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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Spectral synthesis in the kernel of a convolution operator on weighted spaces


Author: R. S. Yulmukhametov
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 21 (2009), nomer 2.
Journal: St. Petersburg Math. J. 21 (2010), 353-363
MSC (2000): Primary 32A50, 45E10, 46E10
Published electronically: January 26, 2010
MathSciNet review: 2553049
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Abstract: Weighted spaces of analytic functions on a bounded convex domain $ D\subset \mathbb{C}^p$ are treated. Let $ U =\{ u_n\} _{n=1}^\infty $ be a monotone decreasing sequence of convex functions on $ D$ such that $ u_n(z)\longrightarrow \infty $ as $ \operatorname{dist}(z,\partial D) \longrightarrow 0$. The symbol $ H(D,U)$ stands for the space of all $ f\in H(D)$ satisfying $ \vert f(z)\vert\exp (-u_n(z))\longrightarrow 0$ as $ \operatorname{dist}(z,\partial D)\longrightarrow 0$, for all $ n\in \mathbb{N}$. This space is endowed with a locally convex topology with the aid of the seminorms $ p_n(f)=\sup_{z\in D}\vert f(z)\vert\exp (-u_n(z))$, $ n=1, 2, \dots$. Clearly, every functional $ S\in H^*(D)$ is a continuous linear functional on $ H(D,U)$, and the corresponding convolution operator $ M_S : f\longrightarrow S_w(f(z+w))$ acts on $ H(D,U)$. All elementary solutions of the equation

 

$\displaystyle M_S[f]=0, \leqno(*) $

 

i.e., all solutions of the form $ z^\alpha e^{\langle a,z\rangle}$, $ \alpha \in \mathbb{Z}_+^p$, $ a\in \mathbb{C}^p$, belong to $ H(D,U)$. It is shown that the system $ E(S)$ of elementary solutions is dense in the space of solutions of equation $ (*)$ that belong to $ H(D,U)$.


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Additional Information

R. S. Yulmukhametov
Affiliation: Institute of Mathematics with Computing Centre, 112 Chernyshevsky Street, Ufa 450077, Russia
Email: Yulmukhametov@mail.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-10-01098-8
PII: S 1061-0022(10)01098-8
Keywords: Weighted spaces of analytic functions, convolution operator, spectral synthesis
Received by editor(s): April 2, 2007
Published electronically: January 26, 2010
Additional Notes: Supported by RFBR (grant 06-01-00516-a.)
Article copyright: © Copyright 2010 American Mathematical Society