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Pairs of Dual Wavelet Frames From Any Two
Refinable Functions

### Ingrid Daubechies and Bin Han

## Abstract

Starting from any two compactly supported refinable functions with
dilation factor $d$, we show that it is always possible to construct
$2d$ wavelet functions with compact support such that they
generate a pair of dual wavelet frames in $L_2(\bR)$.
Moreover, the number of vanishing
moments of each of these wavelet frames
is equal to the approximation order of the dual MRA; this is
the highest possible.
In particular, when we consider symmetric refinable functions, the
constructed dual wavelets are also symmetric or
antisymmetric.
As a consequence, for any compactly supported refinable function
$\phi$,
it is possible to construct explicitly and easily wavelets that are
finite linear combinations of translates $\phi(d\cdot-k)$, and that
generate
a wavelet frame with arbitrarily preassigned number of vanishing
moments.
We illustrate the general theory by
examples of such pairs of dual wavelet frames derived from $B$-spline
functions.

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