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Gap and Density Theorems
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Colloquium Publications
1940; 246 pp; softcover
Volume: 26
Reprint/Revision History:
fifth printing 2011
ISBN-10: 0-8218-1026-X
ISBN-13: 978-0-8218-1026-2
List Price: US$42 Member Price: US$33.60
Order Code: COLL/26

Fourier Transforms in the Complex Domain - Raymond E A C Paley and Norbert Wiener

A typical gap theorem of the type discussed in the book deals with a set of exponential functions $${ \{e^{{{i\lambda}_n} x}\} }$$ on an interval of the real line and explores the conditions under which this set generates the entire $$L_2$$ space on this interval. A typical gap theorem deals with functions $$f$$ on the real line such that many Fourier coefficients of $$f$$ vanish.

The main goal of this book is to investigate relations between density and gap theorems and to study various cases where these theorems hold. The author also shows that density- and gap-type theorems are related to various properties of zeros of analytic functions in one variable.

Reviews

"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."

-- Mathematical Reviews

• On the closure of $${ \{e^{{{i\lambda}_n} x}\} }$$, I
• On the closure of $${ \{e^{{{i\lambda}_n} x}\} }$$, II
• Zeros of entire functions of exponential type
• On non-harmonic Fourier series
• Fourier transforms of nonvanishing functions
• A density theorem of Pólya
• Determination of the rate of growth of analytic functions from their growth on sequences of points
• An inequality and functions of zero type
• Existence of functions of zero type bounded on a sequence of points
• The general higher indices theorem
• The general unrestricted Tauberian theorem for larger gaps
• On restrictions necessary for certain higher indices theorems
• Appendix