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The Russian edition of this book appeared 5 years ago. Since that time, many results have been improved upon and new approaches to the problems investigated in the book have appeared. But the greatest surprise for us was to discover that there exists a large group of mathematicians working in the area of the so-called White Noise Analysis which is closely connected with the essential part of our book, namely, with the theory of generalized functions of infinitely many variables. The first papers dealing with White Noise Analysis were written by T. Hida in Japan in 1975. Later, this analysis was devel oped intensively in Japan, Germany, U.S.A., Taipei, and in other places. The related problems of infinite-dimensional analysis have been studied in Kiev since 1967, and the theory of generalized functions of infinitely many variables has been in vestigated since 1973. However, due to the political system in the U.S.S.R., contact be tween Ukrainian and foreign mathematicians was impossible for a long period of time. This is why, to our great regret, only at the end of 1988 did one of the authors meet L. Streit who told him about the existence of White Noise Analysis. And it become clear that many results in these two theories coincide and that, in fact, there exists a single theory and not two distinct ones.

Spectral Methods in Infinite-Dimensional Analysis: Volume I Volume II

Contents Volume I: Preface to the English Edition. Preface. Introduction. 1. Rigged Spaces. 2. Generalized Functions of Infinitely Many Variables. Gaussian Measures. 3. Spectral Theorem. 4. Representations by Commuting Operators. Bibliographical Notes. References. Subject Index. List of Notations. Contents Volume II: 5. Application of the Theory of Expansions to the Harmonic Analysis. 6. Infinite-Dimensional Elliptic Differential Operators of the Second Order. 7. Infinite-Dimensional Differential Operators in the Models of Quantum Statistical Physics and Field Theory. Bibliographical Notes. References. Subject Index. List of Notations.