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Pseudo-splines of integer order $(m,\ell)$ were introduced by Daubechies,Han, Ron, and Shen as a family which allows interpolation between the classicalB-splines and the Daubechies' scaling functions. The purpose of this paper isto generalize the pseudo-splines to fractional and complex orders $(z, \ell)$with $\alpha:=\re z > 1$. This allows increased flexibility in regard tosmoothness: 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 theH\"older spaces $C^{\alpha-1}$. The presence of the imaginary part of $z$allows for direct utilization in complex transform techniques for signal andimage analyses. We also show that in analogue to the integer case, thegeneralized pseudo-splines lead to constructions of Parseval wavelet frames viathe unitary extension principle.

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