Fourier orthogonal series on a paraboloid

Abstract

We study orthogonal structures and Fourier orthogonal series on the surface of a paraboloidThe reproducing kernels of the orthogonal polynomials with respect to t β (1 − t) γ on V d+1 0 are related to the reproducing kernels of the Jacobi polynomials on the parabolic domainThis connection serves as an essential tool for our study of the Fourier orthogonal series on the surface of the paraboloid, which allow us, in particular, to study the convergence of the Cesàro means on the surface. Analogous results are also established for the solid paraboloid bounded by V d+1 0 and the hyperplane t = 1. 0 be a quadratic surface in R d+1 , parametrized in (x, t), x ∈ R d and t ∈ R, that is a surface of revolution around

Bernstein inequality on conic domains and triangle

“…Our analysis relies on orthogonal structures on the conic domains [25]. It is part of an ongoing program that aims at extending the results in approximation theory and harmonics analysis from the unit sphere to quadratic surfaces of revolution [25,26,27,28]. Our main tools are the closed-form formula for the reproducing kernels in [25] and the highly localized kernel studied in [27].…”