Gap and Density Theorems (Colloquium Publications (Amer Mathematical Soc))

Description

The author has collected in this book his original contributions on a series of related subjects; some of these contributions were published earlier in various periodicals, others are published here for the first time. The book doubtless has a certain unity, but it seems to me difficult, if not impossible, to circumscribe the subject or characterize the methods in simple general terms. Subject and methods alike are greatly influenced by the work of Wiener, and, especially, by Paley and Wiener's book "Fourier Transforms in the Complex Domain" [New York, 1934], of which the present book constitutes, in some respects, a continuation. The combination of "complex variables" and "real variables" in methods and results also recalls the work of Paley and Wiener. A good part of the book is connected with the work of G. Pólya [Math. Z. 29, 549--640 (1929), and problems 105--108, Jber. Deutsch. Math. Verein. 40, 80--81 (1931)]. Some of the subjects treated here were considered earlier by V. Bernstein, Miss Cartwright and others. The last part of the book is devoted to a wide generalization of a theorem of Hardy and Littlewood. The author contributes something essential to all his subjects, obtains very precise results and gives new proofs. Some of his proofs are long, difficult and highly technical, but the details are presented with much care and precision. (Reviewed by G. Pólya)




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