多元正交多项式。I

Some notes on orthogonally additive polynomials

We provide two new characterizations of bounded orthogonally additive polynomials from a uniformly complete vector lattice into a convex bornological space using separately two polynomial identities of Kusraeva involving the root mean power and the geometric mean. Furthermore, it is shown that a polynomial on a vector lattice is orthogonally additive whenever it is orthogonally additive on the positive cone. These results improve recent characterizations of bounded orthogonally additive polynomials by G. Buskes and the author.

多元正交多项式

Multi-variable orthogonal polynomials

刻画了有限原子数的$\mathbb{R}^d$上的原子概率测度.我们进一步证明了与多重Hermite（resp. Laguerre，resp. Jacobi）正交多项式是对角矩阵。最后，作为多Jacobi正交多项式情形的推广，给出了Gegenbauer，Chebyshev和Legendre正交多项式的Jacobi序列.

Self-inversive polynomial and quasi-orthogonality on the unit circle

In this paper we study quasi-orthogonality on the unit circle based on the structural and orthogonal properties of a class of self-invariant polynomials. We discuss a special case in which these polynomials are represented in terms of the reversed Szegő polynomials of consecutive degrees and illustrate the results using contiguous relations of hypergeometric functions. This work is motivated partly by the fact that recently cases have been made to establish para-orthogonal polynomials as the unit circle analogues of quasi-orthogonal polynomials on the real line so far as spectral properties ar…

离散变量的正交多项式与Dirichlet核的有界性

Orthogonal polynomials of discrete variable and boundedness of Dirichlet kernel

对于系数指数衰减的紧Jacobi矩阵所定义的正交多项式，给出了正交性测度的精确性质。这允许显示均匀有界的正交展开式的部分和关于$L^\infty$规范，推广了类似的结果，小$q$ -Legendre，小$q$ -Jacobi和小$q$ -Laguerre多项式，作者。

多个非交换变量的正交多项式。II

Orthogonal polynomials in several non-commuting variables. II

本文继续研究了一类Hankel型正定核的正交多项式性质。本文的主要结果是关于这类核的结构。

A Derivation of Classical Orthogonal Polynomials using Generalized Vandermonde Determinants

We present a derivation of classical Hermite, Laguerre, and Jacobi orthogonal polynomials directly through the Gram-Schmidt orthogonization process. The derivation uses certain generalized Vandermonde determinants with entries defined by Gamma and Beta functions. We also provide a geometric formulation of Gram-Schmidt orthogonalization using the Hodge star operator.