A method to compute recurrence relation coefficients for bivariate orthogonal polynomials by unitary matrix transformations

We present an algorithm computing recurrence relation coefficients for bivariate polynomials, orthonormal with respect to a discrete inner product. These polynomials make it possible to give the solution of a discrete least squares approximation problem. To compute these polynomials, we pose the inverse eigenvalue problem and solve it efficiently and in a stable way, using a sequence of Givens rotations. We also show how to generalize the algorithm for the case of polynomials in more variables. Several numerical experiments show the validity of the approach.     

An Inequality for Trigonometric Polynomials and Its Application for Estimating the Entropy Numbers

We prove in the two-dimensional case an inequality for trigonometric polynomials with frequencies between two hyperbolic crosses. This inequality is an analog of Talagrand′s inequality for the Haar polynomials. We use this inequality to prove some new estimates of the entropy numbers of classes of functions with bounded mixed difference or derivative in the most difficult case of the uniform norm.     

Some remarks on a paper by L. Carlitz

We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials.     

Differential equations for the elementary 3-symmetric Chebyshev polynomials

We continue the study of a “compound model of a generalized oscillator” and related elementary 3-symmetric Chebyshev polynomials. For these polynomials, we obtain second-order differential equations which are of Fuchs type and have 13 singular points. In the considered simplest case, the obtained results give us an answer to a more general question: What changes in the differential equations for polynomials of the Askey–Wilson scheme when the Jacobi matrix related to these polynomials is perturbed by a diagonal matrix with a complex diagonal? Bibliography: 8 titles.     

On the condition of algebraic equations

Summary Given an algebraic equation, in which the polynomial in question is expressed in terms of any set of basis polynomials, we study the sensitivity of the roots with respect to small perturbations in the coefficients of the equation. The degree of sensitivity of each root is measured by an appropriate condition number. We analyze this condition number first in the case where the basis polynomials are the powers, and then, in less detail, in the case where the basis is a set of orthogonal polynomials. Several examples are treated, allowing for a comparative study.Work performed in part at the U.S.A.F. Aerospace Research Laboratories under contract F33615-71-C-1463 with Technology Incorporated.     

Some Orthogonal Polynomials Related to Elliptic Functions

We characterize the orthogonal polynomials in a class of polynomials defined through their generating functions. This led to three new systems of orthogonal polynomials whose generating functions and orthogonality relations involve elliptic functions. The Hamburger moment problems associated with these polynomials are indeterminate. We give infinite families of weight functions in each case. The different polynomials treated in this work are also polynomials in a parameter and as functions of this parameter they are orthogonal with respect to unique measures, which we find explicitly. Through a quadratic transformation we find a new exactly solvable birth and death process with quartic birth and death rates.     

Factorizations of tropical and sign polynomials

In this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call tropical polynomials and sign polynomials, respectively. We classify all irreducible polynomials in either case. We show that tropical polynomials factor uniquely into irreducible factors, but that unique factorization fails for sign polynomials. We describe division algorithms for tropical and sign polynomials by linear terms that correspond to roots of the polynomials.     

On Paravector Valued Homogeneous Monogenic Polynomials with Binomial Expansion

The aim of this note is to study a set of paravector valued homogeneous monogenic polynomials that can be used for a construction of sequences of generalized Appell polynomials in the context of Clifford analysis. Therefore, we admit a general form of the vector part of the first degree polynomial in the Appell sequence. This approach is different from the one presented in recent papers on this subject. We show that in the case of paravector valued polynomials of three real variables, there exist essentially two different types of such polynomials together with two other trivial types of polynomials. The proof indicates a way of obtaining analogous results in the case of polynomials of more than three variables.     

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.     

Radial orthogonality and Lebesgue constants on the disk

In polynomial interpolation, the choice of the polynomial basis and the location of the interpolation points play an important role numerically, even more so in the multivariate case. We explore the concept of spherical orthogonality for multivariate polynomials in more detail on the disk. We focus on two items: on the one hand the construction of a fully orthogonal cartesian basis for the space of multivariate polynomials starting from this sequence of spherical orthogonal polynomials, and on the other hand the connection between these orthogonal polynomials and the Lebesgue constant in multivariate polynomial interpolation on the disk. We point out the many links of the two topics under discussion with the existing literature. The new results are illustrated with an example of polynomial interpolation and approximation on the unit disk. The numerical example is also compared with the popular radial basis function interpolation.     

A Littlewood–Richardson rule for Macdonald polynomials

Macdonald polynomials are orthogonal polynomials associated to root systems, and in the type A case, the symmetric Macdonald polynomials are a common generalization of Schur functions, Macdonald spherical functions, and Jack polynomials. We use the combinatorics of alcove walks to calculate products of monomials and intertwining operators of the double affine Hecke algebra. From this, we obtain a product formula for Macdonald polynomials of general Lie type.     

Polynomial functions on upper triangular matrix algebras

There are two kinds of polynomial functions on matrix algebras over commutative rings: those induced by polynomials with coefficients in the algebra itself and those induced by polynomials with scalar coefficients. In the case of algebras of upper triangular matrices over a commutative ring, we characterize the former in terms of the latter (which are easier to handle because of substitution homomorphism). We conclude that the set of integer-valued polynomials with matrix coefficients on an algebra of upper triangular matrices is a ring, and that the set of null-polynomials with matrix coefficients on an algebra of upper triangular matrices is an ideal.     

Asymptotics for the ratio and the zeros of multiple Charlier polynomials

We investigate multiple Charlier polynomials and in particular we will use the (nearest neighbor) recurrence relation to find the asymptotic behavior of the ratio of two multiple Charlier polynomials. This result is then used to obtain the asymptotic distribution of the zeros, which is uniform on an interval. We also deal with the case where one of the parameters of the various Poisson distributions depends on the degree of the polynomial, in which case we obtain another asymptotic distribution of the zeros.     

Polynomials,random walks and risk processes: a multivariate framework

This paper is concerned with two families of multivariate polynomials: the Appell polynomials and the Abel-Gontcharoff polynomials. Both families are well-known in the univariate case, but their multivariate version is much less standard. We first provide a simple interpretation of these polynomials through particular constrained random walks on a lattice. We then derive nice analytical results for two special cases where the parameters of the polynomials are randomized. Thanks to the interpretation and randomization of the polynomials, we can derive new results and give other insights for the study of two different risk problems: the ruin probability in a multiline insurance model and the size distribution in a multigroup epidemic.     

Varying Jacobi–Krall orthogonal polynomials: local asymptotic behaviour and zeros

Krall orthogonal polynomials are well known and they constitute a generalization of classical orthogonal polynomials obtained by addition of positive masses located at some points on the real line. In this contribution we consider two families of Krall polynomials already known in the literature, but now the corresponding absolutely continuous measure is perturbed by a sequence of nonnegative masses located at the point 1 in the Jacobi case and at the end points of the interval of orthogonality in the Gegenbauer case. We analyze the asymptotic behaviour of these varying Krall orthogonal polynomials in the neighbourhood of the points where the perturbation has been done. To do this we use Mehler–Heine type asymptotic formulae. As a consequence we can establish limit relations between the zeros of these polynomials and the ones of the Bessel functions of the first kind (or linear combinations of them). We do some numerical experiments to illustrate the results.     

Isometries between spaces of homogeneous polynomials

We derive Banach-Stone theorems for spaces of homogeneous polynomials. We show that every isometric isomorphism between the spaces of homogeneous approximable polynomials on real Banach spaces E and F is induced by an isometric isomorphism of E^′ onto F^′. With an additional geometric condition we obtain the analogous result in the complex case. Isometries between spaces of homogeneous integral polynomials and between the spaces of all n-homogeneous polynomials are also investigated.     

Edgeworth–Cornish–Fisher–Hill–Davis expansions for normal and non-normal limits via Bell polynomials

Cornish and Fisher gave expansions for the distribution and quantiles of asymptotically normal random variables whose cumulants behaved like those of a sample mean. This was extended by Hill and Davis to the case, where the asymptotic distribution need not be normal. Their results are cumbersome as they involve partition theory. We overcome this using Bell polynomials. The three basic expansions (for the distribution and its derivatives, for the inverse of the quantile, and for the quantile) involve three sets of polynomials. We give new ways of obtaining these from each other. The Edgeworth expansions for the distribution and density rest on the Charlier expansion. We give an elegant form of these as linear combinations of generalized Hermite polynomials, using Bell polynomials.     

An alternative proof of Pellet’s theorem for matrix polynomials

We propose an alternative proof of Pellet’s theorem for matrix polynomials that, unlike existing proofs, does not rely on Rouché’s theorem. A similar proof is provided for the generalization to matrix polynomials of a result by Cauchy that can be considered as a limit case of Pellet’s theorem.     

Multiplication of a Schubert polynomial by a Schur polynomial

Schur polynomials are a special case of Schubert polynomials. In this paper, we give an algorithm to compute the product of a Schubert polynomial with a Schur polynomial on the basis of Schubert polynomials. This is a special case of the general problem of the multiplication of two Schubert polynomials, where the corresponding algorithm is still missing. The main tools for the given algorithm is a factorization property of a special class of Schubert polynomials and the transition formula for Schubert polynomials.     

Optimally Space Localized Polynomials with Applications in Signal Processing

For the filtering of peaks in periodic signals, we specify polynomial filters that are optimally localized in space. The space localization of functions f having an expansion in terms of orthogonal polynomials is thereby measured by a generalized mean value ε(f). Solving an optimization problem including the functional ε(f), we determine those polynomials out of a polynomial space that are optimally localized. We give explicit formulas for these optimally space localized polynomials and determine in the case of the Jacobi polynomials the relation of the functional ε(f) to the position variance of a well-known uncertainty principle. Further, we will consider the Hermite polynomials as an example on how to get optimally space localized polynomials in a non-compact setting. Finally, we investigate how the obtained optimal polynomials can be applied as filters in signal processing.