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DOI: 10.46698/i3178-1119-0009-t
One-Sided Dual Schemes
Shishkin, A. B.
Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 3.
Abstract: The phenomenon of duality appears in all areas of mathematics and is closely related to the phenomenon of equivalence. These phenomena complement each other and are used to transfer various mathematical statements from one area of mathematics to another and vice versa (dual and equivalent transitions). The main difference between duality and equivalence is the use of involution. An involution of an object is a transformation of an object whose action is eliminated by an inverse transformation, that is, an inverse transformation restores an object. Any involution generates its duality, which is affirmed by the corresponding duality theorem. Duality theorems are two-sided. They allow for dual transitions to one and the other. Weaken the conditions for involution and assume that its repeated action restores the object only by half (instead of equality, we obtain inequality). In this case, for such a complete restoration of an object, two such involutions are required. This article is about weakened (one-sided) involutions. As such, completely isotonic mappings are considered (they are defined in the second section). The properties of these mappings and their conditionally inverse mappings allow half-dual transitions - transitions in only one direction. Duality theorems claiming the possibility of such transitions are called one-sided duality schemes. The content of the work is an attempt to bring a unified mathematical base for all possible one-sided schemes of duality, which allows us to reformulate each of them in accordance with a single standard. This possibility is presented by the interpretation of dual transitions that arose under the conditions of the theory of spectral synthesis in the complex field as transitions from an injective (internal) description of some mathematical objects to a projective (external) description of other mathematical objects. The involutions used in one-sided schemes of duality, in turn, are one-sided and the restrictions imposed on them are much weaker. This leads to a significant expansion of the scope of the possible application of dual schemes in research practice.
1. Shishkin, A. B. Spectral Synthesis for Systems of Differential Operators with Constant Coefficients. Duality Theorem, Sbornik: Mathematics, 1998, vol. 189, no. 9, pp. 1423-1440.
DOI: 10.1070/SM1998v189n09ABEH000355.
2. Shishkin, A. B. Spectral Synthesis for Systems of Differential Operators with Constant Coefficients, Sbornik: Mathematics, 2003, vol. 194, no. 12, pp. 1865-1898.
DOI: 10.1070/SM2003v194n12ABEH000789.
3. Shishkin A. B. Proektivnoe i in'ektivnoe opisaniya v kompleksnoi oblasti. Spektralnyi sintez i lokalnoe opisanie analiticheskikh funktsii, Slavyansk-na-Kubani, Izdatelskii tsentr KubGU, 2013, 304 p. (in Russian).
4. Shishkin A. B. Projective and Injective Descriptions in the Complex Domain. Duality, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2014, vol. 14, no. 1, pp. 47-65.
5. Ehrenpreis L. Mean Periodic Functions I, American Journal of Mathematics, 1955, vol. 77, pp. 293-328.
6. Krasichkov I. F. Closed Ideals in Locally Convex Algebras of Entire Functions, Mathematics of the USSR-Izvestiya, 1967, vol. 1, no. 1, pp. 35-55. DOI: 10.1070/IM1967v001n01ABEH000546.
7. Krasichkov I. F. Closed Ideals in Locally Convex Algebras of Entire Functions. II. Mathematics of the USSR-Izvestiya, 1968, vol. 2, no. 5, pp. 979-986. DOI: 10.1070/IM1968v002n05ABEH000683.
8. Ehrenpreis L. Fourier Analysis in Several Complex Variables. Pure and Applied Mathematics, 1970, vol. 17,
N.Y.-London-Sydney, John Wiley & Sons Inc., 1970.
9. Krasichkov-Ternovsky I. F. Invariant Subspaces of Analytic Functions. I. Spectral Analysis on Convex Regions, Mathematics of the USSR-Izvestiya, 1972, vol. 16, no. 4, pp. 471-500. DOI: 10.1070/SM1972v016n04ABEH001436.
10. Krasichkov-Ternovsky I. F. Invariant Subspaces of Analytic Functions. II. Spectral Synthesis of Convex Domains, Mathematics of the USSR-Izvestiya, vol. 17, no. 1, pp. 1-29. DOI: 10.1070/SM1972v017n01ABEH001488.
11. Korobeinik Yu. F. Representing Systems, Mathematics of the USSR-Izvestiya, 1978, vol. 12, no. 2, pp. 309-335. DOI: 10.1070/IM1978v012n02ABEH001856.
12. Korobeinik Yu. F. Representing Systems, Russian Mathematical Surveys, 1981, vol. 36, no. 1, pp. 75-137. DOI: 10.1070/RM1981v036n01ABEH002542.
13. Korobeinik Yu. F., Melikhov S. N. A Continuous Linear Right Inverse of the Representation Operator and Applications to the Convolution Operators, Siberian Mathematical Journal, 1993, vol. 34, no. 1, pp. 59-72. DOI: 10.1007/BF00971241.
14. Shishkin A. B. Spectral synthesis for systems of differential operators with constant coefficients, Sbornik: Mathematics, 2003, vol. 194, no. 12, pp. 1865-1898. DOI: 10.1070/SM2003v194n12ABEH000789.
15. Trutnev V. M. Convolution Equations in Spaces of Entire Functions of Exponential Type, Journal of Mathematical Sciences (N.Y.), 2004, vol. 120, no 6, pp. 1901-1915. DOI: 10.1023/B:JOTH.0000020709.31698.80.
16. Shishkin A. B. Exponential Synthesis in the Kernel of a Symmetric Convolution, Journal of Mathematical Sciences (N.Y.), 2018, vol. 229, no. 5, pp. 572-599. DOI: 10.1007/s10958-018-3700-9.