A Review of The Disteributional Derivatives in The Littlewood-paley Inequalities

We known that from the univariate wavelet , we can construct efficient bases for (ℝ) and other function spaces by dilation and shifts. Also using Given a univariate function , we can obtain a multivariate family of functions by taking tensor products. In this paper by using Littlewood-paley inequalities we will make for our approach the some assumptions a bout the multivariate basis and its dual basis.We study the wavelet bases formed by tensor products of univariate wavelets. From that the Littlewood-paley theory applies to many other orthogonal and nonorthogonal expansions, in this paper first by using Littlewood-paley inequalities we stated the assumptions a bout multivariate basis. Then under concideration univariate wavelet we define the seminorm for the set of all functions in whose distributional derivatives are in space, also in format a theorem we acquired Necessary and suffic

Keywords:

Littlewood-paley inequalities , hyperbolic wavelet , univariate function , tensor product , dyadic rectangles