On a Particular Solution of a Nonhomogeneous Convolution Equation in Spaces of Ultradifferentiable Functions

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.23671/VNC.2018.4.23389 On a Particular Solution of a Nonhomogeneous Convolution Equation in Spaces of Ultradifferentiable Functions Polyakova, D. A. Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 4. Abstract: We consider the Beurling spaces of ultradifferentiable functions of mean type on the real axis determined by special weight functions. These spaces are the general projective analogs of the well-known Gevrey classes. In these spaces we investigate a nonhomogeneous convolution equation (differential equation of infinite order with constant coefficients) generated by the symbol which has only simple zeros and satisfies some natural growth estimates. Given the zeros of a symbol, a symmetric sequence of real numbers is explicitly constructed, in each of which the module of the symbol has a suitable lower estimate. This sequence determines a system of exponentials with imaginary indexes which is absolutely representing in the corresponding space. This allows us to represent the right-hand side of the equation as an absolutely convergent series with respect to this system. Then we establish a particular solution of the equation under considering as an absolutely convergent series with respect to this system,too. The coefficients of the series are naturally determined by the right-hand side of the equation. The proof is essentially based on the analogous results which were earlier obtained in the case of spaces on finite interval. We also use the stability property of weakly sufficient sets and absolutely representing systems. Some concrete examples of constructing the desired sequences are also given in the paper. Keywords: space of ultradifferentiable functions, nonhomogeneous convolution equation. Language: Russian Download the full text For citation: Polyakova, D. A. On a Particular Solution of a Nonhomogeneous Convolution Equation in Spaces of Ultradifferentiable Functions, Vladikavkaz Math. J., 2018, vol. 20, no. 4, pp. 68-75 (in Russian). DOI 10.23671/VNC.2018.4.23389 + References 1. Abanin, A. V. and Abanina, D. A. Division Theorem in Some Weighted Spaces of Entire Functions, Vladikavkaz. Math. J., 2010, vol. 12, no. 3, pp. 3-21 (in Russian). 2. Abanina, D. A. Representation of Solutions of Convolution Equations in Nonquasianalytic Beurling Classes of Ultradifferentiable Functions of Mean Type, Russian Mathematics, 2011, vol. 55, no. 6, pp. 1-8. DOI: 10.3103/S1066369X11060016. 3. Polyakova, D. A. On Solutions of Convolution Equations in Spaces of Ultradifferentiable Functions, St. Petersburg Math. J., 2015, vol. 26, no. 6, pp. 949-963. DOI: 10.1090/spmj/1369. 4. Abanin, A. V. On the Continuation and Stability of Weakly Sufficient Sets, Soviet Mathematics (Izvestiya VUZ. Matematika), 1987, vol. 31, no. 4, pp. 1-10. 5. Zharinov, V. V. Compact Families of Locally Convex Topological Vector Spaces, Frechet-Schwartz and Dual Frechet-Schwartz Spaces, Russian Mathematical Surveys, 1979, vol. 34, no. 4, pp. 105-143. DOI: 10.1070/RM1979v034n04ABEH002963. 6. Abanin, A. V. and Filip'ev I. A. Analytic Implementation of the Duals of Some Spaces of Infinitely Differentiable Functions, Siberian Mathematical Journal, 2006, vol. 47, no. 3, pp. 397-409. DOI: 10.1007/s11202-006-0052-3. 7. Korobeinik, Yu. F. Representing Systems, Mathematics of the USSR-Izvestiya, 1978, vol. 12, no. 2, pp. 309-335. DOI: 10.1070/IM1978v012n02ABEH001856. 8. Schneider, D. M. Sufficient Sets for Some Spaces of Entire Functions, Trans. Amer. Math. Soc., 1974, vol. 197, pp. 161-180. DOI: 10.2307/1996933. 9. Korobeinik, Yu. F. Inductive and Projective Topologies. Sufficient Sets and Representing Systems, Mathematics of the USSR-Izvestiya, 1987, vol. 28, no. 3, pp. 529-554. DOI: 10.1070/IM1987v028n03\linebreak \noindent ABEH000896. 10. Levin, B. Ja. Distribution of Zeros of Entire Functions, Transl. Math. Monogr., vol. 5, Amer. Math. Soc., Providence, RI, 1972, 523 p. 11. Leont'ev, A. F. Ryady ehksponent [Exponential Series], Moscow, Nauka Publ., 1976, 536 p. (in Russian). 12. Abanin, A. V. Nontrivial Expansions of Zero and Absolutely Representing Systems, Mathematical Notes, 1995, vol. 57, no. 4, pp. 335-344. DOI: 10.1007/BF02304161. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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