Localization for Fourier series on 𝑆𝑈(2)

Abstract

Let N(b) be the space of functionsf E L1(SU(2)) which vanish on a neighborhood Vf of b. The main results of this paper are Theorems A-C below. THEOREM A. Let b E SU(2), fE N(b). If all of the first derivatives off (in the distribution sense) are functions in LP(SU(2)) for some p > 3/2, then limn_ , Sn(f: b) = 0. If p < 3/2, there is a function f E N(b) whose first derivatives are all functions in LP(SU(2)) and such that limn_.o Sn(f: b) does not exist. THEOREM B. Iff fE N(b) n N(b) and the first derivatives of f are functions in L1(SU(2)), then limn_ c, Sn(f b)=0. Say a functionf E L1(SU(2)) is of bounded variation if its first derivatives are all measures.