Geodesic
A~dyadic analogue of Wiener's Tauberian theorem and some related questions
Izvestiya. Mathematics , Tome 67 (2003) no. 1, pp. 29-53.

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A dyadic analogue is proved of Wiener's Tauberian convolution theorem for two functions. Closedness criteria are established for the linear span of the set of binary shifts $\{f(\,\circ\oplus y)\colon y\geqslant 0\}$ for a given function $f\in L(\mathbb R_+)$ or $f\in L^2(\mathbb R_+)$. A consequence of these criteria is that the linear span of the set of binary shifts $\{f(\,\circ\oplus y)\colon 0\leqslant y\leqslant 1\}$ for a given function $f\in L([0,1))$ ($f\in L^2([0,1))$) is dense in the space $L([0,1))$ ($L^2([0,1))$) if and only if all the Fourier coefficients of $f$ with respect to the orthonormalized Walsh system on $[0,1)$ are non-zero. 
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B. I. Golubov. A~dyadic analogue of Wiener's Tauberian theorem and some related questions. Izvestiya. Mathematics , Tome 67 (2003) no. 1, pp. 29-53. http://geodesic.mathdoc.fr/item/IM2_2003_67_1_a2/

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