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.