Approximation and localized polynomial frame on conic domains

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