Darian M. Onchis

Constructive realization of dual systems for generators of multi-window spline-type spaces

Multi-window spline-type spaces arise naturally in many areas. Among others they have been used as model spaces in the theory of irregular sampling. This class of shift-invariant spaces is characterized by possessing a Riesz basis which consists of a set of translates along some lattice @L of a finite family of atoms. Part of their usefulness relies on the explicit knowledge of the structure of the projection operator on such a space using the existence of a finite family of dual atoms. The main goal of this paper is to address the problems arising from the discrepancy between a constructive description and an implementable approximate realization of such concepts. Using function space concepts (e.g. Wiener amalgam spaces) we describe how approximate dual atoms can be computed for any given degree of precision. As an application of our result we describe the best approximation of Hilbert-Schmidt operators by generalized Gabor multipliers, using smooth analysis and synthesis windows. The Kohn-Nirenberg symbols of the rank-one operators formed from analysis and synthesis windows satisfy our general assumptions. Applications to irregular sampling are given elsewhere.

Approximate dual Gabor atoms via the adjoint lattice method

Regular Gabor frames for L2(ℝd)

An optimally concentrated Gabor transform for localized time-frequency components

{\boldsymbol {L}{^{2}}(\mathbb {R}^d)}

Construction of approximate dual wavelet frames

are obtained by applying time-frequency shifts from a lattice in Λ◃ℝd×ℝ̂

Generalized Goertzel algorithm for computing the natural frequencies of cantilever beams

\boldsymbol {\Lambda } \vartriangleleft {\mathbb {R}^{d} \times \mathbb {\widehat {R}}}

Face and marker detection using Gabor frames on GPUs

to some decent so-called Gabor atom g, which typically is something like a summability kernel in classical analysis, or a Schwartz function, or more generally some g∈S0(ℝd)

A parallel Homological Spanning Forest framework for 2D topological image analysis

g \in {\boldsymbol {S}_{0}(\mathbb {R}^{d})}

Observing damaged beams through their time-frequency extended signatures

. There is always a canonical dual frame, generated by the dual Gabor atom g~

Increasing the image resolution using multi-windows spline-type spaces

{\widetilde g}

Signal Reconstruction in Multi-Windows Spline-Spaces Using the Dual System

. The paper promotes a numerical approach for the efficient calculation of good approximations to the dual Gabor atom for general lattices, including the non-separable ones (different from aℤd×bℤd