Approximation and localized polynomial frame on double hyperbolic and conic domains

TL;DR

This paper develops approximation techniques and localized polynomial frames on double hyperbolic and conic domains, extending recent frameworks for homogeneous spaces with localized kernels and providing tools for best polynomial approximation.

Contribution

It introduces a construction of semi-discrete localized tight frames and characterizes polynomial approximation on complex double hyperbolic and conic domains, using orthogonal polynomials and reproducing kernels.

Findings

Existence of highly localized kernels via closed form formulas.

Construction of semi-discrete localized tight frames in weighted L^2 norm.

Validation of Marcinkiewicz-Zygmund inequalities, cubature rules, and Bernstein inequalities on the domains.

Abstract

We study approximation and localized polynomial frames on a bounded double hyperbolic or conic surface and the domain bounded by such a surface and hyperplanes. The main work follows the framework developed recently in \cite{X21} for homogeneous spaces that are assumed to contain highly localized kernels constructed via a family of orthogonal polynomials. The existence of such kernels will be established with the help of closed form formulas for the reproducing kernels. The main results provide a construction of semi-discrete localized tight frame in weighted $L^2$ norm and a characterization of best approximation by polynomials on our domains. Several intermediate results, including the Marcinkiewicz-Zygmund inequalities, positive cubature rules, Christoeffel functions, and Bernstein type inequalities, are shown to hold for doubling weights defined via the intrinsic distance on the domain.