Littlewood-Paley Type Estimates for Besov Spaces on a Cube by Wavelet Coefficients

Abstract

This paper deals with Littlewood-Paley type estimates of the Besov spaces {{{{ { B}`_{p,q } ^{ α } }}}} on the d-dimensional unit cube for 0< p,q< ∞ by two certain classes. These classes are including biorthogonal wavelet systems or dual multiscale systems but not necessarily obtained as the dilates or translates of certain fixed functions. The main assumptions are local supports of both classes, sufficient smoothness for one class, and sufficiently many vanishing moments for the other class. With these estimates, we characterize the Besov spaces by coefficient norms of decompositions with respect to biorthogonal wavelet systems on the cube.