Localized polynomial frames on the ball

Abstract

Almost exponentially localized polynomial kernels are constructed on the unit ball $B^d$ in $\RR^d$ with weights %functions $W_μ(x)= (1-|x|^2)^{μ-1/2}$, $μ\ge 0$, by smoothing out the coefficients of the corresponding orthogonal projectors. These kernels are utilized to the design of cubature formulae on $B^d$ with respect to $W_μ(x)$ and to the construction of polynomial tight frames in $L^2(B^d, W_μ)$ (called needlets) whose elements have nearly exponential localization.