Ebook: The Convolution Transform

From the Preface:
The operation of convolution applied to sequences or functions is basic
in analysis. It arises when two power series or two Laplace (or Fourier)
integrals are multiplied together. Also most of the classical integral
transforms involve integrals which define convolutions. For the present
authors the convolution transform oame as a natural generalization of the
Laplace transform. It was early reoognized that the now familiar real
inversion of the latter is essentially accomplished by a particular linear
differential operator of infinite order (in which translations are allowed).
When one studies general operators of the same nature one encounters
immediately general convolution transforms as the objects which they
invert. This relation between differential operators and integral transforms
is the basio theme of the present study.
The book may be read easily by anyone who has a working knowledge
of real and complex variable theory. For such a reader it should be
complete in itself, exoept that certain fundamentals from the Laplace
Tranform (number 6 in this series) are assumed. However, it is by no
means necessary to have read that treatise completely in order to under-
stand this one. Indeed some of those earlier results can now be better
understood as special oases of the newer developments.