Polynomial Wavelet-Type Expansions on the Sphere

We present a polynomial wavelet-type system on $S^d$ such that any continuous function can be expanded with respect to these “wavelets”. The order of the growth of the degrees of the polynomials is optimal. The coefficients in the expansion are the inner products of the function and the corresponding element of a “dual wavelet system”. The “dual wavelets system” is also a polynomial system with the same growth of degrees of polynomials. The system is redundant. A construction of a polynomial basis is also presented. In contrast to our wavelet-type system, this basis is not suitable for implementation, because, first, there are no explicit formulas for the coefficient functionals and, second, the growth of the degrees of polynomials is too rapid.