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Mat. Sb. (N.S.), 1985, Volume 128(170), Number 1(9), Pages 133–142 (Mi msb2022)  

Nonlocal almost differential operators and interpolation by functions with sparse spectrum

P. P. Kargaev


Abstract: 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 $A$.
Bibliography: 7 titles.

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English version:
Mathematics of the USSR-Sbornik, 1987, 56:1, 131–140

Bibliographic databases:

UDC: 517.53
MSC: Primary 47B38; Secondary 42A15, 46E20, 31A15, 30D60
Received: 05.05.1984

Citation: P. P. Kargaev, “Nonlocal almost differential operators and interpolation by functions with sparse spectrum”, Mat. Sb. (N.S.), 128(170):1(9) (1985), 133–142; Math. USSR-Sb., 56:1 (1987), 131–140

Citation in format AMSBIB
\Bibitem{Kar85}
\by P.~P.~Kargaev
\paper Nonlocal almost differential operators and interpolation by functions with sparse spectrum
\jour Mat. Sb. (N.S.)
\yr 1985
\vol 128(170)
\issue 1(9)
\pages 133--142
\mathnet{http://mi.mathnet.ru/msb2022}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=805700}
\zmath{https://zbmath.org/?q=an:0622.42007}
\transl
\jour Math. USSR-Sb.
\yr 1987
\vol 56
\issue 1
\pages 131--140
\crossref{https://doi.org/10.1070/SM1987v056n01ABEH003028}


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