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JOURNALS // Itogi Nauki i Tekhniki. Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory // Archive

Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 2018 Volume 153, Pages 94–107 (Mi into366)

This article is cited in 1 paper

Operators Whose Resolvents Have Convolution Representations and Their Spectral Analysis

B. E. Kanguzhin

Al-Farabi Kazakh National University

Abstract: In this paper, we study spectral decompositions with respect to a system of generalized eigenvectors of second-order differential operators on the interval whose resolvents possess convolution representations. We obtain the convolution representation of resolvents of second-order differential operators on an interval with integral boundary conditions. Then, using the convolution generated by the initial differential operator, we construct the Fourier transform. A connection between the convolution operation in the original functional space and the multiplication operation in the space of Fourier transforms is established. Finally, the problem on the convergence of spectral expansions generated by the original differential operator is studied. Examples of convolutions generated by operators are also presented.

Keywords: convolution, spectral decomposition, resolvent, boundary-value problem, differential operator, boundary form.

UDC: 517.984, 517.927.2

MSC: 34B05, 34L05


 English version:
Journal of Mathematical Sciences (New York), 2021, 252:3, 384–398

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© Steklov Math. Inst. of RAS, 2025