61 research outputs found
Exponential Splines of Complex Order
We extend the concept of exponential B-spline to complex orders. This
extension contains as special cases the class of exponential splines and also
the class of polynomial B-splines of complex order. We derive a time domain
representation of a complex exponential B-spline depending on a single
parameter and establish a connection to fractional differential operators
defined on Lizorkin spaces. Moreover, we prove that complex exponential splines
give rise to multiresolution analyses of and define wavelet
bases for
Coxeter Groups and Wavelet Sets
A traditional wavelet is a special case of a vector in a separable Hilbert
space that generates a basis under the action of a system of unitary operators
defined in terms of translation and dilation operations. A
Coxeter/fractal-surface wavelet is obtained by defining fractal surfaces on
foldable figures, which tesselate the embedding space by reflections in their
bounding hyperplanes instead of by translations along a lattice. Although both
theories look different at their onset, there exist connections and
communalities which are exhibited in this semi-expository paper. In particular,
there is a natural notion of a dilation-reflection wavelet set. We prove that
dilation-reflection wavelet sets exist for arbitrary expansive matrix
dilations, paralleling the traditional dilation-translation wavelet theory.
There are certain measurable sets which can serve simultaneously as
dilation-translation wavelet sets and dilation-reflection wavelet sets,
although the orthonormal structures generated in the two theories are
considerably different
On the Invalidity of Fourier Series Expansions of Fractional Order
The purpose of this short paper is to show the invalidity of a Fourier series
expansion of fractional order as derived by G. Jumarie in a series of papers.
In his work the exponential functions are replaced by the
Mittag-Leffler functions over
the interval where and
is the period of the function i.e.,
$E_\alpha \left( ix^\alpha\right)=E_\alpha \left( i(x+M_\alpha)^\alpha\right).
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