On the Structure of Functions, Universal for Weighted Spaces

摘要

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