Approximation and localized polynomial frame on double hyperbolic and conic domains

2021/05/23 by Yuan Xu, Xu, Yuan · 1 citation
Engineering · #41A10 #41A63 #42C10 #42C40 #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #Elasticity and Material Modeling #FOS: Mathematics #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2105.10958

openalex publication_date 2021/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \citeX21 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 L² 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.

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