Approximation and localized polynomial frame on conic domains

Abstract

Highly localized kernels constructed by orthogonal polynomials have been fundamental in the recent development of approximation and computational analysis on the unit sphere, unit ball, and several other regular domains. In this work, we first study homogeneous spaces that are assumed to contain highly localized kernels and establish a framework for approximation and localized tight frames in such spaces, which extends recent works on bounded regular domains. We then show that the framework is applicable to homogeneous spaces defined on bounded conic domains, which consists of conic surfaces and the solid domains bounded by such surfaces and hyperplanes. The highly localized kernels on conic domains require precise estimates that rely on recently discovered addition formulas for orthogonal polynomials with respect to special weight functions on each domain and an intrinsic distance that takes into account the boundary of the domain, the latter is not comparable to the Euclidean distance at around the apex of the cone.The main results provide construction of semi-discrete localized tight frame in weighted L2 norm and characterization of best approximation by polynomials on conic domains. The latter is achieved by using a K-functional, defined via the differential operator that has orthogonal polynomials as eigenfunctions, as well as a modulus of smoothness defined via a multiplier operator that is equivalent to the K-functional. Several intermediate results are of interest in their own right, including the Marcinkiewicz-Zygmund inequalities, positive cubature rules, Christoeffel functions, and Bernstein type inequalities. Moreover, although the highly localizable kernels hold only for special families of weight functions on each domain, many intermediate results are shown to hold for doubling weights defined via the intrinsic distance on the domain.