Speaker: John J. Benedetto, University of Maryland

Title: Finite frames and non-harmonic Fourier series

Abstract: Non-harmonic Fourier series are a generalization of Fourier series, and they are naturally associated with non-uniform sampling problems. They are formulated in the context of Fourier frames. These frames go back to the 19th century and ideas of Riemann, Weber, and Dini. In the 20th century, Fourier frames were the basis of profound analysis by Paley and Wiener, Beurling and Malliavin, and Landau. They also led to the general theory of frames for Hilbert spaces formulated by Duffin and Schaeffer.

We describe the finite frame special case of the general theory, and several contemporary applications for this setting, e.g., the construction of constant amplitude zero autoccorrelation sequences for phase-coded waveforms used in radar and communications, and Sigma-Delta modulation for A/D conversion, to name but two.

In a different direction, and originating in Christoffel's notion of balayage (1871), which was developed in the area of potential theory, we see how balayage is related to Fourier frames (Beurling). We extend this point of view into a theory generalizing Fourier frames in terms of parametrization by Radon measures. The Wiener-Beurling theory of spectral synthesis plays an intrinsic role.

The balayage results are a collaboration with Enrico Au-Yeung.

Place: Science and Technology Building I, Room 242

Refreshments will be served before the talk at 3:00 p.m. in Room 222.

Department of Mathematical Sciences
George Mason University
4400 University Drive, MS 3F2
Fairfax, VA 22030-4444
http://math.gmu.edu/
Tel. 703-993-1460, Fax. 703-993-1491