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The Hilbert and Bessel properties of systems of eigenfunctions of second-order operators

The Hilbert and Bessel properties of systems of eigenfunctions of second-order operators Differential Equations, Vol. 36, No. 2, 2000, pp. 169-174. Translated from Differentsial'nye Uravneniya, Vol. 36, No. 2, 2000, pp. 147-151. Original Russian Text Copyright (~ 2000 by Assonova, Budaev. ORDINARY DIFFERENTIAL EQUATIONS The Hilbert and Bessel Properties of Systems of Eigenfunctions of Second-Order Operators N. V. Assonova and V. D. Budaev Smolensk State Pedagogical Institute, Russia Received October 8, 1997 1. BASIC NOTIONS. STATEMENT OF RESULTS A system {en}+_~ of elements of a Hilbert space H is called a Hilbert system if (3c~ > 0) (Vf 9 H) ~ I(f, en)l 2 _> c~llfll 2, (1) n--1 where (., .) is the inner product and I1" II is the norm in H. +oo In a similar way, a system {en}n=l of elements of H is called a Bessel system if (sZ > 0) (vf 9 H) ~ I(f,*n)l ~ < Zllfll ~. (2) n=l Inequalities (1) and (2) are called the Hilbert and Bessel inequalities, respectively, and a and/3 are called the Hilbert and Bessel constants. In the following, we set H = L2[0, 1] and denote the inner product and norm in L2[0, 1] by (., .) and I1" tl, respectively. On G = [0, 1] we consider http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

The Hilbert and Bessel properties of systems of eigenfunctions of second-order operators

Assonova, N.; Budaev, V.
Differential Equations , Volume 36 (2) – Nov 15, 2007
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Publisher
Springer Journals
Copyright
Copyright © 2000 by MAIK “Nauka/Interperiodica”
Subject
Mathematics; Ordinary Differential Equations; Partial Differential Equations; Difference and Functional Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1007/BF02754201
Publisher site
See Article on Publisher Site

Abstract

Differential Equations, Vol. 36, No. 2, 2000, pp. 169-174. Translated from Differentsial'nye Uravneniya, Vol. 36, No. 2, 2000, pp. 147-151. Original Russian Text Copyright (~ 2000 by Assonova, Budaev. ORDINARY DIFFERENTIAL EQUATIONS The Hilbert and Bessel Properties of Systems of Eigenfunctions of Second-Order Operators N. V. Assonova and V. D. Budaev Smolensk State Pedagogical Institute, Russia Received October 8, 1997 1. BASIC NOTIONS. STATEMENT OF RESULTS A system {en}+_~ of elements of a Hilbert space H is called a Hilbert system if (3c~ > 0) (Vf 9 H) ~ I(f, en)l 2 _> c~llfll 2, (1) n--1 where (., .) is the inner product and I1" II is the norm in H. +oo In a similar way, a system {en}n=l of elements of H is called a Bessel system if (sZ > 0) (vf 9 H) ~ I(f,*n)l ~ < Zllfll ~. (2) n=l Inequalities (1) and (2) are called the Hilbert and Bessel inequalities, respectively, and a and/3 are called the Hilbert and Bessel constants. In the following, we set H = L2[0, 1] and denote the inner product and norm in L2[0, 1] by (., .) and I1" tl, respectively. On G = [0, 1] we consider

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Differential Equations – Springer Journals

Published: Nov 15, 2007

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