Wavelet characterization of Sobolev norms∗

Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. We begin with the classical definition of Sobolev spaces. Definition 1. Let k be a nonnegative integer and let 1 < p < ∞ . The Sobolev space W k,p(Rn) is defined as the space of functions f in Lp(Rn) all of whose distributional derivatives ∂αf are also in Lp(Rn) for all multi-indices α that satisfy |α| ≤ k. This space is normed by the expression ‖f‖W k,p = ∑