Nonlocal almost differential operators and interpolation by functions with sparse spectrum

Let $k$ be a measurable function on $\mathbf R$. Define an operator $\mathscr L_k\colon f\to\mathscr F^{-1}(k\mathscr F(f))$, where $f\in L^2(\mathbf R)$ and $\mathscr F$ is the Fourier transform. Let $\mathscr D_k=\{f\in L^2(\mathbf R):k\mathscr F(f)\in L^2(\mathbf R)\}$ be its domain. The operator $\mathscr L_k$ is called local if $f|E=0$ implies $\mathscr L_k(f)|E=0$ for $E\subset\mathbf R$ with $\operatorname{mes} E>0$. An entire function $g$ of order zero is constructed for which the operator $\mathscr L_g$ is not local. Let $W$ be the Wiener algebra of absolutely convergent trigonometric series. We prove a theorem on correction in the spirit of Luzin's theorem: a condition is exhibited on a set $A$ of integers under which each function of $W$ can be corrected on a set of arbitrarily small measure so that the spectrum of the corrected function (also in $W$) is contained in $A$.
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