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St. Petersburg Mathematical Journal

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Some functional-difference equations solvable in finitary functions

Author: E. A. Gorin
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 18 (2006), nomer 5.
Journal: St. Petersburg Math. J. 18 (2007), 779-796
MSC (2000): Primary 34K99, 32A15
Published electronically: August 9, 2007
MathSciNet review: 2301043
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Abstract: The following equation is considered: $ q(-i\partial/\partial x)u(x)=(f*u)(Ax)$, where $ q$ is a polynomial with complex coefficients, $ f$ is a compactly supported distribution, and $ A:\mathbb{R}^n\to\mathbb{R}^n$ is a linear operator whose complexification has no spectrum in the closed unit disk. It turns out that this equation has a (smooth) solution $ u(x)$ with compact support. In the one-dimensional case, this problem was treated earlier in detail by V. A. Rvachev and V. L. Rvachev and their numerous students.

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Additional Information

E. A. Gorin
Affiliation: Department of Mathematics, Moscow State Pedagogical University, Ulitsa Malaya Pirogovskaya 1, Moscow 119882, Russia

Keywords: Finitary solution, entire function of exponential type, refinement equation
Received by editor(s): April 22, 2006
Published electronically: August 9, 2007
Dedicated: Dedicated to the 100th anniversary of B. Ya. Levin’s birth
Article copyright: © Copyright 2007 American Mathematical Society

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