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Complex analysis and special topics in harmonic analysis. (English) Zbl 0837.30001
Berlin: Springer-Verlag. x, 482 p. (1995).
This is the second volume on complex analysis and it follows: Complex variables; an introduction by the same authors. Here is its Contents: Preface; (1) Boundary values of holomorphic functions and analytic functionals; (2) Interpolation and the algebras \(A_p\); (3) Exponential polynomials; (4) Integral-valued entire functions; (5) Summation methods; Harmonic analysis; References; Notations and an index. Chapter 1 provides the reader some useful tools of complex variable analysis: Hardy spaces in the unit disk, hyperfunctions, elements of analytic functionals. Here is achieved a fair connection between the distributions on the real line and the analytic functional. In chapter 2 a general study of Hörmander algebra \(A_p\) is presented. Here, the entire function with the growth of the type \(p(z)= |\text{Im}(z)|+ \log(1+ |z|^2)\) plays a central role. Chapter 3 provides an exposition of the properties of exponential polynomials, whose typical element is \(f(z)= \sum_j P_j(z)\exp(\alpha_j z)\), where \(j\) runs a finite set of integers.
Here \(P_j(z)\) are complex polynomials and \(\alpha_j\) are distinct complex numbers. This family plays an important role in many applications. The so-called \(G\)-transform introduced in chapter 4 allows to derive in an easy way many properties of entire functions. The chapter 5 is devoted to a systematic study of some summation methods allowing to find the analytic continuation of a given power series. Much attention is paid to the Mittag-Leffler method and to the Lindelöf indicator function.
Also, the Fourier-Borel transform of order \(\rho\) of an analytic functional is discussed. In chapter 6 some actual problems of harmonic analysis are discussed. They are of great interest in applications (computerized tomography based on the Radon transform). Among them we remark: convolution equations in \(\mathbb{R}\) and in \(\mathbb{C}\), difference equations in \(\mathbb{C}\) and an always of interest subject the deconvolution. The present book has a great extent (482 pages) with a clear and high definition typing. If the form is excellent, the content is also exceptional. The two authors have many contributions in the discussed problems.
Clearly, they are excellent theorists, but the practical material included in the book (many interesting extensions and exercises extracted from various applications) show that they feel rightly the areas, where the theory is useful. Therefore, the book will be an invaluable “instrument de travail” for applied mathematicians, for research engineers and those students which want to master the harmonic analysis.

MSC:
30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30D15 Special classes of entire functions of one complex variable and growth estimates
44A35 Convolution as an integral transform
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