On the universal and strong \((L^1,L^\infty)\)-property related to Fourier-Walsh series. (English) Zbl 1376.42037

Grigoryan, Martin G.

Banach J. Math. Anal. 11, No. 3, 698-712 (2017).

Let \(r(x)\) be equal to \(1\) on \([0,1/2)\) and to \(-1\) on \([1/2,1)\) and be \(1\)-periodic. If \(n\in\mathbb Z_+=\{0,1,2,\dots\}\) is represented in the dyadic form \(n=\sum^\infty_{k=0}n_k2^k\), \(n_k\in\{0,1\}\), then the \(n\)-th Walsh-Paley function is defined by \[ \varphi_n(x)=\prod^\infty_{k=0}r^{n_k}_k(x), \quad r_k(x)=r(2^kx), \quad x\in [0,1). \] Since \(\{\varphi_n\}^\infty_{n=0}\) is orthonormal on \([0,1]\), the Fourier-Walsh coefficients \(c_n(f)\) are defined for \(f\in L^1[0,1)\) in a standard way. The main result of paper is
Theorem 1.4. There exists a function \({U\in L^1}[0,1)\) with strictly decreasing Fourier-Walsh coeffcients \(\{c_k(U)\}^\infty_{k=0}\) such that, for any \(\delta\in(0,1)\) and for every almost everywhere finite measurable function \(f\) on \([0,1]\), one can find a function \(g\in L^\infty[0,1)\) with \(|\{x\in[0,1): g(x)\neq f(x)\}| <\delta\) such that \(c_k(g)=c_k(U)\), \(k\in\mathrm{spec}(g)\) and the Fourier-Walsh series of \(g\) converges uniformly on \([0,1)\).
Here \(\mathrm{spec}(g)=\{k\in\mathbb Z_+: c_k(g)\neq 0\}\).

Reviewer: Sergey S. Volosivets (Saratov)

MSC:

42C10	Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42A65	Completeness of sets of functions in one variable harmonic analysis
42A20	Convergence and absolute convergence of Fourier and trigonometric series

Keywords:

universal function; Fourier-Walsh coefficients; monotone coefficients