Fractional and Complex Pseudo-Splines and the Construction of Parseval Frames

TL;DR

This paper generalizes pseudo-splines to fractional and complex orders, enhancing smoothness control and enabling complex transform applications, while also constructing Parseval wavelet frames using these generalized functions.

Contribution

It introduces fractional and complex pseudo-splines, broadening the family of functions for wavelet frame construction and signal analysis.

Findings

Generalized pseudo-splines to fractional and complex orders.

Constructed Parseval wavelet frames using the generalized pseudo-splines.

Enhanced smoothness control and applicability in complex signal processing.

Abstract

Pseudo-splines of integer order $(m,\ell)$ were introduced by Daubechies, Han, Ron, and Shen as a family which allows interpolation between the classical B-splines and the Daubechies' scaling functions. The purpose of this paper is to generalize the pseudo-splines to fractional and complex orders $(z, \ell)$ with $\alpha:=\re z > 1$. This allows increased flexibility in regard to smoothness: instead of working with a discrete family of functions from $C^m$, $m\in \N_0$, one uses a \emph{continuous} family of functions belonging to the H\"older spaces $C^{\alpha-1}$. The presence of the imaginary part of $z$ allows for direct utilization in complex transform techniques for signal and image analyses. We also show that in analogue to the integer case, the generalized pseudo-splines lead to constructions of Parseval wavelet frames via the unitary extension principle.