Applied Harmonic Analysis

Advances in communication, sensing, and computational power have led to an explosion of data. The size and varied formats for these datasets challenge existing techniques for transmission, storage, querying, display, and numerical manipulation. It is thus vital to develop efficient and robust data representations that lend themselves to scientific analysis and computation.

Harmonic analysis studies the decomposition of signals and data as the superposition of basic waves. This provides an efficient framework for compact representation of data, signals and operators using multiscale and other decompositions. Multiscale methods allow for rapid access to good coarse resolution of the data while retaining the flexibility for increasingly fine representations.

Another point of view is to represent the data using redundant function systems. Redundant representations allow for a multitude of data decompositions. While this appears contrary to the need for efficiency, the redundancy gives the flexibility of choosing `best representations' from a unified family of representers and thereby provides efficiency and robustness.

Topics of recent and current interest include

  • Generalized wavelet systems,such as brushlets, framelets, Gabor frames and wavelet packets.
  • Jackson and Bernstein inequalities for non-linear approximation.
  • Greedy algoritms and related constructive methods of approximation.
  • Compressed sensing.
  • Representation of function spaces using redundant systems.
  • Statistical models for feature identification.
  • Pseudodifferential operators.