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 Izv. Akad. Nauk SSSR Ser. Mat., 1980, Volume 44, Issue 5, Pages 1131–1149 (Mi izv1956)

Extension of convergence of quasipolynomials

A. M. Sedletskii

Abstract: The system $\{\exp(i\lambda_nx)\}$, minimal in $L^p(-a,a)$ ($a<\infty$, $1\leqslant p\leqslant\infty$), is called a system of extension of $L^p$-convergence if any sequence of linear combinations of this system converging in $L^p(-a,a)$ converges in $L^p$-norm on every finite interval. A complete description of systems of extension of convergence is given in the class of systems $\{\exp(i\lambda_nx)\}$ generated by sequences of zeros of entire functions of the form
$$L(z)=\int_{-a}^a \frac{e^{izt}k(t)}{(a-|t|)^\alpha} dt,\quad0<\alpha<1,\quad\operatorname{var}k(t)<\infty,\quad k(\pm a\mp0)\ne0,$$
where $k(t)$ has, in addition, a certain smoothness in a neighborhood of the points $\pm a$. Specifically, for $1<p<\infty$ this property is realized if and only if $\alpha\ne1-1/p$, while for $p=1$ or $\infty$ there is no extension of convergence. This result is applied to the question of bases of exponential functions in $L^p(-a,a)$, $1<p<\infty$.
Bibliography: 13 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1981, 17:2, 353–368

Bibliographic databases:

UDC: 517.5
MSC: Primary 30C15, 46E30; Secondary 26A99, 30D15, 42A45, 45D05

Citation: A. M. Sedletskii, “Extension of convergence of quasipolynomials”, Izv. Akad. Nauk SSSR Ser. Mat., 44:5 (1980), 1131–1149; Math. USSR-Izv., 17:2 (1981), 353–368

Citation in format AMSBIB
\Bibitem{Sed80} \by A.~M.~Sedletskii \paper Extension of convergence of quasipolynomials \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1980 \vol 44 \issue 5 \pages 1131--1149 \mathnet{http://mi.mathnet.ru/izv1956} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=595261} \zmath{https://zbmath.org/?q=an:0509.42040|0458.42023} \transl \jour Math. USSR-Izv. \yr 1981 \vol 17 \issue 2 \pages 353--368 \crossref{https://doi.org/10.1070/IM1981v017n02ABEH001363} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1981MW12400007} 

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This publication is cited in the following articles:
1. A. M. Sedletskii, “Biorthogonal expansions of functions in series of exponents on intervals of the real axis”, Russian Math. Surveys, 37:5 (1982), 57–108
2. A. M. Sedletskii, “On the summability and convergence of non-harmonic Fourier series”, Izv. Math., 64:3 (2000), 583–600
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