Colloquium Publications 1940; 246 pp; softcover Volume: 26 Reprint/Revision History: fifth printing 2011 ISBN10: 082181026X ISBN13: 9780821810262 List Price: US$45 Member Price: US$36 Order Code: COLL/26
 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 gaptype 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 Table of Contents  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 nonharmonic 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
