The periodic complex method in interpolation spaces

Abstract

In this paper we consider the “periodic” variant of the complex interpolation method, apparently first studied by Peetre (Rend Sem Mat Univ Padova 46:173–190, 1971). Cwikel showed (Indiana Univ Math J 27:1005–1009, 1978) that using functions with a given period $$i\lambda $$ in the complex method construction introduced and studied by Calderón (Studia Math 24:113–190, 1964), one may construct the same interpolation spaces as in the “regular” complex method, up to equivalence of norms. Cwikel also showed that one of the constants of this equivalence will, in some cases, “blow up” as $$\lambda \rightarrow 0$$ . (The other constant is obviously bounded by 1.) We show that this same equivalence constant approaches $$1$$ as $$\lambda \rightarrow \infty $$ . Intuitively, this means that when applying the complex method of Calderón, it makes a very small difference if one restricts oneself to periodic functions, provided that the period is very large.