The \(m\)th difference operator applied to \(L^{2}\) functions on a finite interval
type: Item
instance of: scholarly article
title: The \(m\)th difference operator applied to \(L^{2}\) functions on a finite interval
zbMATH Open document ID: 1029.39020
DOI: 10.1016/S0024-3795(02)00470-6
published in: Linear Algebra and its Applications
publication date: 2003-05-25T00:00:00Z
review text: The author introduces the continuous \(m\)th difference operator with amplitude \(\Delta\) as \(\mathcal D_m: L^2(\tilde I) \to L^2(\tilde I)\) such that \[ (\mathcal D_m f)(x) = \sum_{j=-\lfloor m/2\rfloor}^{\lceil m/2\rceil} (-1)^j b_j f(x+j\Delta), x \in [0,X], \] where \(b_j\) are binomial coefficients, \(X>0\), \(\tilde I = [-\lfloor m/2\rfloor \Delta, X + \lceil m/2\rceil \Delta]\), \(-\lfloor m/2\rfloor\) and \(\lceil m/2\rceil\) denote the lower and upper integer parts of \(m/2.\) The null space and the singular system of \(\mathcal D_m\) are characterized. In parallel, discretizations \({\mathbb D}_m^{(\delta)}\) of operator \(\mathcal D_m\) are considered. The null space and the singular system of \({\mathbb D}_m^{(\delta)}\) are compared to their continuous analogues. Under some simplifying assumption, there is a strong correspondence between \((\mathcal D_m f)(x)\) and \({\mathbb D}_m^{(\delta)}\): the two operators share the same singular values and the null space of \({\mathbb D}_m^{(\delta)}\) contains discretizations of elements of \(N(\mathcal D_m f)(x).\) This correspondence is lost if the simplifying assumption is not verified.
Mathematics Subject Classification ID: 39A70
Mathematics Subject Classification ID: 65D25
Mathematics Subject Classification ID: 47B39
Mathematics Subject Classification ID: 39A12
zbMATH DE Number: 1915804
zbMATH Keywords: continuous difference operator
zbMATH Keywords: discretization
zbMATH Keywords: spectral analysis
zbMATH Keywords: null space
zbMATH Keywords: singular system
author: F. Di Benedetto
reviewed by: Victor I. Tkachenko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1915804