viii

PREFACE

of this volume is especially appealing. Wavelet theory can be used with Radon

transforms to develop directional wavelets, and these can be used in microlocal

analysis. Current researchers in Blashke-Gelfand-Santalo-style integral geometry

use Fourier and microlocal analysis to understand classical problems. The interplay

between Lie theory and Radon transforms is illustrated by several articles in this

proceedings. Ideas from wavelet theory give rise to questions in the theory of group

representation theory. As represented in this volume, problems on symmetric spaces

and geometry can be understood and analyzed using integral geometry and integral

transforms. The Fourier, wavelet and similar transforms discussed in this volume

are used in PDEs. Some new trends include the use of geometries and diagrams in

the extension of the concept of wavelet sets, e.g., Bratteli diagrams, Coxeter, and

Dynkin diagrams!

This volume of Contemporary Mathematics is based on two special sessions at

the annual AMS meeting in New Orleans, January 2007 and a satellite workshop

in Baton Rouge, January 4-5, 2007, on Harmonic Analysis and Applications. The

sections are: Special Session on Radon Transforms, Convex Geometry, and Geo-

metric Analysis organized by Eric L. Grinberg, Peter Kuchment, Gestur Olafsson,

Eric Todd Quinto, and Boris S. Rubin; Special Session on Frames and Wavelets

in Harmonic Analysis, Geometry, and Applications organized by Palle E. T. Jor-

gensen, David R. Larson, Peter R. Massopust, and Gestur Olafsson. The workshop

at LSU was organized by Gestur Olafsson and Boris Rubin and was supported by

NSF grant DMS-0637383. All articles in this volume have been peer-reviewed.

This volume consists of invited expositions which together represent a broad

spectrum of fields, stressing surprising interactions and connections between areas

that are normally thought of as disparate. On the relatively pure side are harmonic

analysis, convex geometry, symmetric spaces, representation theory (the groups

include continuous and discrete, finite and infinite, compact and non-compact), op-

erator theory, PDE, and mathematical probability. Moving in the applied direction

are wavelets, fractals, ergodic theory, engineering topics such as frames, signal and

image processing, including medical imaging, and mathematical physics. Work on

Radon transforms and their applications spans almost a century, and it continues to

inspire students and to interact with both pure and applied mathematics (e.g., to-

mography, PDE, and Gabor time-frequency analysis). Although the articles cover a

broad range in harmonic analysis, the main themes are related to integral geometry,

the Radon transform, wavelets and frame theory. We group them in the following

way, but any such classification reflects only some aspects of the individual articles.

Frame Theory and applications

(1) J.J. Benedetto, 0. Oktay, A. Tangboondouangjit: Complex Sigma-Delta

Quantization Algorithms for Finite Frames.

(2) B. Johnson and

K.

Okoudjou: Frame Potential and Finite Abelian Groups.

(3) G. Kutyniok and Casazza: Robustness of Fusion Frames under Erasures

of Subspaces and of Local Frame Vectors.