NATO Advanced Study Institute

Fourier Analysis and Its Applications

July 16-29, 1989,   Il Ciocco, Tuscany, Italy

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	Purpose
	Fourier Analysis, founded upon 18th century mathematics, has remained a central topic in real and complex analysis and in applied mathematics for over two hundred years. Although the fundamentals of this intriguing area of mathematics have long been well developed, recent important advances make a NATO ASI on Fourier Analysis and its applications especially significant at this time. One major advance, which is an exciting subtopic at the conference, is the solution of the "Phase Reconstruction From Amplitude Problem," for which Herbert Hauptman won a 1985 Nobel Prize in Chemistry. Although Dr. Hauptman is a mathematician of the first rank, the fact that most of his research is published in the crystallographic literature has meant that many other Fourier analysts are only vaguely aware of his work. This unfortunate circumstance will be redressed at the ASI. Other important advances have recently occurred in both the theory and application of polynomials with restricted coefficients. Foremost among these is the remarkable result of J.P. Kahane, who improved upon T. Körner's answer to the Littlewood Conjecture and solved the Erdös Conjecture for polynomials with coefficients of modulus one. D.J. Newman has also made major contributions to the state of mathematical knowledge in this area. Furthermore, the work of M.R. Schroeder in concert hall acoustics and other applications of reflection phase gratings has added new importance to these classical questions in Fourier Analysis. The current work of Prometheus Inc. in notch filtering, null steering, and the design of low peak factor signals offers still more examples of applications of polynomials with restricted coefficients. A third particularly exciting current research area in both pure and applied Fourier Analysis is wavelet construction, which is a traditional goal in signal processing. Present research of the "French school," including the work of J. Benedetto, is focusing on the construction of custom-made waveforms with which to provide a spectral analysis of complicated signals arising in time-varying environments. Clearly, advances in both pure and applied Fourier Analysis are continuing, and there is a need for further interaction between these two intimately related disciplines. A major purpose of our ASI, in addition to the stated NATO purpose of promoting international scientific cooperation, is to effect such an interaction.
	Acknowledgements
	We wish to thank the following for their contribution to the success of this conference:
	NATO Scientific & Environmental Affairs Division
	U.S. Air Force EOARD
	National Science Foundation
	Prometheus Inc.