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Acta Applicandae

Book review

Author(s) Szmydt, Z.
Ziemian, B.
Title The Mellin transformation and Fuchsian type partial differential equations
Publisher Kluwer
Year of publication 1992
Reviewed by Günter Vojta

In recent years there has been growing interest in the theory and the applications of the Mellin transformation. Besides the Fourier and the Laplace transformations, this transformation has been established as the third of the great linear integral transformations. It operates very effectively in the solution of certain types of integral equations, in the fractional calculus, and in the treatment of problems e.g. of fractal physics and of the theory of chaotic systems. The interest has centered around one-dimensional Mellin transforms. Only recently problems of the multidimensional Mellin transformation have been attacked to a larger extent, and apparently up to now there existed no textbook on this topic.

The book under review fills this gap. It is written as a textbook on graduate level, and at the same time it guides to the frontiers of research. It provides a systematic and rigorous introduction into the theory on the basis of distributions. As an important application it presents results on singular partial differential equations of the Fuchsian type.

The volume consists of three chapters. Basic facts on complex topological vector spaces and on the theory of distributions are given in the short first chapter. The second chapter (100 pp) represents the main part of the work. In turn are treated the fundamentals of the Mellin transformation in several dimensions, the spaces and subspaces of Mellin distributions and operations in these spaces, examples of Mellin transforms, the structure of Mellin distributions, and mod)fied Cauchy and Hilbert transformations (in dimension 1). In the final third chapter (65 pp) various Fuchsian type differential equations and the existence and regularity of their solutions are discussed. Using the Mellin transformation this chapter constitutes a link between the theory of micro-localization and the theory of the generalized Borel transformation. An appendix explains elementary facts of the theory of resurgent functions of J. Ecalle used in this chapter.

The text is supplemented by exercises (with hints). There is a bibliography (4 pp) with titles, a list of symbols, and a aufficient subject index. The volume is well produced.

This work can be recommended to all who are interested in differential equations, integral equations, and integral transforms, particularly mathematicians, physicists, and engineers. Libraries are well advised to purchase the book.