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

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 $α:=\re z > 1$. 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 Hölder spaces $C^{α-1}$. 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.

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