The first edition of this book has been out of print for some time and I
have decided to follow the publisher's kind suggestion to prepare a new
edition. Many examples with explicit inversion formulas and range theo-
rems have been added, and the group-theoretic viewpoint emphasized. For
example, the integral geometric viewpoint of the Poisson integral for
the disk leads to interesting analogies with the X-ray transform in
Euclidean 3-space. To preserve the introductory flavor of the book the
short and self-contained Chapter Von Schwartz' distributions has been
added. Here §5 provides proofs of the needed results about the Riesz
potentials while §§3-4 develop the tools from Fourier analysis following
closely the account in Hormander's books (1963] and [1983]. There is
some overlap with my books (1984] and [1994b] which however rely
heavily on Lie group theory. The present book is much more elementary. I
am indebted to Sine Jensen for a critical reading of parts of the
manuscript and to Hilgert and Schlichtkrull for concrete contributions
men- tioned at specific places in the text. Finally I thank Jan Wetzel
and Bonnie Friedman for their patient and skillful preparation of the
manuscript.