Approximation and localized polynomial frame on conic domains

Highly localized kernels constructed by orthogonal polynomials have been fundamental in 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 frame 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 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 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 several 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.