Explicit construction of matrix-valued orthogonal polynomials of arbitrary size

Some examples of orthogonal matrix polynomials satisfying odd order differential equations

When an attempt is made to construct an even particular solution of Mathieu's differential equation (written in a slightly modified notation), either with period 2π or with period 4π, by inserting into it an appropriate trigonometric series of the cosine type, it appears that the coefficients of this series satisfy a recurrence relation of a kind encountered in the theory of orthogonal polynomials. After splitting off a certain proportionality factor in each coefficient, there ultimately result two infinite sequences of parameter-dependent orthogonal polynomials constituting, respectively, a generalization of the Chebyshev polynomials of the first kind {Tn(x)} and a generalization of another special case of the Jacobi polynomials, namely {P(1/2,−1/2)n(x)}. The weight function corresponding to each of these orthogonal sequences consists of an infinite series of Dirac δ-peaks (mass-points) whose locations and (positive) coefficients are given by quantities appearing in the standard theory of Mathieu functions. With the aid of the author's theory of recursive generation of systems of orthogonal polynomials, each of the two orthogonal sequences may be complemented with infinitely many systems of associated orthogonal polynomials whose recurrence relations present a positive integer shift on the discrete independent variable compared to the recursion of the orthogonal sequence from which they originate. The treatment of Mathieu's differential equation with trigonometric series of the sine type, in view of constructing an odd particular solution with 2π or 4π as period, does not produce any sequence of orthogonal polynomials which is in essence not already comprised in those previously obtained.

In this work, we give some criteria that allow us to decide when two sequences of matrix-valued orthogonal polynomials are related via a Darboux transformation and how to build such a transformation explicitly. In particular, they allow us to see when and how any given sequence of polynomials is Darboux-related to classic orthogonal polynomials. We also explore the notion of Darboux-irreducibility and study some sequences that are not a Darboux transformation of classical orthogonal polynomials.

When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?

A method of composition orthogonality and new sequences of orthogonal polynomials and functions for non-classical weights

This paper is concerned with energy-to-peak state estimation on static neural networks (SNNs) with interval time-varying delays. The objective is to design suitable delay-dependent state estimators such that the peak value of the estimation error state can be minimized for all disturbances with bounded energy. Note that the Lyapunov-Krasovskii functional (LKF) method plus proper integral inequalities provides a powerful tool in stability analysis and state estimation of delayed NNs. The main contribution of this paper lies in three points: 1) the relationship between two integral inequalities based on orthogonal and nonorthogonal polynomial sequences is disclosed. It is proven that the second-order Bessel-Legendre inequality (BLI), which is based on an orthogonal polynomial sequence, outperforms the second-order integral inequality recently established based on a nonorthogonal polynomial sequence; 2) the LKF method together with the second-order BLI is employed to derive some novel sufficient conditions such that the resulting estimation error system is globally asymptotically stable with desirable energy-to-peak performance, in which two types of time-varying delays are considered, allowing its derivative information is partly known or totally unknown; and 3) a linear-matrix-inequality-based approach is presented to design energy-to-peak state estimators for SNNs with two types of time-varying delays, whose efficiency is demonstrated via two widely studied numerical examples.

A connection between orthogonal polynomials on the unit circle and matrix orthogonal polynomials on the real line

Continuous symmetrized Sobolev inner products of order N (I)

This research paper delves into the properties and convergence behaviors of various sequences of orthogonal polynomials, reproducing kernels, and bases within Hilbert spaces governed by norm-attainable operators. Through rigorous analysis, the study establishes the completeness of the sequences of monic orthogonal polynomials and orthonormal polynomials, highlighting their comprehensive representation and approximation capabilities in the Hilbert space. The paper also demonstrates the completeness and density attributes of the sequence of normalized reproducing kernels, showcasing its effective role in capturing the intrinsic structure of the space. Additionally, the research investigates the uniform convergence of these sequences, revealing their convergence to essential operators within the Hilbert space. Ultimately, these results contribute to both theoretical understanding and practical applications in various fields by providing insights into function approximation and representation within this mathematical framework.

New sequences of orthogonal polynomials with ultra-exponential weight functions are discovered. In particular, we give an explicit solution to the Ditkin–Prudnikov problem (1966). The 3-term recurrence relations, explicit representations, generating functions and Rodrigues-type formulae are derived. The method is based on differential properties of the involved special functions and their representations in terms of the Mellin–Barnes and Laplace integrals. A notion of the composition polynomial orthogonality is introduced. The corresponding advantages of this orthogonality to discover new sequences of polynomials and their relations to the corresponding multiple orthogonal polynomial ensembles are shown.

A construction of real weight functions for certain orthogonal polynomials in two variables

Given any sequence of orthogonal polynomials, satisfying the three term recurrence relation xpn(x) = ?n+1pn+1(x) + ?npn(x) + ?npn-1(x), p-1(x)=0 po(x)=1 with ?n ? 0, n ? N, ?0 = 1, an infinite Jacobi matrix can be associated in the following way ? ? ??0 ?1 0...? ? ? ??1 ?1 ?2...? ? ? J = ?0 ?2 ?2...? ?.... ? ?.... ? ?....? ? ? In the general case if the sequences {?n} or {?n} are complex the associated Jacobi matrix is complex. Under the condition that both sequences {?n}and {?n} are uniformly bounded, the associated Jacobi matrix can be understood as a linear operator J acting on ?2, the space of all complex square-summable sequences, where the value of the operator J at the vector x is a product of an infinite vector x and an infinite matrix J in the matrix sense. The case when the sequences {?n} and {?n} are not uniformly bounded, an operator acting on ?2 can not be defined that easily. Additional properties of the sequence of orthogonal polynomials are needed in order to be able to define the operator uniquely. The case when the sequences ?n and ?n are real is very well understood. The spectra of the Jacobi matrix J equals the support of the measure of orthogonality for the given sequence of orthogonal polynomials. All zeros of orthogonal polynomials are real, simple and interlace, contained in the convex hull of the spectra of the Jacobi operator associated with the infinite Jacobi matrix J. Every point in ?(J) attracts zeros of orthogonal polynomials. An application of orthogonal polynomials is the construction of quadrature rules for the approximation of integration with respect to the measure of orthogonality. For arbitrary sequences {?n} and {?n} the situation is changed dramatically. Zeros of orthogonal polynomials need not be simple; they are not real and they do not necessarily lie in the convex hull of ?(J). There is also a little known about convergence results of related quadrature rule. Only in recent years a connection between complex Jacobi matrices and related orthogonal polynomials is interesting again (see [2]). Studies of complex Jacobi operators should lead to a better understanding of related orthogonal polynomials, but also the study of orthogonal polynomials with the complex Jacobi matrices should put more light on the non-hermitian banded symmetric matrices. In this lecture some results are given about complex Jacobi matrices and related quadrature rules, and also some interesting examples are presented.

Recurrence coefficients and difference equations of classical discrete orthogonal and q-orthogonal polynomial sequences

The aim of paper is to give some limit relationships between finite and infinite sequences of orthogonal polynomials in two variables and to obtain the well-known relations of some infinite sets of two-variable orthogonal polynomials by taking limit of the properties verified by finite classes of two-variable orthogonal polynomials. Furthermore, the fourth-order partial differential equations satisfied by the polynomials, which are the products of finite orthogonal polynomials, are presented and by taking limit of the derived fourth-order equations, partial differential equations for some infinite sequences of two-variable orthogonal polynomials are found.

Let (P ν) be a sequence of monic polynomials orthogonal on the unit circle with respect to a nonnegative weight function, let (Ωυ) the monic associated polynomials of (P v), and letA andB be self-reciprocal polynomials. We show that the sequence of polynomials (APυλ+BΩυλ)/Aλ, λ stuitably determined, is a sequence of orthogonal polynomials having, up to a multiplicative complex constant, the same recurrence coefficients as theP ν's from a certain index value onward, and determine the orthogonality measure explicity. Conversely, it is also shown that every sequence of orthogonal polynomials on the unit circle having the same recurrence coefficients from a certain index value onward is of the above form. With the help of these results an explicit representation of the associated polynomials of arbitrary order ofP ν and of the corresponding orthogonality measure and Szego function is obtained. The asymptotic behavior of the associated polynomials is also studied. Finally necessary and suficient conditions are given such that the measure to which the above introduced polynomials are orthogonal is positive.