Pairs of dual Gabor frame generators with compact support and desired frequency localization

Pairs of Dual Gabor Frame Generators with Compact Support and Desired Frequency Localization

Let g ∈ L2(R) be a compactly supported function, whose integertranslates {Tkg}k∈Z form a partition of unity. We prove that for certain translationand 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 {Tkg}k∈Z; the coefficients in the linear combination are given explicitly. Thus, h has compac...

Constructing pairs of dual bandlimited framelets with desired time localization

On dual Gabor frame pairs generated by polynomials ∗

We provide explicit constructions of particularly convenient dual pairs of Gabor frames. We prove that arbitrary polynomials restricted to sufficiently large intervals will generate Gabor frames, at least for small modulation parameters. Unfortunately, no similar function can generate a dual Gabor frame, but we prove that almost any such frame has a dual generated by a B-spline. Finally, for fr...

Characterization of Oblique Dual Frame Pairs

Given a frame for a subspace W of a Hilbert space H , we consider all possible families of oblique dual frame vectors on an appropriately chosen subspace V. In place of the standard description, which involves computing the pseudoinverse of the frame operator, we develop an alternative characterization which in some cases can be computationally more efficient. We first treat the case of a gener...

Time-Frequency Localization and the Gabor Transform

The time-frequency content of a signal can be measured by the Gabor transform or windowed Fourier transform. This is a function defined on phase space that is computed by taking the Fourier transform of the product of the signal against a translate of a fixed window. The problem of finding signals with Gabor transform that are maximally concentrated within a given region of phase space is discu...