Best time localized trigonometric polynomials and wavelets
As analogon to the Heisenberg uncertainty principle for \(L^2\)-functions, \textit{E. Breitenberger} [Found. Phys. 15, 353--364 (1983)] has introduced an uncertainty principle for \(2\pi\)-periodic functions which is formulated in terms of angular variance and frequency variance. The angular variance is a time localization measure for \(2\pi\)-periodic functions. In this paper, the author determines the minimum of the angular variance in the space \({\mathbb T}_n\) of all trigonometric polynomials of order \(\leq n\) and in the space \({\mathbb T}_m^n = {\mathbb T}_n \ominus {\mathbb T}_m\) \((1\leq m \leq n)\). Best time localized trigonometric polynomials are determined. Furthermore, trigonometric wavelets and wavelet packets are constructed with these trigonometric polynomials.
scientific article; zbMATH DE number 2162392