
Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2015, Volume 19, Number 1, Pages 63–77
(Mi vsgtu1369)




Differential Equations and Mathematical Physics
The multiple interpolation de La Vallée Poussin problem
V. V. Napalkov^{a}, A. U. Mullabaeva^{b}^{†} ^{a} Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa, 450077, Russian Federation
^{b} Bashkir State University, Ufa, 450074, Russian Federation
Abstract:
This article is concerned with the solving of multiple interpolation de La Vallée Poussin problem for generalized convolution operator. Particular attention is paid to the proving of the sequential sufficiency of the set of solutions of the generalized convolution operator characteristic equation. In the generalized Bargmann–Fock space the adjoint operator of multiplication by the variable $z$ is the generalized differential operator. Using this operator we introduce the generalized shift and generalized convolution operators. Applying the chain of equivalent assertions we obtain the fact that the multiple interpolation de La Vallée Poussin problem is solvable if and only if the composition of generalized convolution operator with multiplication by the fixed entire function $\psi(z)$ is surjective. Zeros of the function $\psi(z)$ are the nodes of interpolation. The surjectivity of composition of the generalized convolution operator with the multiplication comes down to the proof of the sequential sufficiency of the set of zeros of a generalized convolution operator characteristic function in the set of solutions of the generalized convolution operator with the characteristic function $\psi(z)$. In the proof of the sequential sufficiency it became necessary to consider the relation of eigenfunctions for different values of $\mu_i.$ The eigenfunction with great value of $\mu_i$ tends to infinity faster than eigenfunction with a lower value for $z$ tends to infinity. The derivative of the eigenfunction of higher order tends to infinity faster than lowerorder derivatives with the same values of $\mu_i$. A significant role is played by the fact that the kernel of the generalized convolution operator with characteristic function $\psi(z)$ is a finite sum of its eigenfunction and its derivatives. Using the Fischer representation, Dieudonne–Schwartz theorem and Michael's theorem on the existence of a continuous right inverse we obtain that if the zeros of the characteristic function of a generalized convolution operator are located on the positive real axis in order of increasing then multiple interpolation de La Vallée Poussin problem is solvable in the interpolation nodes.
Keywords:
generalized convolution operator, eigenfunctions of the generalized differentiation operator, Fischer representation, sequentially sufficient set, de La Vallée Poussin problem, interpolation nodes
Funding Agency 
Grant Number 
Russian Foundation for Basic Research 
140100720а 140197037рповолжьея 
This work has been supported by the Russian Foundation for Basic Research (projects no. 14–01–00720a, no. 14–01–97037rpovolzh'eya). 
^{†} Author to whom correspondence should be addressed
DOI:
https://doi.org/10.14498/vsgtu1369
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Bibliographic databases:
UDC:
517.982.2:517.927
MSC: Primary 46E10; Secondary 30D10, 30E20, 30H20 Original article submitted 15/XII/2014 revision submitted – 10/II/2015
Citation:
V. V. Napalkov, A. U. Mullabaeva, “The multiple interpolation de La Vallée Poussin problem”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 19:1 (2015), 63–77
Citation in format AMSBIB
\Bibitem{NapMul15}
\by V.~V.~Napalkov, A.~U.~Mullabaeva
\paper The multiple interpolation de La Vall\'ee Poussin problem
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2015
\vol 19
\issue 1
\pages 6377
\mathnet{http://mi.mathnet.ru/vsgtu1369}
\crossref{https://doi.org/10.14498/vsgtu1369}
\zmath{https://zbmath.org/?q=an:06968948}
\elib{https://elibrary.ru/item.asp?id=23681742}
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