Integral Operators, Pseudodifferential Operators, and Gabor Frames

Abstract

This chapter illustrates the use of Gabor frame analysis to derive results on the spectral properties of integral and pseudodifferential operators. In particular, we obtain a sufficient condition on the kernel of an integral operator or the symbol of a pseudodifferential operator which implies that the operator is trace-class. This result significantly improves a sufficient condition due to Daubechies and Hörmander.

History and Evolution of the Density Theorem for Gabor Frames

“…For example, due to multipath propagation and Doppler shifts, wireless communications channels can be modeled as superpositions of time-frequency shift operators, which in mathematical terms are pseudodifferential operators. Gabor frames are a natural tool for analyzing pseudodifferential operators; for example see [94,104] for applications to the theory of pseudodifferential operators, [170] for applications to mobile communications, [135] for the use of Gabor frames in pseudodifferential operators in engineering (where they are called time-varying filters), and [89,Chapter 14] for a wealth of discussion on pseudodifferential operators in mathematics, engineering, and physics (where they are called quantization rules).…”

Frame expansions for Gabor multipliers

“…Additional applications of the time-frequency analysis of such operators are found in [30,[40][41][42][43][44][45][46][47][48][49].…”

Pseudodifferential operators and Banach algebras in mobile communications

“…Let I denote the coordinate reflection operator If (x, y) = f (−x, y). By combining the support condition onσ with (21) and using the relation…”

“…Sjöstrand's class can be seen as a member of the family of modulation spaces. Modulation spaces have been used as symbol classes for pseudodifferential operators in, e.g., [15,18,21,29,44,45]. By using the modulation space M p,q w defined in (10) as symbol class we characterize the time-frequency contents of the symbol σ via the decay properties of its STFT, i.e., by considering V Ψ σ L p,q w .…”