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Streszczenie

It is shown that there exists a function $U\in L^1([0,1)^2)$ such that for each $\varepsilon \gt 0$ one can find a measurable set $E_\varepsilon \subset [0,1)^2$ with $|E_\varepsilon | \gt 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 $U$.