Softcover ISBN:  9781470455934 
Product Code:  ULECT/75 
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eBook ISBN:  9781470460099 
Product Code:  ULECT/75.E 
List Price:  $55.00 
MAA Member Price:  $49.50 
AMS Member Price:  $44.00 
Softcover ISBN:  9781470455934 
eBook: ISBN:  9781470460099 
Product Code:  ULECT/75.B 
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Softcover ISBN:  9781470455934 
Product Code:  ULECT/75 
List Price:  $55.00 
MAA Member Price:  $49.50 
AMS Member Price:  $44.00 
eBook ISBN:  9781470460099 
Product Code:  ULECT/75.E 
List Price:  $55.00 
MAA Member Price:  $49.50 
AMS Member Price:  $44.00 
Softcover ISBN:  9781470455934 
eBook ISBN:  9781470460099 
Product Code:  ULECT/75.B 
List Price:  $110.00 $82.50 
MAA Member Price:  $99.00 $74.25 
AMS Member Price:  $88.00 $66.00 

Book DetailsUniversity Lecture SeriesVolume: 75; 2020; 219 ppMSC: Primary 46; 30
The classical \(\ell^{p}\) sequence spaces have been a mainstay in Banach spaces. This book reviews some of the foundational results in this area (the basic inequalities, duality, convexity, geometry) as well as connects them to the function theory (boundary growth conditions, zero sets, extremal functions, multipliers, operator theory) of the associated spaces \(\ell^{p}_{A}\) of analytic functions whose Taylor coefficients belong to \(\ell^p\). Relations between the Banach space \(\ell^p\) and its associated function space are uncovered using tools from Banach space geometry, including BirkhoffJames orthogonality and the resulting Pythagorean inequalities. The authors survey the literature on all of this material, including a discussion of the multipliers of \(\ell^{p}_{A}\) and a discussion of the Wiener algebra \(\ell^{1}_{A}\).
Except for some basic measure theory, functional analysis, and complex analysis, which the reader is expected to know, the material in this book is selfcontained and detailed proofs of nearly all the results are given. Each chapter concludes with some end notes that give proper references, historical background, and avenues for further exploration.
ReadershipGraduate students and researchers interested in the connections between functional analysis and analytic function theory.

Table of Contents

Chapters

The basics of $\ell ^p$

Frames

The geometry of $\ell ^p$

Weak parallelogram laws

Hardy and Bergman spaces

$\ell ^p$ as a function space

Some operators on $\ell ^p_A$

Extremal functions

Zeros of $\ell ^p_A$ functions

The shift

The backward shift

Multipliers of $\ell ^p_A$

The Wiener algebra


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The classical \(\ell^{p}\) sequence spaces have been a mainstay in Banach spaces. This book reviews some of the foundational results in this area (the basic inequalities, duality, convexity, geometry) as well as connects them to the function theory (boundary growth conditions, zero sets, extremal functions, multipliers, operator theory) of the associated spaces \(\ell^{p}_{A}\) of analytic functions whose Taylor coefficients belong to \(\ell^p\). Relations between the Banach space \(\ell^p\) and its associated function space are uncovered using tools from Banach space geometry, including BirkhoffJames orthogonality and the resulting Pythagorean inequalities. The authors survey the literature on all of this material, including a discussion of the multipliers of \(\ell^{p}_{A}\) and a discussion of the Wiener algebra \(\ell^{1}_{A}\).
Except for some basic measure theory, functional analysis, and complex analysis, which the reader is expected to know, the material in this book is selfcontained and detailed proofs of nearly all the results are given. Each chapter concludes with some end notes that give proper references, historical background, and avenues for further exploration.
Graduate students and researchers interested in the connections between functional analysis and analytic function theory.

Chapters

The basics of $\ell ^p$

Frames

The geometry of $\ell ^p$

Weak parallelogram laws

Hardy and Bergman spaces

$\ell ^p$ as a function space

Some operators on $\ell ^p_A$

Extremal functions

Zeros of $\ell ^p_A$ functions

The shift

The backward shift

Multipliers of $\ell ^p_A$

The Wiener algebra