The non-equivalence between the trigonometric system and the system of functions with pointwise restrictions on values in the uniform and L1 norms

Abstract

Let n denote the space of trigonometric polynomials of degree n i.e. n = span(e−ikt : |k| ≤ n) ⊂ Lp() and let (Ω, dx) be any mesurable space with finite measure. In this paper we use the quantitative version of the Helson-Rudin-Cohen idempotent theorem due to Green and Sanders (cf. [3]) to prove the following.