Approximation and localized polynomial frame on double hyperbolic and conic domains
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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 <cit.> 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.
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