This volume contains papers presented at the July, 1989 NATO Advanced Study Institute on Fourier Analysis and its Applications. The conference, held at the beautiful II Ciocco resort near Lucca, in the glorious Tuscany region of northern Italy, created a dynamic in­ teraction between world-renowned scientists working in the usually disparate communities of pure and applied Fourier analysts. The papers to be found herein include important new results in x-ray crystallography by Nobel Laureate Herbert Hauptman, the application of the new concept of bispectrum to system identification by renowned probabilist Athanasios Papoulis, fascinating appli­ cations of number theory in Fourier analysis by eminent electrical engineer Manfred R. Schroeder, and exciting concepts regarding polynomials with restricted coefficients by foremost mathematical problem solver Donald J. Newman. The remaining papers further illustrate the inherent power and beauty of classical Fourier analysis, whether the results presented were sought as an end in themselves, or whether these classical methods were employed as a tool in illustrating and solving a particular applied problem. From antenna design to concert hall acoustics to image and speech processing to unimodular polynomi­ als, each conference participant benefited significantly from his or her exposure, in many cases for the first time, to those scientists on the other end of the spectrum from them­ selves. The purpose of this volume is to pass those benefits on to the reader.

I: Chemical, Physical and Biomedical Applications.- A Minimal Principle in the Direct Methods of X-ray Crystallography.- Fourier Shape Analysis of Anatomic Structures.- Harmonic Analysis of the Semi-Classical Limit.- Fourier Transform in non-Euclidean Space and its Applications to Physics.- Complex Finite Fourier Transform Techniques Applied to Thermal Energy Storage Problems.- Spline Interpolation of Data of Power Growth.- II: Polynomials with Restricted Coefficients.- Properties on the Unit Circle of Polynomials with Unimodular Coefficients.- On the Maximum Modulus for a Certain Class of Unimodular Trigonometric Polynomials.- Number Theory and Fourier Analysis Applications in Physics, Acoustics and Computer Science.- Functions of Modulus 1 on Zn Whose Fourier Transforms Have Constant Modulus, and "Cyclic n-Roots".- III: Fourier Analysis in Engineering.- Uncertainty Principle Inequalities and Spectrum Estimation.- The Generalized Pencil-of-Function Method for Extracting Poles from Transient Responses.- Application of Conjugate Gradient Method and FFT to Electromagnetics and Signal Processing Problems.- On the Application of Fourier Transformation in Antenna Measurements.- Computations in X-ray Crystallography.- A New Approach to Irregular Sampling of Band-Limited Functions.- Auto-Cross Correlation Functions of the Even and the Odd Parts of Fourier Series.- Fourier Methods Applicable in the Numerical Solution of Electromagnetic Time-Domain Scattering Problems.- Results on the Absolutely Convergent Series of Functions and of Distributions.- Invariance Techniques in Spectral Estimation and Signal Processing.- Fourier Analysis in Reflector Antenna Synthesis.- Bispectra and System Identification.- Combined Laplace and Fourier Transformations in the Theory of TransientWave Propagation in Layered Media.- Coherent Frames and Irregular Sampling.- Gabor and Wavelet Expansions.- Translation Invariant Multiscale Analysis.- High-Resolution Spectral Analysis Using Recurrence Relations.- Diophantine Inequalities and Sampling Rates for Multiband Signals.- Second-Harmonic Effects in Schottky-Barrier Diodes.- Holographic Image Coding and Neurocomputer Architectures.- IV: Complex Analysis and Fourier Analysis.- A Weighted L1 Estimate for the Laplace Operator.- Weighted Extensions of Restriction Theorems for the Fourier Transform.- Atomic Decompositions.- The Wiener-Plancherel Theorem on Rn.- Successive Derivatives of Analytic Functions.- Combinatorial Dimension and Measurements of Interdependencies.- Comparison of Spectras in some Functional Spaces.- On a Class of Riesz Products on Metrizable Groups.- Groupoids and Fourier Approximations.- V: Problems.- Miscellanous.- A Problem from the Theory of Wave Propagation in a Curved, Layered Medium.- Problems on Polynomials with Restricted Coefficients Arising From Questions in Antenna Array Theory.