On coefficients of eigenfunction expansions for boundary-value problems with a parameter in the boundary conditions
type: Item
instance of: scholarly article
title: On coefficients of eigenfunction expansions for boundary-value problems with a parameter in the boundary conditions
zbMATH Open document ID: 1062.34092
published in: Mathematical Notes
publication date: 2004-12-06T00:00:00Z
review text: Consider \(l(y,\lambda ):=l_0(y)+\lambda l_1(y)+\dots+\lambda ^nl_n(y)\) on \([0,1]\), where the \(l_\nu\) are differential expressions of order \(n-\nu\) with constant leading coeffient (which may be zero for \(\nu\neq0,n\)), together with boundary conditions which depend polynomially on the eigenvalue parameter \(\lambda \). \textit{A. A. Shkalikov} [Tr. Semin. Im. I. G. Petrovskogo 9, 190--229 (1983; Zbl 0553.34014)] proved that this problem is linearized by an operator \(H\) in a finite-codimensional subspace \(\mathcal W_{2,U}^r\times \mathbb C^{N_r}\) of \(W_2^{n+r-1}\times \dots \times W_2^r\times \mathbb C^{N_r}\) with suitable integers \(r\) and \(N_r\) and that the eigenfunctions and associated functions of \(H\) form a Riesz basis. In the present paper, the dual problem is explicitly calculated, and the expansion coefficients of an element \(\tilde v\in \mathcal W_{2,U}^r\times \mathbb C^{N_r}\) with respect to the above mentioned Riesz basis are calculated explicitly using the biorthogonal system of eigenfunctions and associated functions from the dual problem.
Mathematics Subject Classification ID: 34L10
Mathematics Subject Classification ID: 34B07
Mathematics Subject Classification ID: 47E05
zbMATH DE Number: 2121380
zbMATH Keywords: spectral parameter in boundary conditions
zbMATH Keywords: operator pencil
zbMATH Keywords: eigenfunction expansion
zbMATH Keywords: chain of eigen and associated functions
zbMATH Keywords: Shkalikov linearization
author: A. M. Akhtyamov
MaRDI profile type: Publication
DOI: 10.1023/B:MATN.0000023330.83453.E2
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description: scientific article; zbMATH DE number 2121380