Ole Christensen
Pairs of Dual Gabor Frame Generators with Compact Support and Desired Frequency Localization
Preprint series: ESI preprints

MSC:

42C15 Series of general orthogonal functions, generalized Fourier expansions, nonorthogonal expansions

Abstract: Let $g\in \ltr$ be a compactly supported function,
whose integer-translates $\{T_kg\}_{k\in \mz}$ form a partition of unity.
We prove that for certain translation- and modulation parameters, such
a function $g$ generates a Gabor frame, with a (non-canonical) dual
generated by a finite linear combination $h$ of the functions
$\{T_kg\}_{k\in \mz}$; the coefficients in the linear combination
are given explicitly.
Thus, $h$ has compact support, and the
decay in frequency is controlled by the decay of $\hat{g}$. In particular,
the result allows the construction of dual pairs of Gabor frames,
where both generators are given explicitly, have compact support, and
decay fast in the Fourier domain. We further relate the
construction to wavelet theory. Letting $D$ denote the dilation
operator and $B_N$ be the $N$th order B-spline, our results imply that
there exist dual Gabor frames with generators of the type
$g=\sum c_kDT_kB_N$ and $h=\sum \tilde{c}_kDT_kB_N$, where both
sums are finite. It is known that for $N>1$, such functions can not generate
dual wavelet frames $\{D^jT_kg\}_{j,k\in \mz},\{D^jT_kh\}_{j,k\in \mz}$.

Keywords: Gabor frames, dual frame, dual generator, wavelet frames