Pseudodifferential operators, Gabor frames, and local trigonometric bases

Time-Frequency Analysis of Sjöstrand’s Class

“…By property (iii) σ w is almost diagonalized by the Gabor frame G(g, Λ). It is well-known that certain types of pseudodifferential operators are almost diagonalized with respect to wavelet bases or local Fourier bases [33,35]. What is remarkable in Theorem 3.2 is that the almost diagonalization property actually characterizes a symbol class.…”

“…This may not be surprising, because it is well-known that pseudodifferential operators with classical symbols are almost diagonalized by wavelet bases and local Fourier bases [33,35]. What is remarkable is that the almost diagonalization property with respect to Gabor frames is a characterization of Sjöstrand's class.…”

Localizable operators and the construction of localized frames

“…Pseudodifferential operators. Rochberg and Tachizawa [18] and Gröchenig [16] showed the boundedness of pseudodifferential operators on generalized Sobolev spaces and modulation spaces respectively using this particular technique of looking at the matrix representation with respect to Gabor frames. We focus on the results of Assume σ(z, ζ) ∈ S (R 2d ) and belongs to the Sjöstrand class.…”

“…These operators are a generalization of pseudodifferential operators and are useful in the study of partial differential equations. Rochberg and Tachizawa [18] proved that certain Gabor frames almost diagonalize pseudodifferential operators whose symbols are in a class similar to that of the classical symbol classes used in partial differential equations. In [17], Gröchenig and Heil proved a similar result for pseuodifferential operators with symbol in the modulation space M ∞,1 , also known as Sjöstrand's class.…”

Integral Operators, Pseudodifferential Operators, and Gabor Frames

“…Tachizawa [28], [29] had earlier used Wilson basis expansions to obtain boundedness results for pseudodifferential operators on the modulation spaces. Rochberg and Tachizawa employed Gabor frame expansions in [24], as do Czaja and Rzeszotnik in [2]. Labate applied Gabor analysis to derive results on compositions and compactness of pseudodifferential operators on the modulation spaces in [20], [21].…”