On the Structure of Functions, Universal for Weighted Spaces \(L_\mu ^p\left[ {0,1} \right],p > 1\)

Abstract

The paper is devoted to the questions relating the structure of universal functions for weighted spaces \(L_\mu ^p\left[{0,\;1} \right],\;p > 1\). We prove existence of a measurable set E ⊂ [0, 1] with measure arbitrarily close to 1, and a weight function 0 < µ(x) ≤ 1, equal to 1 on E, such that by suitable continuation of an arbitrary function f ∈ L¹(E) on [0, 1] \ E, a function \(\widetilde{f}\; \in \;{L^1}\left[{0,\;1} \right]\) can be obtained, which is universal for each class \(L_\mu ^p\left[{0,\;1} \right],\;p > 1\), in the sense of subsequences of signs of its Fourier-Walsh coefficients.