On the Fourier-Walsh coefficients (Q987923)

Here the behavior of Fourier-Walsh coefficients after modification of functions is considered. The main result contained in the theorem is given below: Theorem: For any \(0<\varepsilon> 1\), \(p\geq 1\) and each function \(f\in L^p[0,1]\) one can find a function \(g\in L^p[0,1]\), \(\text{mes}\{x\in [0,1]; g\neq f\}<\varepsilon\), such that thesequence \(\{|c_k(g)|, k\in\text{spec}(g)\}\) is monotonically decreasing, where \(\{c_k(g)\}\) is a sequence of Fourier-Walsh coefficients of the function \(g(x)\).