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Softcover ISBN:  9781470473631 
Product Code:  CHEL/322.S 
List Price:  $34.00 
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AMS Member Price:  $30.60 
eBook ISBN:  9781470474942 
Product Code:  CHEL/322.E 
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MAA Member Price:  $30.60 
AMS Member Price:  $30.60 
Softcover ISBN:  9781470473631 
eBook ISBN:  9781470474942 
Product Code:  CHEL/322.S.B 
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Book DetailsAMS Chelsea PublishingVolume: 322; 1986; 188 ppMSC: Primary 41
This is an easily accessible book on the approximation of functions—simple and without unnecessary details, but complete enough to include the main results of the theory. Except for a few sections, only functions of a real variable are treated. This work can be used as a textbook for graduate or advanced undergraduate courses or for selfstudy. Included in the volume are Notes at the end of each chapter, Problems, and a selected Bibliography.
ReadershipGraduate students and advanced undergraduate students interested in analysis.

Table of Contents

Front Cover

Preface to the Second Edition

Preface to the First Edition

Contents

Chapter 1. Possibility of Approximation

1. Basic Notions

2. Linear Operators

3. Approximation Theorems

4. The Theorem of Stone

5. Notes

PROBLEMS

Chapter 2. Polynomials of Best Approximation

1. Existence of Polynomials of Best Approximation

2. Characterization of Polynomials of Best Approximation

3. Applications of Convexity

4. Chebyshev Systems

5. Uniqueness of Polynomials of Best Approximation

6. Chebyshev's Theorem

7. Chebyshev Polynomials

8. Approximation of Some Complex Functions

9. Notes

PROBLEMS

Chapter 3. Properties of Polynomials and Moduli of Continuity

1. Interpolation

2. Inequalities of Bernstein

3. The Inequality of Markov

4. Growth of Polynomials in the Complex Plane

5. Moduli of Continuity

6. Moduli of Smoothness

7. Classes of Functions

8. Notes

PROBLEMS

Chapter 4. The Degree of Approximation by Trigonometric Polynomials

1. Generalities

2. The Theorem of Jackson

3. The Degree of Approximation of Differentiable Functions

4. Inverse Theorems

5. Differentiable Functions

6. Notes

PROBLEMS

Chapter 5. The Degree of Approximation by Algebraic Polynomials

1. Preliminaries

2. The Approximation Theorems

3. Inequalities for the Derivatives of Polynomials

4. Inverse Theorems

5. Approximation of Analytic Functions

6. Notes

PROBLEMS

Chapter 6. Approximation by Rational Functions. Functions of Several Variables

1. Degree of Rational Approximation

2. Further Theorems

3. Periodic Functions of Several Variables

4. Approximation by Algebraic Polynomials

5. Notes

PROBLEMS

Chapter 7. Approximation by Linear Polynomial Operators

1. Sums of de la ValleePoussin. Positive Operators

2. The Principle of Uniform Boundedness

3. Operators that Preserve Trigonometric Polynomials

4. Trigonometric Saturation Classes

5. The Saturation Class of the Bernstein Polynomials

6. Notes

PROBLEMS

Chapter 8. Approximation of Classes of Functions

1. Introduction

2. Approximation in the Space L1

3. The Degree of Approximation of the Classes W*p

4. Distance Matrices

5. Approximation of the Classes Λω

6. Arbitrary Moduli of Continuity; Approximation by Operators

7. Analytic Functions

8. Notes

PROBLEMS

Chapter 9. Widths

1. Definitions and Basic Properties

2. Sets of Continuous and Differentiable Functions

3. Widths of Balls

4. Applications of Theorem 2

5. Differential Operators

6. Widths of the Set R1

7. Notes

PROBLEMS

Chapter 10. Entropy

1. Entropy and Capacity

2. Sets of Continuous and Differentiable Functions

3. Entropy of Classes of Analytic Functions

4. General Sets of Analytic Functions

5. Relations between Entropy and Widths

6. Notes

PROBLEMS

Chapter 11. Representation of Functions of Several Variables by Functions of One Variable

1. The Theorem of Kolmogorov

2. The Fundamental Lemma

3. The Completion of the Proof

4. Functions Not Representable by Superpositions

5. Notes

PROBLEMS

Bibliography

Index

Back Cover


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This is an easily accessible book on the approximation of functions—simple and without unnecessary details, but complete enough to include the main results of the theory. Except for a few sections, only functions of a real variable are treated. This work can be used as a textbook for graduate or advanced undergraduate courses or for selfstudy. Included in the volume are Notes at the end of each chapter, Problems, and a selected Bibliography.
Graduate students and advanced undergraduate students interested in analysis.

Front Cover

Preface to the Second Edition

Preface to the First Edition

Contents

Chapter 1. Possibility of Approximation

1. Basic Notions

2. Linear Operators

3. Approximation Theorems

4. The Theorem of Stone

5. Notes

PROBLEMS

Chapter 2. Polynomials of Best Approximation

1. Existence of Polynomials of Best Approximation

2. Characterization of Polynomials of Best Approximation

3. Applications of Convexity

4. Chebyshev Systems

5. Uniqueness of Polynomials of Best Approximation

6. Chebyshev's Theorem

7. Chebyshev Polynomials

8. Approximation of Some Complex Functions

9. Notes

PROBLEMS

Chapter 3. Properties of Polynomials and Moduli of Continuity

1. Interpolation

2. Inequalities of Bernstein

3. The Inequality of Markov

4. Growth of Polynomials in the Complex Plane

5. Moduli of Continuity

6. Moduli of Smoothness

7. Classes of Functions

8. Notes

PROBLEMS

Chapter 4. The Degree of Approximation by Trigonometric Polynomials

1. Generalities

2. The Theorem of Jackson

3. The Degree of Approximation of Differentiable Functions

4. Inverse Theorems

5. Differentiable Functions

6. Notes

PROBLEMS

Chapter 5. The Degree of Approximation by Algebraic Polynomials

1. Preliminaries

2. The Approximation Theorems

3. Inequalities for the Derivatives of Polynomials

4. Inverse Theorems

5. Approximation of Analytic Functions

6. Notes

PROBLEMS

Chapter 6. Approximation by Rational Functions. Functions of Several Variables

1. Degree of Rational Approximation

2. Further Theorems

3. Periodic Functions of Several Variables

4. Approximation by Algebraic Polynomials

5. Notes

PROBLEMS

Chapter 7. Approximation by Linear Polynomial Operators

1. Sums of de la ValleePoussin. Positive Operators

2. The Principle of Uniform Boundedness

3. Operators that Preserve Trigonometric Polynomials

4. Trigonometric Saturation Classes

5. The Saturation Class of the Bernstein Polynomials

6. Notes

PROBLEMS

Chapter 8. Approximation of Classes of Functions

1. Introduction

2. Approximation in the Space L1

3. The Degree of Approximation of the Classes W*p

4. Distance Matrices

5. Approximation of the Classes Λω

6. Arbitrary Moduli of Continuity; Approximation by Operators

7. Analytic Functions

8. Notes

PROBLEMS

Chapter 9. Widths

1. Definitions and Basic Properties

2. Sets of Continuous and Differentiable Functions

3. Widths of Balls

4. Applications of Theorem 2

5. Differential Operators

6. Widths of the Set R1

7. Notes

PROBLEMS

Chapter 10. Entropy

1. Entropy and Capacity

2. Sets of Continuous and Differentiable Functions

3. Entropy of Classes of Analytic Functions

4. General Sets of Analytic Functions

5. Relations between Entropy and Widths

6. Notes

PROBLEMS

Chapter 11. Representation of Functions of Several Variables by Functions of One Variable

1. The Theorem of Kolmogorov

2. The Fundamental Lemma

3. The Completion of the Proof

4. Functions Not Representable by Superpositions

5. Notes

PROBLEMS

Bibliography

Index

Back Cover