相关论文: Localized polynomial frames on the interval with J…
Radial eigenfunctions of the Laplace-Beltrami operator on compact rank-one symmetric spaces may be expressed in terms of Jacobi polynomials. We use this fact to prove an identity for Jacobi polynomials which is inspired by results of…
The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Jacobi-Sobolev bilinear form with mass point at $-1$ and/or $+1$. In particular, we construct the orthogonal polynomials using…
Multivariate versions of classical orthogonal polynomials such as Jacobi, Hahn, Laguerre and Meixner are reviewed and their connection explored by adopting a probabilistic approach. Hahn and Meixner polynomials are interpreted as posterior…
Let $w$ be a weight on the unit disk $\mathbb{D}$ having the form \[w(z)=|v(z)|^2\prod_{k=1}^s\left|\frac{z-a_k}{1-z\overline{a}_k}\right|^{m_k}\,,\quad m_k>-2,\ |a_k|<1,\] where $v$ is analytic and free of zeros in $\overline{\mathbb{D}}$,…
The linearization coefficients for a set of orthogonal polynomials are given explicitly as a weighted sum of combinatorial objects. Positivity theorems of Askey and Szwarc are corollaries of these expansions.
In this paper, it is shown that every polynomial function is mixed monotone globally with a polynomial decomposition function. For univariate polynomials, the decomposition functions can be constructed from the Gram matrix representation of…
We study multivariate $L_2$-approximation for a weighted Korobov space of analytic periodic functions for which the Fourier coefficients decay exponentially fast. The weights are defined, in particular, in terms of two sequences…
We discuss some properties of the moduli of smoothness with Jacobi weights that we have recently introduced and that are defined as \[ \omega_{k,r}^\varphi(f^{(r)},t)_{\alpha,\beta,p} :=\sup_{0\leq h\leq t} \left\|…
Orthogonal polynomials on the product domain $[a_1,b_1] \times [a_2,b_2]$ with respect to the inner product $$ \langle f,g \rangle_S = \int_{a_1}^{b_1} \int_{a_2}^{b_2} \nabla f(x,y)\cdot \nabla g(x,y)\, w_1(x)w_2(y) \,dx\, dy + \lambda…
We study the problem of space and time efficient evaluation of a nonparametric estimator that approximates an unknown density. In the regime where consistent estimation is possible, we use a piecewise multivariate polynomial interpolation…
We illustrate how Jordan algebras can provide a framework for the interpretation of certain classes of orthogonal polynomials. The big -1 Jacobi polynomials are eigenfunctions of a first order operator of Dunkl type. We consider an algebra…
A ladder structure of operators is presented for the Jacobi polynomials, J_n^(a,b)(x), with parameters n, a and b integers, showing that they are related to the unitary irreducible representation of SU(2,2) with quadratic Casimir…
We develop a theory of Jacobi polynomials for parabolic subgroups of finite reflection groups that specializes to the cases studied by Heckman and Opdam in which the whole group and the trivial group are considered. For the intermediate…
Functional and linear-algebraic approaches to the Delsarte problem of upper bounds on codes are discussed. We show that Christoffel-Darboux kernels and Levenshtein polynomials related to them arise as stationary points of the moment…
A semi-infinite weighted Hankel matrix with entries defined in terms of basic hypergeometric series is explicitly diagonalized as an operator on $\ell^{2}(\mathbb{N}_{0})$. The approach uses the fact that the operator commutes with a…
This work initiates the study of {\it orthogonal} symmetric polynomials in superspace. Here we present two approaches leading to a family of orthogonal polynomials in superspace that generalize the Jack polynomials. The first approach…
Modular, Jacobi, and mock-modular forms serve as generating functions for BPS black hole degeneracies. By training feed-forward neural networks on Fourier coefficients of automorphic forms derived from the Dedekind eta function, Eisenstein…
In this paper, models that approximate stochastic processes from the space $Sub_\varphi(\Omega)$ with given reliability and accuracy in $L_p(T)$ are considered for some specific functions $\varphi(t)$. For processes that are decomposited in…
We give formulas for the density of the measure of orthogonality for orthonormal polynomials with unbounded recurrence coefficients. The formulas involve limits of appropriately scaled Tur\'an determinants or Christoffel functions. Exact…
We establish dispersive time-decay estimates for periodic Jacobi operators on the discrete half-line, $\N$. Specifically, we prove $t^{-1/2}$ decay in the weighted $\ell^\infty_{-1}$ norm for all such operators. For the global $\ell^1 \to…