A numerical-analytic method of differentiating functions with a bounded spectrum based on Kotel'nikov's formula (Q1803107)

Let \(C_ \omega\) denote the class of functions with bounded spectrum having the form \[ f(x)={1\over 2\pi}\int_{-\omega_ c}^{\omega_ c} F(i\omega) e^{i\omega x}d\omega,\;x\in\mathbb{R}^ 1,\;\omega\in\Omega\subset\mathbb{R}^ 1,\;\Omega=[-\omega_ c,\omega_ c], \] where \(F(i\omega)\neq 0\) for \(-\omega_ c<\omega<\omega_ c\), and vanishes outside of this (closed) interval. Using some differential-matrix transformations within an orthogonal basis, the authors present an effective, numerical-analytical method of \(N\)-fold differentiation of a function \(f(x)\in C_ \omega\), given as the table of values on a finite set of equidistant points. The authors also display applications of the method to construct Taylor series for functions of the class \(C_ \omega\) and to solve some partial differential equations of mathematical physics.