Localized Polynomial Frames on the Ball

Abstract

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