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On the existence of universal functions with respect to the double Walsh system for classes of integrable functions. (English) Zbl 1450.42016
Summary: It is shown that there exists a function \(U\in L^1([0,1)^2)\) such that for each \(\varepsilon > 0\) one can find a measurable set \(E_\varepsilon \subset [0,1)^2\) with \(|E_\varepsilon | > 1-\varepsilon\) such that \(U\) is universal for the space \(L^1(E_\varepsilon)\) with respect to the double Walsh system \(\{W_k(x) W_s(y)\}\) in the sense of signs of Fourier coefficients, i.e. any function \(f\in L^1(E_\varepsilon)\) is a limit (over rectangles and over spheres) of \(\sum\delta_{k,s}a_{k,s}(U)W_k(x)W_s(y)\) for some signs \(\delta_{k,s}=\pm 1\), where \(a_{k,s}(U)\) are the Fourier-Walsh coefficients of \(U\).
MSC:
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
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