Abstract
We identify a class of continuous compactly supported functions for which the known part of the Gabor frame set can be extended. At least for functions with support on an interval of length two, the curve determining the set touches the known obstructions. Easy verifiable sufficient conditions for a function to belong to the class are derived, and it is shown that the B-splines $B_{N}$ , $N\ge2$ , and certain ‘continuous and truncated’ versions of several classical functions (e.g., the Gaussian and the two-sided exponential function) belong to the class. The sufficient conditions for the frame property guarantees the existence of a dual window with a prescribed size of the support.
Highlights
Frames is a functional analytic tool to obtain representations of the elements in a Hilbert space as a superposition of building blocks
Frames lead to decompositions that are similar to those obtained via orthonormal bases, but with much greater flexibility, due to the fact that the definition is significantly less restrictive
One of the main manifestations of frame theory is within Gabor analysis, where the aim is to obtain efficient representations of signals in a way that reflects the time-frequency distribution
Summary
Frames is a functional analytic tool to obtain representations of the elements in a Hilbert space as a (typically infinite) superposition of building blocks. Given g ∈ L (R), the collection of functions {EmbTnag}m,n∈Z is called a (Gabor) frame if there exist constants A, B > such that. For applications of Gabor frames, it is essential that the window g is a continuous function with compact support. Among others, consider a class of functions for which we can extend the known set of parameters (a, b) yielding a Gabor frame. Let us first collect some of the known results concerning frame properties for continuous compactly supported functions; (i) is classical, and we refer to [ ] for a proof. We will introduce the window class that will be used in the current paper; it is a subset of the set of functions g considered in Proposition .
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