## Localized polynomial frames on the ball

**A 2006 Preprint by
P. Petrushev and
Y. Xu
**

- 2006:08
Almost exponentially localized polynomial kernels are constructed on the unit ball $B^d$ in $R^d$ with weights $W _ \mu(x)=(1-|x|^2)^{\mu-\frac{1}{2}},$ $\mu\geq0$, 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 $W _ \mu(x)$ and to the constriction of polynomial tight frames in $L^2(B^d,W _ \mu)$ (called needlets) whose elements have nearly exponential localization.