A pedestrian’s approach to pseudodifferential operators. (English) Zbl 1098.47043
Heil, Christopher (ed.), Harmonic analysis and applications. In Honor of John J. Benedetto. Basel: Birkhäuser (ISBN 0-8176-3778-8/hbk). Applied and Numerical Harmonic Analysis, 139-169 (2006).
Summary: Pseudodifferential operators are an indispensable tool for the study of partial differential equations and are therefore a branch of classical analysis. In this chapter [see Zbl 1095.00007 for the whole collection], we offer an approach using time-frequency methods. In this approach, time-frequency representations that are standard in signal analysis are used to set up the formalism of pseudodifferential operators, and certain classes of function spaces and symbols, the so-called modulation spaces, arise naturally in the investigation. Although the approach is “pedestrian” and based more on engineering intuition than on “hard” analysis, strong results on boundedness and Schatten class properties are within its scope.
For the entire collection see [Zbl 1095.00007].
For the entire collection see [Zbl 1095.00007].
MSC:
| 47G30 | Pseudodifferential operators |
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 46F12 | Integral transforms in distribution spaces |
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
| 47-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory |
