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 Mat. Sb. (N.S.), 1977, Volume 102(144), Number 4, Pages 475–498 (Mi msb2687)

Compound operator equations in generalized derivatives and their applications to Appell sequences

Yu. F. Korobeinik

Abstract: Let $E$ be a vector space of sequences of numbers, containing all of the basis vectors $e_k$, with the Köthe topology $\nu$; let $\{f_k\}$ be a fixed sequence of nonzero complex numbers; let $D$ be a Gel'fond–Leont'ev generalized differentiation operator:
$$(Dc)_k=\frac{f_k}{f_{k+1}}c_{k+1},\qquad k=0,1,2,…,$$
and let $p$ be an operator of the form $(p_c)_m=(-1)^m, m=0,1,…$ .
In this work there is an investigation of an infinite-order operator
$$Lc=\sum_{k=0}^\infty a_kD^kc+\sum_{k=0}^\infty b_kD^kP_c.$$

Under rather general assumptions it is shown that $L_0$ is an epimorphism of $(E,\nu)$, and the kernel is described; conditions are established for $L_0$ to be an isomorphism of $(E,\nu)$.
On the basis of these results criteria are found for an Appell sequence to be a quasi-power basis or representing system in $(E,\nu)$.
Bibliography: 16 titles.

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English version:
Mathematics of the USSR-Sbornik, 1977, 31:4, 425–443

Bibliographic databases:

UDC: 517.947.35
MSC: 46A45, 47A50, 46A35

Citation: Yu. F. Korobeinik, “Compound operator equations in generalized derivatives and their applications to Appell sequences”, Mat. Sb. (N.S.), 102(144):4 (1977), 475–498; Math. USSR-Sb., 31:4 (1977), 425–443

Citation in format AMSBIB
\Bibitem{Kor77} \by Yu.~F.~Korobeinik \paper Compound operator equations in generalized derivatives and their applications to Appell sequences \jour Mat. Sb. (N.S.) \yr 1977 \vol 102(144) \issue 4 \pages 475--498 \mathnet{http://mi.mathnet.ru/msb2687} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=467383} \zmath{https://zbmath.org/?q=an:0355.47030|0388.47026} \transl \jour Math. USSR-Sb. \yr 1977 \vol 31 \issue 4 \pages 425--443 \crossref{https://doi.org/10.1070/SM1977v031n04ABEH003714} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1977GB39600001} 

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This publication is cited in the following articles:
1. Yu. F. Korobeinik, “Representing systems”, Russian Math. Surveys, 36:1 (1981), 75–137
2. M. MALDONADO, J. PRADA, M. J. SENOSIAIN, “APPELL BASES ON SEQUENCE SPACES”, J. Nonlinear Math. Phys, 18:supp01 (2011), 189
3. M. Maldonado, J. Prada, M. J. Senosiain, “Generalized Appell bases”, Math. Nachr, 2013, n/a
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