**Colloquium Publications**

Volume: 26;
1940;
246 pp;
Softcover

MSC: Primary 30;

Print ISBN: 978-0-8218-1026-2

Product Code: COLL/26

List Price: $45.00

Individual Member Price: $36.00

**Electronic ISBN: 978-1-4704-3174-7
Product Code: COLL/26.E**

List Price: $45.00

Individual Member Price: $36.00

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# Gap and Density Theorems

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*N. Levinson*

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.

#### Table of Contents

# Table of Contents

## Gap and Density Theorems

- Cover Cover11
- Title page i2
- Dedication iii4
- Preface v6
- Contents vii8
- On the closure of {๐^{{}{{๐๐}_{๐}}๐ฅ}}, I 110
- On the closure of {๐^{{}{{๐๐}_{๐}}๐ฅ}}, II 1221
- Zeros of entire functions of exponential type 2534
- On non-harmonic Fourier series 4756
- Fourier transforms of nonvanishing functions 7382
- A density theorem of Pรณlya 8897
- Determination of the rate of growth of analytic functions from their growth on sequences of points 100109
- An inequality and functions of zero type 127136
- Existence of functions of zero type bounded on a sequence of points 153162
- The general higher indices theorem 186195
- The general unrestricted Tauberian theorem for larger gaps 203212
- On restrictions necessary for certain higher indices theorems 229238
- Appendix 243252
- Back Cover Back Cover1256

#### 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