Taylor Series as Wide-sense Biorthogonal Wavelet Decomposition

TL;DR

This paper introduces a novel wavelet-based framework using Taylor series and generalized wavelets for signal analysis, establishing energy identities and new signal representations called derivagrams, enhancing understanding of wavelet applications.

Contribution

It develops a generalized biorthogonal wavelet analysis based on Taylor series, introducing Dual-Taylor series and derivagrams for improved signal representation.

Findings

Derivation of a Parseval-like identity for Taylor series

Introduction of derivagrams as new signal representations

Connection between wavelets and Taylor series in signal analysis

Abstract

Pointwise-supported generalized wavelets are introduced, based on Dirac, doublet and further derivatives of delta. A generalized biorthogonal analysis leads to standard Taylor series and new Dual-Taylor series that may be interpreted as Laurent Schwartz distributions. A Parseval-like identity is also derived for Taylor series, showing that Taylor series support an energy theorem. New representations for signals called derivagrams are introduced, which are similar to spectrograms. This approach corroborates the impact of wavelets in modern signal analysis.