Preface

In recent years a number of specialties in mathematical analysis, geometry and num-

ber theory have come together. We have witnessed ideas from one field connecting

with and being used in surprising ways in another fields, and making contacts to

applications involving the Radon transforms, frames, wavelets, and fractals; and

to applications outside of mathematics as well: engineering, physics, and medical

imaging. This works in reverse as well, including even applications and basic ideas

from signal processing to mathematics! This multifaceted enterprise has a com-

mon core: harmonic analysis and symmetry. The purpose of the present volume

of Contemporary Mathematics is to present some of this fascinating material and

connections in a form that is understandable and attractive to a wider readership,

including graduate students in mathematics and its neighboring fields.

Harmonic analysis is related to-and inspired by-several branches of mathe-

matics and science: geometry, Lie groups, representation theory, signal processing,

quantum physics, medicine, and engineering, just to name few influential connec-

tions. In particular, considerable progress has been made on Radon and wavelet

transforms that is motivated by their wide range of applications. New questions in

applications have inspired new questions and problems in mathematics.

Two main themes of the book are integral geometry and integral transforms,

including the Radon transforms and the wavelet transform. Wavelets and related

basis constructions, such as frames from engineering, are a little more than two

decades old (not counting Haar's wavelet). Their theory and applications are pre-

sented from different angles: fractal analysis, harmonic analysis, operator theory,

geometry, computations, special tilings, and algorithms. The use of wavelets is com-

pared with other transform tools: Fourier, Radon, Gabor, multiscale and more.

Modern Integral geometry is the study of Radon transforms-integral trans-

forms that are defined geometrically, as integrals over subsets. They are named

after the Austrian mathematician, Johann Radon (1887-1956), who studied the

transform that integrates planar functions over all lines in the plane. The goal is

to learn about the function

f

from its integrals. This example is the mathemati-

cal model of X-ray tomography, and related transforms are appear in many other

important tomography problems. Radon transforms are used in partial differential

equations and group theory as well. The interplay between groups and Fourier and

Radon transforms, including microlocal and harmonic analysis, provides the bridge

between this theme and the other themes of this book.

The subjects covered in this book form a unified whole, and they stand at

the crossroads of pure and applied mathematics. Together the topics of Radon

transforms, geometry and wavelets represent a panoramic view of what is now called

Applied and Numerical Harmonic Analysis. The interplay between the themes

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