Christopher Heil 
Harmonic Analysis and Applications 
In Honor of John J. Benedetto

This volume is dedicated to John Benedetto. It seems just yesterday that we celebrated his 60th birthday in a memorable conference in College Park. Yet that was October of 1999, and already more than six years have passed. But John is still too young to be fully honored by a single foreword, or even a singlevolume, thatattemptstosummarizetheimpactofhisworkonharmonic analysis, his students, and his coworkers.Given his continuing high (and even increasing) level of activities, his list of “lifetime achievements” is surely far from complete. Even so, we will make an attempt in this foreword to take a look back, to see the major lines of his work and activities during the past 40 years of his life as a scientist, and to learn from his biography (and bibliography) how the ?eld of harmonic analysis has changed over the years, and in particular to see the vibrant role that John has taken in this process. John’s ?rst paper appeared in 1965, when he was 25 years old, and his ?rst book (the Springer Lecture Notes on Harmonic Analysis on Totally D- connected Sets) when he was 31. By that time he had already published on the subjects of Tauberian algebras, in the theory of generalized functions, and on questions related to spectral synthesis. His work on this latter topic continued through the 1970s, culminating in the insightful volume Spectral Synthesis (1975). Only a year later, his text Real Variables and Integration with Historical Notes appeared.

विषयसूची

Harmonic Analysis.- The Gibbs Phenomenon in Higher Dimensions.- Weighted Sobolev Inequalities for Gradients.- Semidiscrete Multipliers.- Frame Theory.- A Physical Interpretation of Tight Frames.- Time-Frequency Analysis.- Recent Developments in the Balian-Low Theorem.- Some Problems Related to the Distributional Zak Transform.- Gabor Duality Characterizations.- A Pedestrian’s Approach to Pseudodifferential Operators.- Linear Independence of Finite Gabor Systems.- Wavelet Theory.- Explicit Cross-Sections of Singly Generated Group Actions.- The Theory of Wavelets with Composite Dilations.- Sampling Theory and Shift-Invariant Spaces.- Periodic Nonuniform Sampling in Shift-Invariant Spaces.- Sampling on Unions of Shifted Lattices in One Dimension.- Learning the Right Model from the Data.- Redundancy in the Frequency Domain.- Density Results for Frames of Exponentials.