SPLINE-TYPE SPACES IN GABOR ANALYSIS

Abstract:

The central topic in Gabor analysis is the set of Weyl-Heisenberg families, obtained from a “window” g, by means of time-frequency shifts along some lattice Λ ◃ ℝ^2d. As a starting point we consider tight Gabor frames, i.e., we choose g ∈ L²(ℝ^d) in such a way that the corresponding frame operator represents the identity:

Using such expansions one can define Gabor multipliers via The main results of this paper will answer the questions under which conditions the mapping from the upper symbol (m(λ)) to the Gabor multiplier G_m (for given (g, Λ)) is injective on ℓ^∞, and how the best approximation of a given operator (e.g. some Hilbert-Schmidt operator) by Gabor multipliers (with ℓ²-symbol) can be described. In order to prove these results we shall make use of properties of spline-type spaces over locally compact Abelian groups, generated by means of atoms in the modulation space , also known as Feichtinger's Segal algebra S₀(G). The corresponding results make an original part of this paper.