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Softcover ISBN:  9780821810262 
Product Code:  COLL/26 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 
eBook ISBN:  9781470431747 
Product Code:  COLL/26.E 
List Price:  $60.00 
MAA Member Price:  $54.00 
AMS Member Price:  $48.00 
Softcover ISBN:  9780821810262 
eBook ISBN:  9781470431747 
Product Code:  COLL/26.B 
List Price:  $125.00 $95.00 
MAA Member Price:  $112.50 $85.50 
AMS Member Price:  $100.00 $76.00 

Book DetailsColloquium PublicationsVolume: 26; 1940; 246 ppMSC: Primary 30;
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. 
Table of Contents

Chapters

Chapter 1. On the closure of $\{ e^{i\lambda _nx} \}$, I

Chapter 2. On the closure of $\{ e^{i\lambda _nx} \}$, II

Chapter 3. Zeros of entire functions of exponential type

Chapter 4. On nonharmonic Fourier series

Chapter 5. Fourier transforms of nonvanishing functions

Chapter 6. A density theorem of Pólya

Chapter 7. Determination of the rate of growth of analytic functions from their growth on sequences of points

Chapter 8. An inequality and functions of zero type

Chapter 9. Existence of functions of zero type bounded on a sequence of points

Chapter 10. The general higher indices theorem

Chapter 11. The general unrestricted Tauberian theorem for larger gaps

Chapter 12. On restrictions necessary for certain higher indices theorems

Appendix


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


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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.

Chapters

Chapter 1. On the closure of $\{ e^{i\lambda _nx} \}$, I

Chapter 2. On the closure of $\{ e^{i\lambda _nx} \}$, II

Chapter 3. Zeros of entire functions of exponential type

Chapter 4. On nonharmonic Fourier series

Chapter 5. Fourier transforms of nonvanishing functions

Chapter 6. A density theorem of Pólya

Chapter 7. Determination of the rate of growth of analytic functions from their growth on sequences of points

Chapter 8. An inequality and functions of zero type

Chapter 9. Existence of functions of zero type bounded on a sequence of points

Chapter 10. The general higher indices theorem

Chapter 11. The general unrestricted Tauberian theorem for larger gaps

Chapter 12. On restrictions necessary for certain higher indices theorems

Appendix

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