; O&§ 0 ; § I $ ' & Ê$ '& ª $F a@ & À Á ; 0 r$ ¨ \ Ð ¨ ^ Ð Ð ö \ Ð ö ^ & ; I I is holomorphic in 0 r$ ¨ \ Ð ¨ ^ Ð S« Ð ö'\ Ð ö'^ & , and we get for the followingI problem:I w $ '&% $ '&}Ø¬ $ '& , with satisfies the jump relation Ñ I ; if Ê A ; 7Ñ I Ò ÌÍÍ w ÍÍ Q LAI N ; if Ê $ Aö'\ § 0 & Ð $ ö'A ^ § 0 & ; Î S I+Ò Ø<¬ $ '&% ÍÍ Ñ 0 & Ð $ Aö ^ 0 w & ; Í LaNI ; if Ê $ Aö \ § § Q S ÍÍ Ñ Q N S IsÒ ÍÍ Q N ; if Ê ¨ \ Ð ¨ ^ X S Ò ÍÍ I ÍÏ local behavior as % : Moreover, has the following Ñ 7% 7% ; if d9ÈA OÐ ÈA ; I $ '&% Î Ñ 7% 7% Ò ÍÌÍ 7% 79% 7% 79% ; if d ÈA ^ Ð ÈA \cX Ò ÍÍ Consider in r$±7:9);AÏ & the transformation 3P4,6 ® 7 ¥.q | $ Fu % & ; (3.10) (we omit the explicit reference to the dependence of ® from % and q in the notation), where we 0 take the main branch of the logarithm. This is a conformal 1-1 map of onto a neighborhood ¨ ¨ ^ are mapped on of the origin. Moreover, ! is mapped onto ¯ oriented positively, \ Ð 6 of the contours, the imaginary axis, and we may use the freedom in the selection deforming them in such a way that ©g$ öA \ & and ©g$ Aö ^ & follow the rays G ® ° NZ ° K . After this ( that has been studied for transformation we get a Riemann-Hilbert problem on the ® -plane (3.9) def the local analysis of the generalized Jacobi weight on the real line. We take advantage of ETNA Kent State University etna@mcs.kent.edu 388 A. MARTÍNEZ-FINKELSHTEIN U $F @ ® & the results proved therein in order to abbreviate the exposition, and refer the reader to [20, Theorem 4.2] where the solution is explicitly written in terms of the Hankel and modified Bessel functions. Since a left multiplication by a holomorphic function has no influence on the jumps, and taking into account (3.9), we see that matrix can be built of the form U Ê$ '&h ²± $ & F$ @ ® & ª $@ @A'& ÀsÁ ; 0 . An adequate selection of ± is motivated by the where ± is any holomorphic function in *³ matching requirement Ê$ '& á $ &] » $°q & on the boundary U , and is constructed analyzing the asymptotic behavior of the matrix-valued function $@ @ ® & at infinity. Let us summarize the results of this analysis: P ROPOSITION 3.1. Let ± $ & (3.11) Ñ A A@ '& f À Á L ¥ â_r $8 L ¥ % I G I¥W ; 7 7 Ú Y | I Ò def â A $ +@ & has been defined in (3.8), and we take the main branch of the square root. Ê$ &% Ê$@%[;E @A'& , Ê$8%[;E @a& ¦± $ & U $@ @ ® & ª $@ @A'& ÀsÁ ; (3.12) with ® given by (3.10), & satisfies: 0 (i) Éd$ & $ is holomorphic in ; (ii) for d , (3.13) L f A @A'& ´ ¥ / 7 Ú % $8 ¶ *! GjI * Ê$ & á $ '&% :» |® ®« L [f A q L W¹X Ú % $ @A'& 7B '&% » *c & d In particular, Ê$ '& á $ $Rq for . S TEP 5: asymptotic analysis. With the notation introduced in (3.3) and with Ê$8%[;E @a& where Then matrix def defined by (3.11)–(3.12) let us take Ê$ & Ê$@%F';&[ @a& def and put for d 5¤,$8%' & F$Ü Ð ö ^ Ð ö \ & ;y I ;+XsX+X+;AnÆ; '& á $ ' & ; É d $ ' & ¢ â/$ Éd$ ' & $ '& ; d r $@5 Ð ! Ð ö ^ Ð ö \ & ; d 5]F $Ü Ð ö ^ Ð ö \ & X í , where Matrix â is holomorphic in the whole plane cut along ö Ð l ö $àö'^ Ð ö\ & ´5 and í Ð ø. (see Fig. 3.3), â/$ '& » as 9 , and if we orient all s ’s clockwise, â w $R¯ &/ â $R¯ &}Øã with í ; ÊÑ $ & á $ '& ; if d Ø ã $R¯ & Î ÍÌ Ú L u $ f :I $ @A'&& IsÒ ; if d ö^D©5; ÍÍ Ñ f L I 7 :$8 @a&au Ú ; if d ö\°©5X ÍÍ I Ò ÍÏ def def for for def , ETNA Kent State University etna@mcs.kent.edu SZEGŐ POLYNOMIALS: A VIEW FROM THE RIEMANN-HILBERT WINDOW 389 µ ö\ ö^ µ} µv ¶ F IG . 3.3. Jumps of . Øã * $ u q & ö'\à´5 d í ö'^D©5 Ø ã $ '&% » ? Ø ã » * · $ I u q & Ê I ö Ð í â 0 Øã & »& 0 & »'& Øã & »'& â/$ '&% )» * | KI ¥ $ $R¯¯ 7 7 "#¯ * | KI ¥ $câ $R¯ 7 ¯.7$ $°¯ 7 ",¯ ; í with the orientation shown in Fig. 3.3. In particular, where we integrate along contours ö Ð l & á $R¯ & 7 »'& "#¯ *!· G I ÷ @ $ Ê R $ ¯ ' h & ) » I K ¥ k¸ â/$ 7 ¯7 qLW ø | F$°ö Ð í & X locally uniformly for d F$°ö Ð 5 & , Now formula (3.13) and the residue theorem yield for Ê L f¥ ´ l / 7 Ú % ª ¶ *!· G I ÷ % ® [f â·$ '&g » * q I (3.14) ¥ L q L WX ø. %F²7 « Ú % 7BÇ It is clear that the off-diagonal terms of on and decay exponentially fast. On the other hand, by (3.13), for . So the conclusion is that the jump matrix uniformly for . Then arguments such as in [4, 5, 7] lead to the following conclusion: P ROPOSITION 3.2. Matrix satisfies the following singular integral equation: We are ready for the asymptotic analysis of the original matrix entries and ). Unraveling our transformations we have $I ;I& $Ü|rª ; I & $ '&% ¢ /â $ '' && áM$ '&&±Ë Ë $ '' & & Â $ ' '& & ; â/$ Ê $ $ Â $ ; if if (and in particular, of its d ´ 5; d 5X ETNA Kent State University etna@mcs.kent.edu 390 A. MARTÍNEZ-FINKELSHTEIN µ È \ È O È È ^ ö\ ö^ µ} µv v F IG . 3.4. Domains for the asymptotic analysis. È O © 5 ª @a&au áÃ$ '&% ¨G /7 Ú u >]\c$ @A'& >]\Û$8 Ú W ; Ë $ &h ÂÃ$ '&% » X ª ª ' % & & ' & Hence, $ â/$ áÃ$ , so that ª a,$ '&% 7 >]\c$ Ú @A'& â L $ '& ; L v$ '&% 7 >]\c$8 Ú @A'& â LaL $ & X Taking into account (3.6) and (3.14), and recalling that t f a we obtain Ñ l ¥ ft $ &h >Å\Û$8 @@a&& I ÷ % fv w *¹· G I W ; > \ $8 q ø. %F:7 q Ò We must analyze the consequences of these formulas in each domain (see Fig. 3.4). We will do it only for the interior domain , where Ë Æ n<7 I valid uniformly in this domains. It shows in particular that for every compact set there exists such that for every , each has at most zeros on , and these zeros should be asymptotically close to those of the rational fraction Ë á áÃ$ Ëp&: q 1 á l ÷ [ ¥ fvw % X ø. %F² 7 (3.15) t f $ & ¸ t f xf Evaluating at we can obtain asymptotics for the Verblunsky coefficients ; I leave this as an exercise for an interested reader. R EMARK 3.3. Orthogonal polynomials with respect to non-analytic (but smooth) and non-vanishing weights on were studied in [13], using a different (but complementary) ETNA Kent State University etna@mcs.kent.edu 391 SZEGŐ POLYNOMIALS: A VIEW FROM THE RIEMANN-HILBERT WINDOW º method based on the problem. There are very interesting similarities with our case. For instance, both zeros of the weight and the jump discontinuities of its derivatives have the same effect on the asymptotics of ’s inside the unit disk . More precisely, according to [13], if , and has jump discontinuities at , , then with a proper normalization, for on compact subsets of will be asymptotically close to rational fractions of the form (3.15), with the basic difference that now coefficient stands for the magnitude of the jump of at (cf. formula (3.23) in [13] and the fact that for the scattering function on , » 53 4#6 $ & í È f » " t" r% I ;+XsX+XD;An ft $ '& » " " F% ú jÖ @ Q SPT × Q¼ ùPT ; _ » where is defined by (1.21) in [13]). These similarities might have as a common ground the duality of both cases: for in (3.1), the imaginary part of has jumps proportional to ’s, while in [13] the finite jumps correspond to . Finally, recall that the leading coefficient of the orthonormal polynomial is expressed in terms of by . This yields immediately the following asymptotic formula: ª ª L `$Ü & ²7 | K L Ñ i Lf' Ú K I 7 I | q X B i$ »f " " & e%f i L'f l ÷ L *! G I q L W Ò ;oq? 9)X ø This result has a consequence for the behavior of the Toeplitz determinants for define the moments . If we 1 [f '" k º þ ¹ $ & " ; def ½ then the Toeplitz determinants are .¤¾ f $8 & ý B $R" & f ø O¿ X ê S Sê ½ that It is known (see e.g. [18, Theorem 1.5.11]) ½ f & $8 f' `$ & i I Lf X Taking into account the asymptotics of i f , (2.8) and (3.6), we arrive at def T HEOREM 3.4. Under the assumption above there exists a constant such that ½ f $ &% À$ Þ | K (/ß°& f q,Á`Â 2 EÃ À depending on * Ä $ && ;oq? 9<X $I ¹ I This formula is in accordance with the well known Fisher-Hartwig conjecture (see e.g. [2]), proved for this case (but using totally different approach and giving an expression for ) by Widom in [21]. À Acknowledgements. The research of this author was supported, in part, by a grant from the Ministry of Education and Science of Spain, project code MTM2005-08648-C02-01, by Junta de Andalucı́a, grants FQM229 and FQM481, by “Research Network on Constructive Complex Approximation (NeCCA)”, INTAS 03-51-6637, and by NATO Collaborative Linkage Grant “Orthogonal Polynomials: Theory, Applications and Generalizations”, ref. PST.CLG.979738. ETNA Kent State University etna@mcs.kent.edu 392 A. MARTÍNEZ-FINKELSHTEIN I wish to acknowledge also the contribution of both anonymous referees, whose careful reading of the manuscript and kind suggestions helped to improve the text. A vast portion of the results exposed here is an outcome of a highly enjoyable collaboration with K. T.-R. McLaughlin, from the University of Arizona, USA, and with E. B. Saff, from the Vanderbilt University, USA. In particular, this celebration is a good occasion to express once again my gratitude to Ed Saff for his support and friendship. 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