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GENERALIZED INTEGRAL TRANSFORMATIONS |
| PREFACE The subject of this book arises from the confluence of two mathematical disciplines, the theory of integral transformations and the theory of generalized functions. The former theory is a classical subject in mathematics, whose literature can be traced back through at least 150 years. The latter subject, on the other hand, is of recent origin, its advent being the publication of Laurent Schwartz's books, which appeared from 1944 onward; the most notable is his two-volume work, ``Theorie des Distributions,'' published in1950 and 1951. Some fragments of the theory appeared still earlier in the works of S. Bochner around 1927 and of S.L.Soboleff around 1936. An important achievement was the extension to generalized functions of the Fourier transformation, which became thereby a remarkably powerful tool especially in the theory of partial differential equations. The theory and applications of the generalized Fourier transformation has been an active research area ever since. In 1952, L. Schwartz also extended the Laplace transformation to generalized functions, and since then research into this subject has also proceeded apace. Still another integral transformation, whose generalization has been investigated quite thoroughly, is the Hilbert transformation. However, research into the extension to generalized functions of other integral transformations remained dormant until recently despite the fact that the various types of such transformations are numerous, It is the purpose of this book to present an account of the recent generalizations of some of the simpler and more commonly encountered integral transformations, in particular, the Laplace, Mellin, Hankel, K, Weierstrass, and convolution transformations, as well as those arising from a variety of orthogonal series expansions. The convolution transformation is especially interesting in that it encompasses as special cases a number of specific transformations such as the one-sided Laplace, Stieltjes, and K transformations. We do not discuss either the generalized Fourier or generalized Hilbert transformations since their theories have already appeared ina number of books, and we have nothing new to add. On the other hand, we do discuss the Laplace transformation since the Mellin and Weierstrass transformations can be obtained from it by certain changes of variables; the theory presented her does not rely upon the Fourier transformation, and in this way it is distinct from (but equivalent to) Schwartz's approach to the Laplace transformation. Actually, for any of the integral transformations mentioned above, it is not at all difficult to define a generalized transformation if one restricts sufficiently the type of generalized function on which it is to act. The difficulty arises in obtaining either an inversion formula or a uniqueness theorem, and this must be accomplished if the generalized integral transformation is to become significant as an analytical tool. Such results are obtained for every integral transformation considered herein. As is the case in the theory of distributions and generalized functions, one meets in this book quite a variety of testing-functions spaces and their duals. This may be disconcerting to the reader especially since the resulting notation becomes somewhat formidable. However, there is no avoiding this situation if one wishes to achieve the generality aimed for in this book. Each integral transformation requires a different testing-function space, which is tailored to suit certain properties of the kernel function of the transformation. However, there is one unifying concept for all cases: Let I be the open interval of integration for the conventional integral transformation under consideration, and let E'(I) be the space of distribution whose supports are compact subsets of I. Then, it turns out that the corresponding generalized integral transformation is always defined on the members of E'(I). Thus, we have in all cases this simple criterion under which the transformation may be applied. Incidentally, even in the classical theory of integral transformations there is at least implicitly an abundance of function spaces. Indeed, every time one states a set of conditions under which a transformation can be applied to a function, one is indeed specifying a space of functions in the domain of the transformation. However, in the classical theory there is in general no need to denote these function spaces by symbols, in contrast to the generalized theory. This book is based upon a graduate course given at the State University of New York at Stony Brook; the course is usually addressed to both mathematics and engineering students. This is reflected in the fact that a substantial part of the book is devoted to the applications of generalized integral transformations to various initial-value and boundary-value problems, as well as to certain problems in systems theory. Nevertheless, the emphasis of this work is on the theory of these transformations. It is presumed that the reader has had a first course in advanced calculus and is therefore familiar with the standard theorems on the interchange of limit processes. Some knowledge of Lebesgue integration including Fubini's theorem and functions of a complex variable is also assumed. On the other hand, those results concerning topological linear spaces and generalized functions that will be needed are discussed in the first two chapters. On a few occasions we refer to our prior book, ``Distribution Theory and Transform Analysis'' for certain results concerning distributions. We freely use various properties of special functions, which appear in standard reference works such as those by Jahnke, Emde, and Losch, and by Erdelyi. We also use without proving a number of classical results from the conventional theory of integral transformations, in particular, the complex inversion formula for the Hankel transformation, orthonormal series expansions in the space L_{2}(a,b), and the Riesz-Fischer theorem. Since the proofs of these results appear in so many books, still another presentation seems hardly warranted. Finally, we also present two special cases of the generaized convolution transformation, and in doing so, we borrow several results from Hirschman and Widder's book on the convolution transformation. A.H. Zemanian Stony Brook, New York |
