The \(m\)th difference operator applied to \(L^{2}\) functions on a finite interval
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.
scientific article; zbMATH DE number 1915804