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
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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