Frames and Time-Frequency Analysis

Reading Materials:

A very short survey on frames for general mathematical audiences: WHAT IS a Frame?, C. Heil, Notices Amer. Math. Soc., 60 (2013), 748-750.

Main reference: A Basis Theory Primer, C. Heil, Birkhauser, Boston (2011) - expanded edition; password protected and available to the Graduate Program participants only.
• Most of the lectures will be based on the material in Chapters 7, 8 (Sections 8.1-8.4), and 11.
• Chapters 1 and 2 contain background on Banach and Hilbert spaces, operator theory, and functional analysis.
• Chapter 9 contains a short review of the Fourier transform.
• An Errata List is available (and please notify the author if you find additional typos!)
• The original 1998 unexpanded version is available here. However, that version covers only a limited portion of the material that will be presented in the lectures.

An old survey paper that covers some of the material on frames and Gabor frames (more concisely than the Basis Primer): Continuous and discrete wavelet transforms, C. E. Heil and D. F. Walnut, SIAM Review, 31 (1989), 628-666.

A survey on Gabor systems and the HRT Conjecture: Linear independence of finite Gabor systems, C. Heil, in: "Harmonic Analysis and Applications," Birkhäuser, Boston (2006), 171-206.   Errata.

A survey on an application of Gabor systems to pseudodifferential operators: Integral operators, pseudodifferential operators, and Gabor frames, C. Heil, in: "Advances in Gabor Analysis," H. G. Feichtinger and T. Strohmer, eds., Birkhäuser, Boston (2003), 153-169.

A (rather long) survey paper on Gabor frames (without proofs): History and evolution of the Density Theorem for Gabor frames, C. Heil, J. Fourier Anal. Appl., 13 (2007), 113-166.

Please visit Christopher Heil's papers page for other surveys and papers (some related to the lectures, others not so much).

LECTURE NOTES

• Day 1: Frames.
• Day 2: Gabor Frames.
• Day 3: Gabor Frames at the Critical Density (and beyond).
• Day 4: The STFT and the Modulation Spaces.
• Day 5: Modulation Spaces and Applications.

Other Useful References:

An introduction to frames in finite dimensions: Frames for Undergraduates, D. Han, K. Kornelson, D. Larson, and E. Weber, AMS, Providence, 2007.

The title says it all: Foundations of Time-Frequency Analysis, K. Gröchenig, Birkhäuser, Boston (2001).

A very nice volume complementary to the Basis Primer: An Introduction to Frames and Riesz Bases, O. Christensen, Birkhäuser, Boston (2003).