An algorithm for nonharmonic signal analysis using Dirichlet series on convex polygons

Preprint series: ETNA (to appear),

MSC 2000

42C15 Series of general orthogonal functions, generalized Fourier expansions, nonorthogonal expansions

30B50 Dirichlet series and other series expansions, exponential series

37M10 Time series analysis

Abstract
This article presents a new algorithm for nonharmonic signal analysis using Dirichlet series $$ f(z) = \sum_{\lambda\in\Lambda} \kappa_{f}(\lambda)\frac{e^{\lambda z}}{L^{\prime}(\lambda)}, \quad z\in D, $$ on a convex polygon $D$ as a generalization of Fourier series. Here $L$ denotes a quasipolynomial whose set of zeros $\Lambda$ generates a Riesz basis ${\cal E}(\Lambda) := \left\{\frac{e^{\lambda z}}{L^{\prime}(\lambda)}\right\}_{\lambda\in\Lambda}$ of the Smirnov space $E^{2}(D)$. The algorithm is based on a simple form of $L$ and on numerical properties of the dual basis of ${\cal E}(\Lambda)$.

Keywords: nonharmonic Fourier series, Dirichlet series, signal analysis, time series analysis