Generation and evaluation of orthogonal polynomials in discrete Sobolev spaces I: algorithms

In this paper, we study theoretically the determination and evaluation of polynomials that are orthogonal with respect to a general discrete Sobolev inner product, that is, an ordinary inner product on the real line plus a finite sum of atomic inner products involving a finite number of derivatives. This Sobolev inner product has the property that the orthogonal polynomials with respect to it satisfy a linear recurrence relation of fixed order. We provide a complete set of formulas to compute the coefficients of this recurrence. Besides, we study the determination of the Fourier–Sobolev coefficients of a finite approximation of a function and the numerical evaluation of the resulting finite series at a general point.     

Generation and evaluation of orthogonal polynomials in discrete Sobolev spaces II: numerical stability

In this paper, we concern ourselves with the determination and evaluation of polynomials that are orthogonal with respect to a general discrete Sobolev inner product, that is, an ordinary inner product on the real line plus a finite sum of atomic inner products involving a finite number of derivatives. In a previous paper we provided a complete set of formulas to compute the coefficients of this recurrence. Here, we study the numerical stability of these algorithms for the generation and evaluation of a finite series of Sobolev orthogonal polynomials. Besides, we propose several techniques for reducing and controlling the rounding errors via theoretical running error bounds and a carefully chosen recurrence.     

New steps on Sobolev orthogonality in two variables

Sobolev orthogonal polynomials in two variables are defined via inner products involving gradients. Such a kind of inner product appears in connection with several physical and technical problems. Matrix second-order partial differential equations satisfied by Sobolev orthogonal polynomials are studied. In particular, we explore the connection between the coefficients of the second-order partial differential operator and the moment functionals defining the Sobolev inner product. Finally, some old and new examples are given.     

Zeros of Gegenbauer-Sobolev Orthogonal Polynomials: Beyond Coherent Pairs

Iserles et al. (J. Approx. Theory 65:151–175, 1991) introduced the concepts of coherent pairs and symmetrically coherent pairs of measures with the aim of obtaining Sobolev inner products with their respective orthogonal polynomials satisfying a particular type of recurrence relation. Groenevelt (J. Approx. Theory 114:115–140, 2002) considered the special Gegenbauer-Sobolev inner products, covering all possible types of coherent pairs, and proves certain interlacing properties of the zeros of the associated orthogonal polynomials. In this paper we extend the results of Groenevelt, when the pair of measures in the Gegenbauer-Sobolev inner product no longer form a coherent pair. This research is supported by grants from CNPq and FAPESP.     

On a class of polynomials orthogonal with respect to a discrete Sobolev inner product

This paper analyzes polynomials orthogonal with respect to the Sobolev inner product with and (x)is a weight function.We study this family of orthogonal polynomials, as linked to the polynomials orthogonal with respect to (x) and we find the recurrence relation verified by such a family. If the weight is semiclassical we obtain a second order differential equation for these polynomials. Finally, an illustrative example is shown.     

On connection coefficients of some perturbed of arbitrary order of the Chebyshev polynomials of second kind

Orthogonal polynomials satisfy a recurrence relation of order two defined by two sequences of coefficients. If we modify one of these recurrence coefficients at a certain order, we obtain the so-called perturbed orthogonal sequence. In this work, we analyse perturbed Chebyshev polynomials of second kind and we deal with the problem of finding the connection coefficients that allow us to write the perturbed sequence in terms of the original one and in terms of the canonical basis. From the connection coefficients obtained, we derive some results about zeros at the origin. The analysis is valid for arbitrary order of perturbation.     

Perturbed recurrence relations II the general case

In this paper, we give some new explicit relations between two families of polynomials defined by recurrence relations of all order. These relations allow us to analyze, even in the Sobolev case, how some properties of a family of orthogonal polynomials are affected when the coefficients of the recurrence relation and the order are perturbed. In a paper we have already given a method which allows us to study the polynomials defined by a three-term recurrence relation. Also here some generalizations are given.     

Higher order hypergeometric Lauricella function and zero asymptotics of orthogonal polynomials

The asymptotic contracted measure of zeros of a large class of orthogonal polynomials is explicitly given in the form of a Lauricella function. The polynomials are defined by means of a three-term recurrence relation whose coefficients may be unbounded but vary regularly and have a different behaviour for even and odd indices. Subclasses of systems of orthogonal polynomials having their contracted measure of zeros of regular, uniform, Wigner, Weyl, Karamata and hypergeometric types are explicitly identified. Some illustrative examples are given.     

Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality

We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same nth root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with respect to a Sobolev inner product with exponential weights of the form e−φ(x), giving a unified treatment for the so-called Freud (i.e., when φ has polynomial growth at infinity) and Erdös (when φ grows faster than any polynomial at infinity) cases. In addition, we provide a new proof for the bound of the distance of the zeros to the convex hull of the support for these Sobolev orthogonal polynomials.     

A generating function for nonstandard orthogonal polynomials involving differences: the Meixner case

In this paper we deal with a family of nonstandard polynomials orthogonal with respect to an inner product involving differences. This type of inner product is the so-called Δ-Sobolev inner product. Concretely, we consider the case in which both measures appearing in the inner product correspond to the Pascal distribution (the orthogonal polynomials associated to this distribution are known as Meixner polynomials). The aim of this work is to obtain a generating function for the Δ-Meixner–Sobolev orthogonal polynomials and, by using a limit process, recover a generating function for Laguerre–Sobolev orthogonal polynomials.     

Companion orthogonal polynomials: some applications

In this paper the recurrence relations of symmetric orthogonal polynomials whose measures are related to each other in a certain way are considered. Many of the relations satisfied by the coefficients of the recurrence relations are exposed. The results are applied to obtain, for example, information regarding certain Sobolev orthogonal polynomials and regarding the measures of certain orthogonal polynomial sequences with twin periodic recurrence coefficients.     

On a general system of orthogonal q-polynomials

A general set of orthogonal q-polynomials {P_m (x); m = 0, 1, 2, …, N} is introduced and characterized by its three-term recursion relation. This set unifies many of the different known systems of orthogonal q-polynomials, e.g. the Stieltjes-Wigert polynomials and their several generalizations, the Brenke-Chihara polynomials, the Al Salam-Carlitz polynomials, the Al Salam-Chihara polynomials, …. Compact expressions of the moments of the asymptotical density of zeros of this global set of q-polynomials are explicitly found in terms of the coefficients of the three-term recurrence relation. As an example the asymptotical density of zeros of the known, above-mentioned systems of orthogonal q-polynomials are calculated through its moments.     

Laurent 多项式及其零点

本文给出了测度dψ为强分布的一个必要条件,并得到了dψ为强分布时的Laurent多项式最大零点的一个表示。     

Turán inequalities for three-term recurrences with monotonic coefficients

We establish some new Turán type inequalities for orthogonal polynomials defined by a three-term recurrence with monotonic coefficients. We deduce as a corollary asymptotic bounds on the extreme zeros of orthogonal polynomials with polynomially growing coefficients of the three-term recurrence.     

Asymptotics and Zeros of Sobolev Orthogonal Polynomials on Unbounded Supports

In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we focus on the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some directions for future research are formulated.Research of Juan José Moreno Balcázar was partially supported by Ministerio de Educación y Ciencia of Spain under grant MTM2005-08648-C02-01 and Junta de Andalucía (FQM 229 and FQM 481).     

Sobolev orthogonal polynomials in the complex plane

Sobolev orthogonal polynomials with respect to measures supported on compact subsets of the complex plane are considered. For a wide class of such Sobolev orthogonal polynomials, it is proved that their zeros are contained in a compact subset of the complex plane and their asymptotic-zero distribution is studied. We also find the nth-root asymptotic behavior of the corresponding sequence of Sobolev orthogonal polynomials.     

Interlacing properties of zeros of multiple orthogonal polynomials

It is well known that the zeros of orthogonal polynomials interlace. In this paper we study the case of multiple orthogonal polynomials. We recall known results and some recursion relations for multiple orthogonal polynomials. Our main result gives a sufficient condition, based on the coefficients in the recurrence relations, for the interlacing of the zeros of neighboring multiple orthogonal polynomials. We give several examples illustrating our result.     

A class of bivariate orthogonal polynomials and cubature formula

Summary. The existence of Gaussian cubature for a given measure depends on whether the corresponding multivariate orthogonal polynomials have enough common zeros. We examine a class of orthogonal polynomials of two variables generated from that of one variable. Received February 9, 1993 / Revised version received January 18, 1994     

Quadratically Convergent Method for Simultaneously Approaching the Roots of Polynomial Solutions of a Class of Differential Equations: Application to Orthogonal Polynomials

This paper investigates the application of the method introduced by L. Pasquini (1989) for simultaneously approaching the zeros of polynomial solutions to a class of second-order linear homogeneous ordinary differential equations with polynomial coefficients to a particular case in which these polynomial solutions have zeros symmetrically arranged with respect to the origin. The method is based on a family of nonlinear equations which is associated with a given class of differential equations. The roots of the nonlinear equations are related to the roots of the polynomial solutions of differential equations considered. Newton's method is applied to find the roots of these nonlinear equations. In (Pasquini, 1994) the nonsingularity of the roots of these nonlinear equations is studied. In this paper, following the lines in (Pasquini, 1994), the nonsingularity of the roots of these nonlinear equations is studied. More favourable results than the ones in (Pasquini, 1994) are proven in the particular case of polynomial solutions with symmetrical zeros. The method is applied to approximate the roots of Hermite–Sobolev type polynomials and Freud polynomials. A lower bound for the smallest positive root of Hermite–Sobolev type polynomials is given via the nonlinear equation. The quadratic convergence of the method is proven. A comparison with a classical method that uses the Jacobi matrices is carried out. We show that the algorithm derived by the proposed method is sometimes preferable to the classical QR type algorithms for computing the eigenvalues of the Jacobi matrices even if these matrices are real and symmetric.     

Polynomials whose coefficients coincide with their zeros

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree N. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about the existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials \(\{x^N\}_{N=0}^\infty \). We propose a family of solvable N-body problems such that their stable equilibria are the zeros of certain Ulam polynomials.