The periodic complex method in interpolation spaces

TL;DR

This paper investigates the periodic complex interpolation method, demonstrating that as the period increases, the associated interpolation spaces become increasingly equivalent to the original spaces, with constants approaching one.

Contribution

It proves that the equivalence constants in the periodic complex interpolation approach one as the period tends to infinity, extending understanding of the method's stability.

Findings

Equivalence constants tend to 1 as period increases

Periodic and original interpolation spaces become similar for large periods

The difference diminishes with increasing annulus size

Abstract

We discuss a question which relates to Calderon's complex interpolation method. More precisely, we will consider the so-called "periodic" complex interpolation method, studied by Peetre. (Which also corresponds to the spaces obtained by Calderon's construction using Banach space valued analytic functions, but defined on an annulus instead of the strip used by Calderon.) Cwikel showed that, using functions with a given period i\lambda in the complex method mechanism, one obtains the same interpolation spaces as in the original version of the complex method, up to equivalence of norms. He also showed that one of the constants of this equivalence will, in some cases, "blow up" as \lambda tends to zero. We will show that the equivalence constants tend to 1 as \lambda tends to infinity. Intuitively, this means that when applying the complex method, it makes a very small difference if one restricts oneself to periodic functions, provided that the period is very large (or the corresponding annulus is very thin).