Behavior of the Fourier–Walsh coefficients of a corrected function

Abstract

We prove that, given a sequence {ak}_k=1^∞ with a_k ↓ 0 and {ak}_k=1^∞ ∉ l₂, reals 0 < ε < 1 and p ∈ [1, 2], and f ∈ L^p(0, 1), we can find f ∈ L^p(0, 1) with mes{f ≠ f < ε whose nonzero Fourier–Walsh coefficients c_k(f) are such that |c_k(f)| = a_k for k ∈ spec(f).