On the minimum norm property of the Fourier projection in $L^1$-spaces and in spaces of continuous functions

Abstract

Introduction, 1. Let C(T) be the Banach space of complex continuous periodic functions on the real line, and L{T) the Banach space of complex periodic functions on the real line which are absolutely integrable on [0, 2w). For simplicity we shall sometimes denote both spaces by E(T). Let then En be the space of trigonometric polynomials ^Ct^-n *i a n d l e t Fn:E(T)—>En be the Fourier projection, defined by