A theorem on equivalent bases for differential operators

Kurbanov, V. M.

doi:10.1134/s1064562406010030

Cited by 10 publications

(11 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Necessary conditions for the Riesz inequality to hold for a system of root functions of a differential operator of arbitrary order were established in [11]. In the present paper, we prove an analog of the Riesz theorem (see [12, p. 154]) and the theorem on the basis property in L p for systems of root functions of a differential operator.Note that the results of the present paper were announced in [13,14]. Here we present a detailed proof.…”

supporting

confidence: 54%

See 2 more Smart Citations

Analog of the Riesz theorem and the basis property in L p of a system of root functions of a differential operator: I

Kurbanov

2013

Diff Equat

Self Cite

View full text Add to dashboard Cite

An ordinary differential operator of arbitrary order is considered. We find necessary conditions for the Riesz property of systems of normalized root functions, prove an analog of the Riesz theorem, and use it to obtain sufficient conditions for the basis property of a system of root functions of this operator in L p . V.A. Il'in is known to be the first to pose the problem as to whether the Riesz inequality [see formula (2) below] holds for a function system that is not complete or orthonormal. He proved the inequality for systems of root functions of the Laplace and Schrödinger operators [1]. The Bessel property and the unconditional basis property were studied for systems of root functions of ordinary differential operators of order ≥ 2 in L 2 in [2-10]. Necessary conditions for the Riesz inequality to hold for a system of root functions of a differential operator of arbitrary order were established in [11]. In the present paper, we prove an analog of the Riesz theorem (see [12, p. 154]) and the theorem on the basis property in L p for systems of root functions of a differential operator.Note that the results of the present paper were announced in [13,14]. Here we present a detailed proof. BASIC NOTIONS AND STATEMENT OF RESULTSOn the interval G = (0, 1), consider a differential operator Lu = u (n) + P 1 (x)u (n−1) + P 2 (x)u (n−2) + · · · + P n (x)u,where P (x) ∈ L 1 (G), = 1, . . . , n. By D we denote the class of functions absolutely continuous together with their derivatives of order ≤ n − 1 on the closed interval G.Following Il'in, we understand the root functions of L in the generalized sense, admitting completely arbitrary boundary conditions [2]. Consider an arbitrary system {u k (x)} of root functions of the operator (1). Let {λ k } be the corresponding system of eigenvalues of this operator. We require that, together with each associated function of order s ≥ 1, the system {u k (x)} also contains the corresponding eigenfunction and all associated functions of order < s. This means (see [15]) that each element u k (x) of the system {u k (x)} other than identical zero belongs to the class D and almost everywhere in G satisfies either the equation Lu k + λ k u k = 0 [in this case, u k (x) is an eigenfunction, and λ k is an eigenvalue] or the equation Lu k + λ k u k = u ν(k) , where the index ν(k) is uniquely determined by the index k and ν(k 1 ) = ν(k 2 ) for k 1 = k 2 . [In the latter case, λ k = λ ν(k) , u k (x) is an associated function of order r ≥ 1, and u ν(k) is an associated function of order r − 1.] 7

show abstract

supporting

confidence: 54%

“…Note that the results of the present paper were announced in [13,14]. Here we present a detailed proof.…”

supporting

confidence: 54%

See 1 more Smart Citation

Analog of the Riesz theorem and the basis property in L p of a system of root functions of a differential operator: I

Kurbanov

2013

Diff Equat

Self Cite

View full text Add to dashboard Cite

show abstract

“…Recently, i.e., in the 2000s, many authors [36,[44][45][46]40,47,50,67,26,49,51] focused on the problem of convergence of eigenfunction (or more generally root function) decompositions in the case of regular but not strictly regular bc.…”

Section: 1mentioning

confidence: 99%

Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators

Djakov

Mityagin

2012

Journal of Functional Analysis

View full text Add to dashboard Cite

We study the system of root functions (SRF) of Hill operator Ly = −y + vy with a singular (complexvalued) potential v ∈ H −1 per and the SRF of 1D Dirac operator Ly = i 1 0Q 0 , subject to periodic or anti-periodic boundary conditions. Series of necessary and sufficient conditions (in terms of Fourier coefficients of the potentials and related spectral gaps and deviations) for SRF to contain a Riesz basis are proven. Equiconvergence theorems are used to explain basis property of SRF in L p -spaces and other rearrangement invariant function spaces.

show abstract

“…In Kerimov and Mamedov, authors showed that eigenfunctions of boundary value problems forms Riesz basis in L 2 (0,1) assuming that q ( x )∈ C 4 [0,1] is complex‐valued function and q (1)− q (0)≠0. In Kurbanov, author studied on the basis property in L p (0,1), where (0< p <2), of the systems of the root functions of nonself‐adjoint different operator of nth order. The basis property of L p (0,1), p >1 of the boundary problem and and when q ( x ) satisfy the condition q (0)= q (1) and q ′ (0)≠ q ′ (1) were considered in the papers .…”

Section: Introductionmentioning

confidence: 99%

On the Riesz basis property of the root functions of a discontinuous boundary problem

Cabri

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we consider the nonself-adjoint discontinuous Sturm Liouville operator with periodic (antiperiodic) boundary condition and compatibility conditions. Asymptotic formulas of eigenvalues and eigenfunctions of the operator are obtained. Using these accurate asymptotic formulas for eigenvalues and eigenfunctions, we prove the basisness of the root functions of the boundary value problem.

show abstract

A theorem on equivalent bases for differential operators

Cited by 10 publications

References 1 publication

Analog of the Riesz theorem and the basis property in L p of a system of root functions of a differential operator: I

Analog of the Riesz theorem and the basis property in L p of a system of root functions of a differential operator: I

Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators

On the Riesz basis property of the root functions of a discontinuous boundary problem

Contact Info

Product

Resources

About