×

Interpolation by series of exponential functions whose exponents are condensed in a certain direction. (English. Russian original) Zbl 1476.30135

J. Math. Sci., New York 257, No. 3, 334-352 (2021); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 162, 62-79 (2019).
Summary: For complex interpolation nodes, we study the problem of interpolation by series of exponential functions whose exponents form a set, which is condensed at infinity in a certain direction. We obtain a criterion for all sets of nodes from a special class. For arbitrary sets of nodes, we obtain a necessary condition for the solvability of a more general problem of interpolation by functions that can be represented as Radon integrals of an exponential function over a set of exponents. The paper also contains well-known results on interpolation, which, in particular, allow studying the multipoint holomorphic Vallée Poussin problem for convolution operators.

MSC:

30E05 Moment problems and interpolation problems in the complex plane
30B50 Dirichlet series, exponential series and other series in one complex variable
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hörmander, L., An Introduction to Complex Analysis in Several Variables (1966), Peinceton, New Jersey: Van Nostrand, Peinceton, New Jersey · Zbl 0138.06203
[2] Krasichkov-Ternovsky, IF, Invariant subspaces of holomorphic functions. Analytical continuation, Izv. Akad. Nauk SSSR. Ser. Mat., 37, 4, 931-945 (1973)
[3] Krivosheev, AS, Fundamental principle for invariant subspaces in convex domains, Izv. Ross. Akad. Nauk. Ser. Mat., 68, 2, 71-136 (2004) · Zbl 1071.30024 · doi:10.4213/im476
[4] Krivosheev, AS, Criterion of the analytic continuation of functions from principal invariant subspaces in convex domains from ℂ^n, Algebra Anal., 22, 4, 137-197 (2010) · Zbl 1230.46022
[5] Krivosheev, AS; Krivosheeva, OA, Closedness of the set of sums of Dirichlet series, Ufim. Mat. Zh., 5, 3, 96-120 (2013)
[6] Krivosheeva, OA, Convergence domain of series of exponential monomials, Ufim. Mat. Zh., 3, 2, 43-56 (2011) · Zbl 1249.30005
[7] Krivosheeva, OA, Convergence domain of series of exponential polynomials, Ufim. Mat. Zh., 5, 4, 84-90 (2013)
[8] Krivosheeva, OA; Krivosheev, AS, Criterion of the fulfillment of the fundamental principle for invariant subspaces in bounded convex domains of the complex plane, Funkts. Anal. Prilozh., 46, 4, 14-30 (2012) · Zbl 1274.46063 · doi:10.4213/faa3086
[9] Leontiev, AF, Series of Exponentials (1976), Moscow: Nauka, Moscow · Zbl 0433.30002
[10] Leontiev, AF, Sequences of Exponential Polynomials (1980), Moscow: Nauka, Moscow · Zbl 0478.30005
[11] Meril, A.; Yger, A., Problèmes de Cauchy globaux, Bull. Soc. Math. France. V., 120, 87-111 (1992) · Zbl 0799.47031 · doi:10.24033/bsmf.2180
[12] Merzlyakov, SG, Invariant subspaces of the multiple differentiation operator, Mat. Zametki, 33, 5, 701-713 (1983) · Zbl 0541.47006
[13] Merzlyakov, SG, Integrals of exponentials with respect to the Radon measure, Ufim. Mat. Zh., 3, 2, 57-80 (2011) · Zbl 1249.32002
[14] Merzlyakov, SG, Cauchy-Hadamard theorem for exponential series, Ufim. Mat. Zh., 6, 1, 75-83 (2014)
[15] Merzlyakov, SG; Popenov, SV, Multiple interpolation by exponential series in H(C) with nodes on the real axis, Ufim. Mat. Zh., 5, 3, 130-143 (2013)
[16] Merzlyakov, SG; Popenov, SV, Interpolation by exponential series in H(D) with real nodes, Ufim. Mat. Zh., 7, 1, 46-58 (2015) · doi:10.13108/2015-7-1-46
[17] S. G. Merzlyakov and S. V. Popenov, “The set of exponents for interpolation by sums of exponential series in all convex domains”, Differential Equations. Mathematical Analysis, Itogi Nauki Tekhn. Ser. Sovr. Mat. Prilozh. Temat. Obzory, 143, Russian Institute for Scientific and Technical Information, Moscow, (2017), 48-62. · Zbl 1450.30055
[18] A. U. Mullabaeva and V. V. Napalkov (jr.), “Solution of the Shapiro problem for the convolution operator,” Izv. Ufim. Nauch. Tsentra Ross. Akad. Nauk, 4, 5-11 (2017).
[19] Napalkov, VV, Convolution Equations in Multidimensional Spaces (1982), Moscow: Nauka, Moscow · Zbl 0582.47041
[20] Napalkov, VV; Mullabaeva, AU, Fisher expansion of the space of entire functions for the convolution operator, Dokl. Ross. Akad. Nauk, 476, 3, 465-467 (2017) · Zbl 1383.30018
[21] Napalkov, VV; Nuyatov, AA, Multipoint Vallée Poussin problem for convolution operators, Mat. Sb., 203, 2, 77-86 (2012) · Zbl 1254.30035 · doi:10.4213/sm7763
[22] Napalkov, VV; Nuyatov, AA, Multipoint Vallée Poussin problem for convolution operators with nodes lying in an angle, Teor. Mat. Fiz., 180, 2, 264-271 (2014) · Zbl 1311.41018 · doi:10.4213/tmf8654
[23] Napalkov, VV; Popenov, SV, Holomorphic Cauchy problem for the convolution operator in analytically uniform spaces and Fisher expansions, Dokl. Ross. Akad. Nauk, 381, 2, 164-166 (2001) · Zbl 1050.46023
[24] Napalkov, VV; Zimens, KR, Multiple Vallée Poussin problem on convex domains in the kernel of a convolution operator, Dokl. Ross. Akad. Nauk, 458, 4, 387-389 (2014) · Zbl 1306.32008
[25] Rudin, W., Functional Analysis (1973), New York: McGraw Hill, New York · Zbl 0253.46001
[26] L. Schwartz, Analyse, Hermann (1998). · Zbl 0920.00002
[27] J. Sebastião e Silva, “Su certe classi di spazi localmente convessi importanti per le applicazioni,” Rend. Mat. Appl. Roma, 14, No. 5, 388-410 (1955). · Zbl 0064.35801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.