Exact bounds for norms of differentiable periodic functions in L metric

Abstract

Exact values are obtained for the upper bounds on the norms,∥f∥L, on the classes W ^(r)_∩ H_ω(r =0, 1, 2,...) of r-fold differentiable functions f(x), of periodicity2π, for which ω(f^(r);t)≤ω(t),where ω(t) is a given convex modulus of continuity.