Geodesic
Algebra of mnemofunctions on a circle
Problemy fiziki, matematiki i tehniki, no. 3 (2018), pp. 55-62.

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It is impossible to define the product of arbitrary generalized functions in the classical theory of distributions. That is an obstacle for applications generalized functions theory to equations with generalized coefficients and nonlinear problems. The common approach for solving the problem of generalized functions multiplication consists in constructing a differential algebra $G$ according to the given space of generalized functions $E$ and building an embedding $R: E\to G$ Such algebras $G$ are called Colombeau type algebras and their elements are called new generalized functions or mnemofunctions. The algebra of mnemofunctions on the circle is constructed in this article. By this example some general questions on algebras of mnemofunctions are formulated.
Keywords: generalized function, space of periodic generalized functions, mnemofunction
Mots-clés : Colombeau type algebra. 
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A. B. Antonevich; T. G. Shahava; E. V. Shkadinskaia. Algebra of mnemofunctions on a circle. Problemy fiziki, matematiki i tehniki, no. 3 (2018), pp. 55-62. http://geodesic.mathdoc.fr/item/PFMT_2018_3_a8/

[1] L. Schwartz, Théorie des distributions, en 2 vol., v. 2, Hermann, Paris, 1950 | MR

[2] I.M. Gelfand, G.E. Shilov, Obobschennye funktsii i deistviya nad nimi, Gosudarstvennoe izdatelstvo fiziko-matematicheskoi literatury, M., 1959, 470 pp.

[3] P. Antosik, Ya. Mikusinskii, R. Sikorskii, Teoriya obobschennykh funktsii. Sekventsialnyi podkhod, Mir, M., 1976, 311 pp.

[4] V.S. Vladimirov, Obobschennye funktsii v matematicheskoi fizike, 2-e izd., Nauka, M., 1979, 320 pp.

[5] J.F. Colombeau, New generalized functions and multiplication of distributions, North-Holland, Amsterdam, 1984, 374 pp. | MR | Zbl

[6] Yu.V. Egorov, “K teorii obobschennykh funktsii”, UMN, 45:5(275) (1990), 3–40

[7] A.B. Antonevich, Ya.V. Radyno, “Ob obschem metode postroeniya algebr obobschennykh funktsii”, Doklady AN SSSR, 43:3 (1991), 680–684 | Zbl

[8] A.B. Antonevich, Ya.V. Radyno, “On the problem of distributions multiplication”, Problems and Methods in Mathematical Physics, Teubner-Texte sur Mathematik, 134, Stuttgart–Leipzig, 1994, 9–14 | DOI | MR

[9] M. Nedeljkov, S. Pilipovic, D. Scarpalezos, Linear Theory of Colombeau's Generalized Functions, Addison-Wesley Longman, Harlow, 1998, 156 pp. | MR

[10] M. Grosser et al., Geometric theory of generalized functions with applications to general relativity, ed. M. Hazewinkel, Springer-Science+Business Media, B.V., 2013, 505 pp.

[11] L.L. Baglini, P. Giordano, “The Category of Colombeau algebras”, Monatsh Math., 182 (2017), 649–674 | DOI | MR | Zbl

[12] I.V. Melnikova, V.A. Bovkun, U.A. Alekseeva, “Reshenie kvazilineinykh stokhasticheskikh zadach v abstraktnykh algebrakh Kolombo”, Differentsialnye uravneniya, 53:12 (2017), 1653–1663 | DOI | Zbl