ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

#### For citation:

Burlutskaya M. S. Jordan–Dirichlet Theorem for Functional Differential Operator with Involution. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2013, vol. 13, iss. 3, pp. 9-14. DOI: 10.18500/1816-9791-2013-13-3-9-14

Published online:
27.08.2013
Full text:
Language:
Russian
UDC:
517.984

# Jordan–Dirichlet Theorem for Functional Differential Operator with Involution

Autors:
Burlutskaya Marija Shaukatovna, Voronezh State University, Russia
Abstract:

In this paper the problem of decomposability of a function f(x) into Fourier series with respect to the system of eigenfunctions of a functional-differential operator with involution Ly = y′(1 − x) + ®y′(x) + p1(x)y(x) + p2(x)y(1−x), y(0) = °y(1) is investigated. Based on the study of the resolvent of the operator easier and using the method of contour integration of the resolvent, we obtain the sufficient conditions for the convergence of the Fourier series for a function f(x) (analogue of the Jordan–Dirichlet’s theorem).

Key words:
References:
1. Khromov A. P. Equiconvergence theorems for integrodifferential and integral operators. Mathematics of the USSR-Sbornik, 1982, vol. 42, no. 3, pp. 331–355.
2. Khromov A. P. Inversion of integral operators with kernels discontinuous on the diagonal. Math. Notes, 1998, vol. 64, no. 5–6, pp. 804–813. DOI: 10.4213/mzm1472.
3. Burlutskaya M. Sh., Kurdyumov V. P., Lukonina A. S., Khromov A. P. A functional-differential operator with involution. Doklady Math., 2007, vol. 75, no 3, pp. 399–402.
4. Burlutskaya M. Sh., Khromov A. P. On the same theorem on a equiconvergence at the whole segment for the functional-differential operators. Izv. Sarat. Univ. N.S. Ser. Math. Mech. Inform., 2009, vol. 9, iss. 4, pt. 1, pp. 3–10 (in Russian).
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