Orthogonal rational functions and rational modifications of a measure on the unit circle

AbstractIn this paper we present formulas expressing the orthogonal rational functions associated with a rational modification of a positive bounded Borel measure on the unit circle, in terms of the orthogonal rational functions associated with the initial measure. These orthogonal rational functions are assumed to be analytic inside the closed unit disc, but the extension to the case of orthogonal rational functions analytic outside the open unit disc is easily made. As an application we obtain explicit expressions for the orthogonal rational functions associated with a rational modification of the Lebesgue measure

Sobre ortogonalidad y cuadratura : desde la circunferencia unidad al eje real

The Theory of Orthogonal Polynomials with respect to a positive measure (or weight function) supported on finite or infinite intervals of the real line, that has become in an important research area in Spain in the last decades, has as one direct application the approximate calculation of integrals with respect to this measure. Indeed, it is very well known that the construction of the so called Gaussian quadrature formulas, depends on the knowledge of the zeros of the corresponding family of orthogonal polynomials or the eigenvalues of the associated Jacobi matrices. Inspired in the Weierstrass theorem, such rules, with the highest degree of algebraic precision, are based on the exact integration of polynomials with the highest possible degree. Their importance and applicability in Approximation Theory have motivated the construction of alternative rules in situations where the use of ordinary polynomials is not appropriate, for example, when the integrand has singularities near the interval of integration. Thus, the so called rational quadrature formulas appear, being the role played by polynomials (rational functions with all the poles in the infinite) replaced by other rational functions with prescribed poles. This therefore leads to the Theory of Orthogonal Rational Functions, or alternatively, orthogonal polynomials with respect to varying measures. All these arguments highlights the intimate relationship between the concepts ``orthogonality'' and ``quadrature''. Moreover, it has been proved also the interest of considering quadrature formulas with some of the nodes fixed in advance. For example, because these points correspond to points where some particular property should hold, or because we know some special values of the integrand in those nodes. Most often these nodes are one or both of the endpoints if the weighted integral is over a finite interval, or the endpoint if the interval is half-infinite, resulting the classical Gauss-Radau or Gauss-Lobatto rules. However, it has been also considered in recent years the situation when one or two nodes are fixed in advance inside the interval of integration, and also possible the corresponding endpoints. Two alternative approaches to this problem lead to the analysis of orthogonal polynomials with respect to variable-signed measures or to quasi-orthogonal polynomials. Note that the quadrature formulas mentioned here are quite different from Gauss-Kronrod formulas where several nodes are added to the classical Gauss nodes. In those formulas the Gauss nodes are fixed and the problem is to add free nodes in an optimal way so as to get best possible degree of accuracy (``improved Gaussian quadrature''). Here, some nodes, usually different from the Gauss nodes, are fixed, and all the remaining nodes are placed in an optimal way to obtain a good quadrature formula, whenever it exists. Since the publication of the famous book ``Orthogonal Polynomials'' of Gabor Szego in 1939, the Theory of Orthogonal Polynomials on the Unit Circle has become an important area of research in Applied Mathematics, with again a direct application in the approximate calculation of weighted integrals on the unit circle. In this respect, W.B. Jones, O. Njastad and W.J. Thron introduced and characterized in 1989 the so called Szego quadrature formulas, whose nodes are obtained as the zeros of para-orthogonal polynomials or via the eigenvalues of Hessenberg or CMV matrices, and where the exactness now is in subspaces of Laurent polynomials with the highest possible dimension, inspired now in the fact that they are dense in the space of the continuous functions defined on the unit circle, contrary in general to the ordinary polynomials. Such rules, that can be alternatively seen as quadrature formulas for periodic integrands with highest possible trigonometric degree of accuracy, have been studied exhaustively by the research group ``Aproximation Theory'' in the Department of Mathematical Analysis in La Laguna University. In analogy to the real situation, extensions to the rational case and the problem of fixing nodes in advance in the quadrature rules have been also analyzed. It seems reasonable to think which results about the concepts ``orthogonality'' and ``quadrature formulas'' for the real line can be passed to the unit circle, and vice versa. Some results are already known. Indeed, if the measures on the finite interval [-1,1] and the unit circle are related by the Joukowsky transformation, then a connection between orthogonal polynomials on [-1,1] and certain para-orthogonal polynomials was given by Szego in the mentioned book (not with the use of this terminology). Also a connection between Gauss-type (Gauss, Gauss-Radau and Gauss-Lobatto) rules and certain symmetric Szego quadrature formulas was established by A. Bultheel, L. Daruis and P. González Vera in 2001. But of course, there are still many questions in Aproximation Theory about these topics that remains to be connected from finite or infinite intervals of the real line to the unit circle, and vice versa. And this is the starting point of this Doctoral Dissertation, structured in four chapters and an appendix on some problems that remain open during it development. The first chapter is devoted to introduce the basics and the preliminary results on orthogonality and quadrature formulas, needed for the rest. Starting from the construction of orthogonal polynomials with respect to a measure supported on a compact set of the complex plane, and recalling the well known result of Féjer about the location of its zeros, it is particularized then the most discussed cases in the literature: the real line and the unit circle. Concerning the real line, interpolatory and Gauss-type quadrature formulas are revised, and their computation detailed. Error bounds for the quadrature rules are also deduced in terms of the corresponding errors of the Padé and Padé-type approximants in the infinity to the Cauchy transform of the measure. For the unit circle case, Szego and para-orthogonal polynomials along with interpolatory and Szego quadrature formulas are introduced. Szego-Radau and Szego-Lobatto rules are also analyzed and some convergence results and error bounds are established, in particular, an error bound in terms of the corresponding errors of two point Padé and modified Padé approximants to the Herglotz-Riesz transform of the measure. The chapter in concluded by considering the Joukowksy transformation, that maps the open unit disk to the extended complex plane with the interval [-1,1] deleted, and the unit circle to the interval [-1,1]. This transform is used to connect orthogonal polynomials on the interval and on the unit circle, Gauss-type quadrature formulas with certain symmetric Szego rules and the corresponding Cauchy and Herglotz-Riesz transforms. The second chapter is about interpolation and approximation questions connected between finite intervals of the real line (namely [-1,1]) and the unit circle. In the first part, the L2 convergence with respect to a positive measure in [-1,1] of certain sequences of interpolating polynomials to a function by taking some special elections of the nodes is studied. The technique used here is again to pass to the unit circle by the Joukowsky transformation, and as a result, an extension of the Erdös-Turán interpolation theorem based on certain Gauss-type nodes is proved. We continue by considering product integration rules on [-1,1] related to a measure that can take now complex values. A complete connection to the product integration rules on the unit circle associated with the complex measure transformed by the Joukowsky transformation is given. An analysis of the convergence for certain particular elections of the nodes is obtained, presenting a more simpler proof of a result due to I.H. Sloan and W.E. Smith (1982), extended now to the Gauss-type nodes. Furthermore, error bounds for the considered product integration rules are proved and several illustrative numerical examples are carried out. Next, we study certain rational approximants to the Cauchy and Herglotz-Riesz transforms of two measures on the interval [-1,1] and on the unit circle, respectively, related by the Joukowsky transformation. As an application, estimations of the Chebyshev weight function of the first kind, the natural logarithm, the inverse of the tangent and some exponential integrals are numerically tested. This chapter concludes with the study of new alternative approach to the construction of Gauss-type quadrature formulas with a node prescribed inside the interval of integration (and also possibly the endpoints of the interval), by passing to the unit circle and considering symmetric Szego-Lobatto quadrature formulas. An algorithm based on the LU decomposition of a certain modified Jacobi matrix for the computation of these rules is presented, by connecting the corresponding entries of the Jacobi matrix with the associated Verblunsky coefficients, and some numerical experiments involving rational modifications of the Chebyshev weight function of the first kind are carried out. The third chapter is dedicated to connect the unit circle with unbounded intervals of the real line, so the Joukowsky transformation will not be used here. We start by considering quadrature formulas for the Hermite weight function (on the real line) associated with the Rogers-Szego weight funtion (on the unit circle), with the aim to approximate certain integrals over the real line for the Hermite weight and a periodic function in the integrand. Thus, a complete characterization of the corresponding Szeg\H{o} quadrature formulas is given. Interpolatory type rules with nodes the roots of a complex number of modulus one are also studied, and an efficient procedure for their computation based on the Fast Fourier Transform algorithm is presented. Error bounds for the quadrature rules are obtained, some numerical examples are illustrated and also the limit cases are analyzed, giving rise on the unit circle by one hand to the Lebesgue measure and by other hand to a mass point located at z=1. These results are nextly generalized for other weight functions different from the Hermite one, under certain conditions on integrability, but by using the same techniques. Some examples and applications including numerical experiments are considered: the Weierstrass operator, the Poisson kernel and some strong Stieltjes distributions. We continue by considering an application of these results in the computation of the Fourrier transform under the presence of nearby polar singularities; some illustrative numerical experiments are carried out. Finally, the problem to estimate weighted integrals over the real line for a function that is not necessarily periodic in the integrand is studied. In this respect, a new transformation connecting the real line and the unit circle is considered: the Cayley transform. The transformed Szego quadrature formulas are also characterized, where now the exactness is imposed in certain spaces of rational funcions with poles in z=i and z=-i, giving thus a motivation for the next chapter. A connection between the orthogonal polynomials on the interval and the orthogonal rational functions transformed on the unit circle is established, and some numerical illustrative examples presented. The last chapter considers orthogonal rational functions and rational quadrature formulas. A complete connection between rational Gauss type quadrature formulas and rational symmetric Szego-type rules is first established, presenting an extension to the rational case of the results by A. Bultheel, L. Daruis and P. González-Vera (2001). Here, the connection by the Joukowsky transformation needs the study of rational para-orthogonal functions, and it is proved here that an appropriate election of the last pole is crucial to determine the existence of such rules. Several illustrative numerical examples are presented to illustrate this. Next, an extension to the rational case of the characterization of rational Gaussian quadrature formulas with a prescribed node inside the interval of integration is investigated in connection via the Joukowsky transformation with certain symmetric rational Szego-Lobatto quadrature formulas. As a consequence, convergence results for those rules are obtained and some illustrative numerical examples carried out. As already said, this Doctoral Dissertation has an appendix where some problems that remain open during it development are proposed for further research

An extension of the associated rational functions on the unit circle

A special class of orthogonal rational functions (ORFs) is presented in this paper. Starting with a sequence of ORFs and the corresponding rational functions of the second kind, we define a new sequence as a linear combination of the previous ones, the coefficients of this linear combination being self-reciprocal rational functions. We show that, under very general conditions on the self-reciprocal coefficients, this new sequence satisfies orthogonality conditions as well as a recurrence relation. Further, we identify the Caratheodory function of the corresponding orthogonality measure in terms of such self-reciprocal coefficients. The new class under study includes the associated rational functions as a particular case. As a consequence of the previous general analysis, we obtain explicit representations for the associated rational functions of arbitrary order, as well as for the related Caratheodory function. Such representations are used to find new properties of the associated rational functions.Comment: 27 page

Bi-orthogonal systems on the unit circle, Regular Semi-Classical Weights and Integrable Systems - II

We derive the Christoffel-Geronimus-Uvarov transformations of a system of bi-orthogonal polynomials and associated functions on the unit circle, that is to say the modification of the system corresponding to a rational modification of the weight function. In the specialisation of the weight function to the regular semi-classical case with an arbitrary number of regular singularities $\{z_1, ..., z_M \}$ the bi-orthogonal system is known to be isomonodromy preserving with respect to deformations of the singular points. If the zeros and poles of the Christoffel-Geronimus-Uvarov factors coincide with the singularities then we have the Schlesinger transformations of this isomonodromic system. Compatibility of the Schlesinger transformations with the other structures of the system - the recurrence relations, the spectral derivatives and deformation derivatives is explicitly deduced. Various forms of Hirota-Miwa equations are derived for the $\tau$-functions or equivalently Toeplitz determinants of the system.Comment: to appear J. Approx. Theor

Spectral methods for orthogonal rational functions

An operator theoretic approach to orthogonal rational functions on the unit circle with poles in its exterior is presented in this paper. This approach is based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure. Two different alternatives are discussed, depending whether we use for the matrix representation the standard basis of orthogonal rational functions, or a new one with poles alternatively located in the exterior and the interior of the unit circle. The corresponding representations are linear fractional transformations with matrix coefficients acting respectively on Hessenberg and five-diagonal unitary matrices. In consequence, the orthogonality measure can be recovered from the spectral measure of an infinite unitary matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions. Besides, the zeros of the orthogonal and para-orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite Hessenberg and five-diagonal matrices. As an application of this operator approach, we obtain new relations between the support of the orthogonality measure and the location of the poles and parameters of the recurrence relation, generalizing to the rational case known results for orthogonal polynomials on the unit circle. Finally, we extend these results to orthogonal polynomials on the real line with poles in the lower half plane.Comment: 62 page

Asymptotic Uniqueness of Best Rational Approximants to Complex Cauchy Transforms in L2{L}^2 of the Circle

For all n large enough, we show uniqueness of a critical point in best rational approximation of degree n, in the L^2-sense on the unit circle, to functions f, where f is a sum of a Cauchy transform of a complex measure \mu supported on a real interval included in (-1,1), whose Radon-Nikodym derivative with respect to the arcsine distribution on its support is Dini-continuous, non-vanishing and with and argument of bounded variation, and of a rational function with no poles on the support of \mu.Comment: 28 page

On perturbed Szegő recurrences

The purpose of the present contribution is to investigate the effects of finite modifications of Verblunsky coefficients on Szegő recurrences. More precisely, we study the structural relations and the corresponding C-functions of the orthogonal polynomials with respect to these modifications from the initial ones.The author thanks the referee for careful reading and valuable comments. The author also wishes to express his thanks to F. Marcellán for suggesting the problem during the 1st Joint Conference of the Belgian, Royal Spanish and Luxembourg Mathematical Societies in Liège, Belgium. The research of the author was supported by CNPq Program/Young Talent Attraction, Ministério da Ciência, Tecnologia e Inovação of Brazil, Project 370291/2013-1 and Dirección General de Investigación, Ministerio de Economía y Competitividad of Spain, Grant MTM2012-36732-C03-01