The Critical Smoothness of Generalized Functions

Abstract

For each integrability parameter p ∈ (0, ∞], the critical smoothness of a periodic generalized function f , denoted by s f (p) is the supremum over the smoothness parameters s for which f belongs to the Besov space B s p,p (T) (or other similar function spaces). This paper investigates the evolution of the critical smoothness with respect to the integrability parameter p. Our main result is a simple characterization of all the possible critical smoothness functions p → s f (p) when f describes the space of generalized periodic functions. We moreover characterize the compressibility of generalized periodic functions in wavelet bases from the knowledge of their critical smoothness function.Proof. Let f ∈ S ′ (T) with p, q 1 , q 2 as stated above, and let s n be an increasing sequence converging to s f (p, q 1 ). Then since f ∈ B sn p,q 1 (T) for all n, we have by the embedding properties1 Kabanava considers the case of tempered generalized functions but the result easily applies to the periodic setting.