A dyadic analogue of Wiener's Tauberian theorem and some related questions

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.