Let T be a tridiagonal operator on ℓ2(ℕ) which has strict row and column dominant property except for some finite number of rows and columns. This matrix is shown to be invertible under certain conditions. This result is also extended to double infinite tridiagonal matrices. Further, a general theorem is proved for solving an operator equation Tx = y using its finite-dimensional truncations, where T is a double infinite tridiagonal operator. Finally, it is also shown that these results can be applied in order to obtain a stable set of sampling for a shift-invariant space. © 2016 Informa UK Limited, trading as Taylor & Francis Group.

R Radha

Department of Mathematics

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IIT Madras

Invertibility of a tridiagonal operator with an application to a non-uniform sampling problem

Linear and Multilinear Algebra

Let H be a separable Hilbert space with an orthonormal basis {en/n∈N}, T be a bounded tridiagonal operator on H and Tn be its truncation on span ({e1, e2, ... , en}). We study the operator equation Tx = y through its finite dimensional truncations T nxn = yn. It is shown that if {∥Tn-1en∥} and {∥Tn*-1en∥} are bounded, then T is invertible and the solution of Tx = y can be obtained as a limit in the norm topology of the solutions of its finite dimensional truncations. This leads to uniform boundedness of the sequence {Tn-1}. We also give sufficient conditions for the boundedness of {∥Tn-1en∥} and {∥Tn*-1en∥} in terms of the entries of the matrix of T. © 2005 Elsevier Inc. All rights reserved.

Fulltext

Linear Algebra and Its Applications

Solution of a tridiagonal operator equation

SIAM Journal on Matrix Analysis and Applications

Spectral Analysis and Spectral Symbol of $d$-variate $mathbb Q_{oldsymbol p}$ Lagrangian FEM Stiffness Matrices

Frames and Bases

SIAM Review

Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces

The Journal of Fourier Analysis and Applications

Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces

Journal	Data powered by TypesetLinear and Multilinear Algebra
Publisher	Data powered by TypesetTaylor and Francis Ltd.
ISSN	03081087
Open Access	No