Set of Exponents for Interpolation by Sums of Exponential Series in All Convex Domains

We study the problem of multiple finite-sum interpolation in all convex domains of the complex plane of absolutely converging exponential series with exponents from a given set Λ. We obtain the following solvability criterion for this problem: each direction at infinity must be a limit direction for...

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Bibliographic Details
Published in:Journal of mathematical sciences (New York, N.Y.) N.Y.), 2020-02, Vol.245 (1), p.48-63
Main Authors: Merzlyakov, S. G., Popenov, S. V.
Format: Article
Language:English
Subjects:
Analytic functions
Approximation
Domains
Exponents
Interpolation
Mathematical analysis
Mathematics
Mathematics and Statistics
Subspaces
Sums
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Summary:We study the problem of multiple finite-sum interpolation in all convex domains of the complex plane of absolutely converging exponential series with exponents from a given set Λ. We obtain the following solvability criterion for this problem: each direction at infinity must be a limit direction for the set Λ. We prove that this problem is equivalent to certain particular problems of simple interpolation and to pointwise approximation of exponential series by sums in some specific domains. The same description is also obtained for problems of simple interpolation and pointwise approximation in all convex domains by functions that belong to subspaces that are invariant with respect to the differentiation operator and admit spectral synthesis in spaces of holomorphic functions on these domains.
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-020-04676-6