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Fractional and complex pseudo-splines and the construction of Parseval frames
Pseudo-splines of integer order 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
with \alpha:=\re z > 1. This allows increased flexibility in regard to
smoothness: instead of working with a discrete family of functions from ,
, one uses a \emph{continuous} family of functions belonging to the
H\"older spaces . The presence of the imaginary part of
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