Address: Vatutina st. 53, Vladikavkaz,
362025, RNO-A, Russia
Phone: (8672)23-00-54
E-mail: rio@smath.ru
DOI: 10.46698/f8294-3012-1428-w
On Kernels of Convolution Operators in the Roumieu Spaces of Ultradifferentiable Functions
Polyakova, D. A.
Vladikavkaz Mathematical Journal 2024. Vol. 26. Issue 3.
Abstract: We consider convolution operators in the Roumieu spaces of ultradiffereniable functions of mean type on the real axis. The famous Gevrey classes are also the Roumieu spaces. As particular cases, convolution operators include the differential equations of infinite order with constant coefficients, difference-differential and integro-differential equations. From recent results for convolution operators in the Beurling spaces of mean type and from the connection between the Roumieu and the Beurling spaces it follows that for the surjectivity of convolution operator it is necessary that the symbol of the operator is slowly decreasing with respect to the weight function. Under this assumption, we obtain the isomorphic description for the kernel of the convolution operator as a sequence space. We also construct an absolute basis in the space of all solutions of the homogeneous convolution equation. These results are of their own interest. On the other hand, they are the necessary step for investigation of the problem of surjectivity of the convolution operator in the Roumieu spaces of mean type.
Keywords: ultradifferentiable functions, convolution operator, the kernel of the operator
For citation: Polyakova, D. A. On Kernels of Convolution Operators in the Roumieu Spaces of Ultradifferentiable Functions, Vladikavkaz Math. J., 2024, vol. 26, no. 3, pp. 72-85 (in Russian). DOI 10.46698/f8294-3012-1428-w
1. Meise, R. Sequence Space Representations For Zero-Solutions of Convolution
Equations on Ultradifferentiable Functions of Roumieu Type, Studia Mathematica, 1989, vol. 92, pp. 211-230. DOI: 10.4064/sm-92-3-211-230.
2. Braun, R. W., Meise, R. and Vogt, D. Existence of Fundamental Solutions and Surjectivity
of Convolution Operators on Classes of Ultradifferentiable Functions, Proceedings
of the London Mathematical Society, 1990, vol. 61, pp. 344-370. DOI: 10.1112/plms/s3-61.2.344.
3. Meyer, T. Surjectivity of Convolution Operators on Spaces of Ultradifferentialble Functions
of Roumieu Type, Studia Mathematica, 1997, vol. 125, no. 2, pp. 101-129. DOI: 10.4064/sm-125-2-101-129.
4. Polyakova, D. A. Solvability of the Inhomogeneous Cauchy-Riemann Equation in Projective
Weighted Spaces, Siberian Mathematical Journal, 2017, vol. 58, no. 1, pp. 142-152. DOI: 10.1134/S0037446617010189.
5. Polyakova, D. A. On the Image of the Convolution Operator in Spaces of
Ultradifferentiable Functions, Algebra i Analiz, 2024, vol. 36, no. 2, pp. 108-130 (in Russian).
6. Polyakova, D. A. General Solution of the Homogeneous Convolution Equation in Spaces
of Ultradifferentiable Functions, St. Petersburg Mathematical Journal, 2020,
vol. 31, no. 1, pp. 85-105. DOI: 10.1090/spmj/1587.
7. Napalkov, V. V. A Basis in the Space of Solutions of a Convolution Equation,
Mathematical Notes, 1988, vol. 43, no. 1, pp. 27-33. DOI: 10.1007/BF01139565.
8. Krivosheev, A. S. The Schauder Basis in the Solution Space of a Homogeneous
Convolution Equation, Mathematical Notes, 1995, vol. 57, no. 1, pp. 41-50. DOI: 10.1007/BF02309393.
9. Abanin, A. V., Ishimura, R. and Le Hai Khoi. Exponential-Polynomial Bases for Null
Spaces of Convolution Operators in \(A^{-\infty}\), Contemporary Mathematics, 2011. vol. 547, pp. 1-16.
10. Brawn, R. W., Meise, R. and Taylor, B. A. Ultradifferentiable Functions and
Fourier Analysis, Results in Mathematics, 1990, vol. 17, pp. 206-237. DOI: 10.1007/BF03322459.
11. 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. 3-20 (in Russian). DOI: 517.547.2+517.982.
12. Abanina, D. A. Solvability of Convolution Equations in the Beurling Spaces of
Ultradifferentiable Functions of Mean Type on an Interval, Siberian Mathematical Journal, 2012,
vol. 53, no. 3, pp. 377-392. DOI: 10.1134/S0037446612020206.
13. 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.
14. Edwards, R. D. Functional Analysis. Theory and Applications, New York, Chicago,
San Francisco, Toronto, and London, Holt Rinehart and Winston, 1965.
15. Robertson, A. P. and Robertson, W. Topological Vector Spaces, Cambridge Tracts in Mathematics,
CUP Archive, 1980.
16. Grothendieck, A. Sur les espaces \((F)\) et \((DF)\), Summa Brasiliensis Mathematicae, 1954, vol. 3, pp. 57-123.
17. Meise R. and Vogt D. Introduction to Functional Analysis,
Oxford Grand. Text. Math., vol. 2, Oxford Univ. Press, New York, 1997.
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.
Сайт использует файлы cookie, необходимые для корректной работы сайта, и сервисы Яндекс-метрики, используемые для анализа статистики посещаемости, которые не содержат сведений, на основании которых можно идентифицировать личность пользователя. Продолжение пользования сайтом является согласием на применение данных технологий.
