The traditional approach to the exponential synthesis problem for the space of solutions of a homogeneous convolution-type equation in a convex domain assumes that this space is invariant under some differential operator. This assumption makes it possible to reduce the problem of exponential synthesis to the problem of spectral synthesis. Is this assumption due to the method used to solve the problem, or is the invariance of the solution space necessary for a positive answer to the exponential synthesis problem? To answer this question, we consider special equations of the convolution type, the equations with q-sided convolution. For these equations, we show that the requirement of invariance for the solution space is necessary and cannot be omitted if we assume that the solution space admits exponential synthesis with a free choice of the convex region and the characteristic function of the equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. V. Napalkov, Convolution Equations in Multidimensional Spaces [in Russian], Nauka, Moscow (1982).
A. B. Shishkin, “Exponential synthesis in the kernel of a symmetric convolution,” Zap. Nauchn. Sem. POMI, 447, 129–170 (2016); English transl., J. Math. Sci., 229, No. 5, 572–599 (2018).
I. F. Krasichkov-Ternovskii, “Invariant subspaces of analytic functions. II. Spectral synthesis on convex domains,” Mat. Sb., 88(130), No. 1, 3–30 (1972); English transl., Math. USSR-Sb., 17, No. 1, 1–29 (1972).
A. B. Shishkin, “Spectral synthesis for an operator generated by multiplication by a power of the independent variable,” Mat. Sb., 182, No. 6, 828–848 (1991); English transl., Math. USSR-Sb., 73, No. 1, 211–229 (1992).
I. F. Krasichkov-Ternovskii, “Spectral synthesis in a complex domain for a differential operator with constant coefficients. IV. Synthesis,” Mat. Sb., 183, No. 8, 23–46 (1992); English transl., Russian Acad. Sci. Sb. Math., 76, No. 2, 407–426 (1993).
S. G. Merzlyakov, “Invariant subspaces of the operator of multiple differentiation,” Mat. Zametki, 33, No. 5, 701–713 (1983); English transl., Math. Notes, 33, No. 5, 361–368 (1983).
A. A. Tatarkin and A. B. Shishkin, “Synthesis in the kernel of the three-way convolution operator,” in: Proceedings of the Voronezh Spring Mathematical School “Modern Methods of the Theory of Boundary-Value Problems. Pontryagin Readings–XXX”. Voronezh, May 3–9, 2019. Part 4, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 193, VINITI, Moscow, 2021, 130–141.
B. N. Khabibullin, “Two general conditions for the inadmissibility of spectral synthesis for invariant subspaces of holomorphic functions,” Vladikavkaz. Mat. Zh., 7, No. 3, 71–78 (2005).
A. B. Shishkin, “Projective and injective descriptions in the complex domain. Duality,” Izv. Saratov Univ. Math. Mech. Inform., 14, No. 1, 47–65 (2014).
A. A. Tatarkin and U. S. Saranchuk, “Elementary solutions of a homogeneous q-sided convolution equation,” Issues Anal., 7(25), Special issue, 137–152 (2018).
Yu. S. Saranchuk and A. B. Shishkin, “General elementary solution of a homogeneous q-sided convolution type equation,” Algebra Analiz, 34, No. 4, 188–213 (2022); English transl., St.Petersburg Math. J., 34, No. 4, 695–713 (2023).
A. B. Shishkin, Projective and Injective Descriptions in the Complex Domain. Spectral Synthesis and Local Description of Ananlytic Funtions, Slaviansk-on-Kuban, KubGU (2013).
A. B. Shishkin, “One-sided dual schemes,” Vladikavkaz. Mat. Zh., 22, No. 3, 124–150 (2020).
A. B. Shishkin, “Symmetric representations of holomorphic functions,” Issues Anal., 7(25), No. 2, 124–136 (2018).
A. F. Leont’ev, Exponential Series [Russian], Nauka, Moscow (1976).
I. F. Krasichkov-Ternovskii, “Invariant subspaces of analytic functions. I. Spectral synthesis on convex regions,” Mat. Sb., 87(129), No. 4, 459–489 (1972); English transl., Math. USSR-Sb., 16, No. 4, 471–500 (1972).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 512, 2022, pp. 191–222.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Tatarkin, A.A., Shishkin, A.B. Exponential Synthesis in the Kernel of a q-Sided Convolution Operator. J Math Sci 282, 581–600 (2024). https://doi.org/10.1007/s10958-024-07200-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-024-07200-2