Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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

For citation:

Rykhlov V. S. Expansion in Eigenfunctions of Quadratic Strongly Irregular Pencils of Differential Operators of the Second Order. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2013, vol. 13, iss. 1, pp. 21-26. DOI: 10.18500/1816-9791-2013-13-1-1-21-26, EDN: SMXXFX

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
15.02.2013
Full text:
download
(downloads: 136)
Language: 
Russian
Heading: 
Mathematics
UDC: 
517.927.25
DOI: 
10.18500/1816-9791-2013-13-1-1-21-26
EDN: 
SMXXFX

Expansion in Eigenfunctions of Quadratic Strongly Irregular Pencils of Differential Operators of the Second Order

Autors: 
Rykhlov Victor Sergeyevich, Saratov State University
Abstract: 

We consider a quadratic strongly irregular pencil of 2-d order ordinary differential operators with constant coefficients and positive roots of the characteristic equation. Both the amounts of double expansions in a series in the derivative chains of such pencils and necessary and sufficient conditions for convergence of these expansions to the decomposed vector-valued function are found.

Key words: 
uadratic pencil of differential operators
strongly irregular pencil
two-fold expansion in the eigenfunctions
References: 
  1. Наймарк М. А. Линейные дифференциальные операторы. М. : Наука, 1969. [Naimark M. A. Linear Differential Operators. Parts I. New York : Ungar Publ. Co., 1967; Naimark M. A. Linear Differential Operators. Parts II. New York : Ungar Publ. Co., 1968.]
  2. Гуревич А. П., Хромов А. П. Операторы дифференцирования первого и второго порядков со знакопеременной весовой функцией // Мат. заметки. 1994. Т. 56, вып. 1. С. 3–15. [Gurevich A. P., Khromov A. P. First and second order differentiation operators with weight functions of variable sign // Math. Notes. 1994. Vol. 56, iss.1. P. 653—661.]
  3. Хромов А. П. Разложение по собственным функциям одной краевой задачи третьего порядка // Исследования по теории операторов. Уфа, 1988. C. 182–193. [Khromov A. P. Expansion in eigenfunctions a boundary value problem of the third order // Issledovaniya po teorii operatorov. Ufa, 1988. P. 182–193.]
  4. Хромов А. П. Теоремы равносходимости для интегро-дифференциальных и интегральных операторов // Мат. сб. 1981. Т. 114(156), № 3. C. 378–405. [Hromov A. P. Equiconvergence theorems for integrodifferential and integral operators // Math. USSR Sb. 1982. Vol. 42, iss. 3. P. 331–355.]
Received: 
19.02.2012
Accepted: 
14.01.2013
Published: 
15.02.2013
  • 847 reads