Symbolic Calculus and Fredholm Property for Localization Operators

We study the composition of time-frequency localization operators (wavepacket operators) and develop a symbolic calculus of such operators on modulation spaces. The use of time-frequency methods (phase space methods) allows the use of rough symbols of ultra-rapid growth in place of smooth symbols in the standard classes. As the main application it is shown that, in general, a localization operator possesses the Fredholm property, and thus its range is closed in the target space.     

Self-Similar Lattice Tilings

We study the general question of the existence of self-similar lattice tilings of Euclidean space. A necessary and sufficient geometric condition on the growth of the boundary of approximate tiles is reduced to a problem in Fourier analysis that is shown to have an elegant simple solution in dimension one. In dimension two we further prove the existence of connected self-similar lattice tilings for parabolic and elliptic dilations. These results apply to produce Haar wavelet bases and certain canonical number systems.     

On Landau's Necessary Density Conditions for Sampling and Interpolation of Band-Limited Functions

We give a new short proof of H. Landau's density result on necessaryconditions for the sampling and interpolation of certain entirefunctions in the case that the boundary of the spectrum hasmeasure 0. For the proof we adapt a method that has been recentlydeveloped by J. Ramanathan and T. Steger in the context of time-frequencyanalysis. As a consequence of a general theorem for the comparisonof densities we obtain several new results about the densityof irregular sampling of band-limited functions with derivativesand differential operators.     

Symmetry of Weighted L1-Algebras and the GRS-Condition

Let G be a compactly generated, locally compact group of polynomialgrowth. Removing a restrictive technical condition from a previouswork, we show that the weighted group algebra is a symmetric Banach *-algebra if and only if theweight function satisfies the GRS-condition. This conditionexpresses in a precise technical sense that grows subexponentially.2000 Mathematics Subject Classification 43A20, 22D15, 22D12.     

Hardy's Theorem and the Short-Time Fourier Transform of Schwartz Functions

The Schwartz space of rapidly decaying test functions is characterizedby the decay of the short-time Fourier transform or cross-Wignerdistribution. Then a version of Hardy's theorem is proved forthe short-time Fourier transform and for the Wigner distribution.