Abstract:
Most known papers on spectral synthesis in complex domains are based on transitions from problems of spectral synthesis to equivalent problems of local description. As a rule, such a transition is carried out in the framework of some special assumptions and meets considerable difficulties. A general method developed in this paper enables one to verify the duality theorem in the setting of several complex variables, when each operator πp(D), p=1,…,q, acts with respect to a single variable. These assumptions cover the case of a system of partial differential operators. The duality transition here breaks into three separate steps. Two of them are connected with classical results of the theory of analytic functions and only one relates to general duality theory. This allows one to speak about singling out the analytic component of the transition from a spectral synthesis problem to an equivalent problem of local description.
Citation:
A. B. Shishkin, “Spectral synthesis for systems of differential operators with constant coefficients. Duality theorem”, Sb. Math., 189:9 (1998), 1423–1440
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\by A.~B.~Shishkin
\paper Spectral synthesis for systems of differential operators with constant coefficients. Duality theorem
\jour Sb. Math.
\yr 1998
\vol 189
\issue 9
\pages 1423--1440
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Linking options:
https://www.mathnet.ru/eng/sm355
https://doi.org/10.1070/sm1998v189n09ABEH000355
https://www.mathnet.ru/eng/sm/v189/i9/p143
This publication is cited in the following 13 articles:
A. B. Shishkin, “On continuous endomorphisms of entire functions”, Sb. Math., 212:4 (2021), 567–591
A. B. Shishkin, “Odnostoronnie skhemy dvoistvennosti”, Vladikavk. matem. zhurn., 22:3 (2020), 124–150
A. B. Shishkin, “Symmetric representations of holomorphic functions”, Probl. anal. Issues Anal., 7(25), spetsvypusk (2018), 124–136
Krantz S., “Harmonic and Complex Analysis in Several Variables”, Harmonic and Complex Analysis in Several Variables, Springer Monographs in Mathematics, Springer, 2017, 1–424
A. B. Shishkin, “Proektivnoe i in'ektivnoe opisaniya v kompleksnoi oblasti. Dvoistvennost”, Izv. Sarat. un-ta. Nov. ser. Ser.: Matematika. Mekhanika. Informatika, 14:1 (2014), 47–65
T. A. Volkovaya, “Sintez v polinomialnom yadre dvukh analiticheskikh funktsionalov”, Izv. Sarat. un-ta. Nov. ser. Ser.: Matematika. Mekhanika. Informatika, 14:3 (2014), 251–262
T. A. Volkovaya, A. B. Shishkin, “Lokalnoe opisanie tselykh funktsii. Podmoduli ranga 1”, Vladikavk. matem. zhurn., 16:2 (2014), 14–28
A. P. Khromov, “Finite-dimensional perturbations of Volterra operators”, Journal of Mathematical Sciences, 138:5 (2006), 5893–6066
I. F. Krasichkov-Ternovskii, “Approximation theorem for a homogeneous
vector convolution equation”, Sb. Math., 195:9 (2004), 1271–1289
I. F. Krasichkov-Ternovskii, “Spectral synthesis and analytic continuation”, Russian Math. Surveys, 58:1 (2003), 31–108
A. B. Shishkin, “Spectral synthesis for systems of differential operators with
constant coefficients”, Sb. Math., 194:12 (2003), 1865–1898
I. F. Krasichkov-Ternovskii, A. B. Shishkin, “Local description of closed submodules of a special module of entire functions of exponential type”, Sb. Math., 192:11 (2001), 1621–1638
I. F. Krasichkov-Ternovskii, “Spectral synthesis and local description for several variables”, Izv. Math., 63:4 (1999), 729–755
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