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Index
Title Page
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AUTHOR’S PREFACE
TRANSLATOR’S PREFACE
Table of Contents
1 - TRIGONOMETRIC FOURIER SERIES
I. Periodic Functions
2. Harmonics
3. Trigonometric Polynomials and Series
4. A More Precise Terminology. Integrability. Series of Functions
5. The Basic Trigonometric System. The Orthogonality of Sines and Cosines
6. Fourier Series for Functions of Period 2π
7. Fourier Series for Functions Defined on an Interval of Length 2π
8. Right-Hand and Left-Hand Limits. Jump Discontinuities
9. Smooth and Piecewise Smooth Functions
10. A Criterion for the Convergence of Fourier Series
II. Even and Odd Functions
12. Cosine and Sine Series
13. Examples of Expansions in Fourier Series
14. The Complex Form of a Fourier Series
15. Functions of Period 2l
PROBLEMS
2 - ORTHOGONAL SYSTEMS
I. Definitions
2. Fourier Series with Respect to an Orthogonal System
3. Some Simple Orthogonal Systems
4. Square Integrable Functions. The Schwarz Inequality
5. The Mean Square Error and its Minimum
6. Bessel’s Inequality
7. Complete Systems. Convergence in the Mean
8. Important Properties of Complete Systems
9. A Criterion for the Completeness of a System
*10. The Vector Analogy
PROBLEMS
3 - CONVERGENCE OF TRIGONOMETRIC FOURIER SERIES
1. A Consequence of Bessel’s Inequality
2. The Limit as n → ∞ of the Trigonometric Integrals
3. Formula for the Sum of Cosines. Auxiliary Integrals
4. The Integral Formula for the Partial Sum of a Fourier Series
5. Right-Hand and Left-Hand Derivatives
6. A Sufficient Condition for Convergence of a Fourier Series at a Continuity Point
7. A Sufficient Condition for Convergence of a Fourier Series at a Point of Discontinuity
8. Generalization of the Sufficient Conditions Proved in Secs. 6 and 7
9. Convergence of the Fourier Series of a Piecewise Smooth Function (Continuous or Discontinuous)
10. Absolute and Uniform Convergence of the Fourier Series of a Continuous, Piecewise Smooth Function of Period 2π
II. Uniform Convergence of the Fourier Series of a Continuous Function of Period 2π with an Absolutely Integrable Derivative
12. Generalization of the Results of Sec. II
13. The Localization Principle
14. Examples of Fourier Series Expansions of Unbounded Functions
15. A Remark Concerning Functions of Period 2l
PROBLEMS
4 - TRIGONOMETRIC SERIES WITH DECREASING COEFFICIENTS
I. Abel’s Lemma
2. Formula for the Sum of Sines. Auxiliary Inequalities
3. Convergence of Trigonometric Series with Monotonically Decreasing Coefficients
*4. Some Consequences of the Theorems of Sec. 3
5. Applications of Functions of a Complex Variable to the Evaluation of Certain Trigonometric Series
6. A Stronger Form of the Results of Sec. 5
PROBLEMS
5 - OPERATIONS ON FOURIER SERIES
I. Approximation of Functions by Trigonometric Polynomials
2. Completeness of the Trigonometric System
3. Parseval’s Theorem. The Most Important Consequences of the Completeness of the Trigonometric System
*4. Approximation of Functions by Polynomials
5. Addition and Subtraction of Fourier Series. Multiplication of a Fourier Series by a Number
*6. Products of Fourier Series
7. Integration of Fourier Series
8. Differentiation of Fourier Series. The Case of a Continuous Function of Period 2π
*9. Differentiation of Fourier Series. The Case of a Function Defined on the Interval [— π, π]
∗10. Differentiation of Fourier Series. The Case of a Function Defined on the Interval [0,π]
II. Improving the Convergence of Fourier Series
12. A List of Trigonometric Expansions
13. Approximate Calculation of Fourier Coefficients
PROBLEMS
6 - SUMMATION OF TRIGONOMETRIC FOURIER SERIES
I. Statement of the Problem
2. The Method of Arithmetic Means
3. The Integral Formula for the Arithmetic Mean of the Partial Sums of a Fourier Series
4. Summation of Fourier Series by the Method of Arithmetic Means
5. Abel’s Method of Summation
6. Poisson’s Kernel
7. Application of Abel’s Method to the Summation of Fourier Series
PROBLEMS
7 - DOUBLE FOURIER SERIES. THE FOURIER INTEGRAL
I. Orthogonal Systems in Two Variables
2. The Basic Trigonometric System in Two Variables. Double Trigonometric Fourier Series
3. The Integral Formula for the Partial Sums of a Double Trigonometric Fourier Series. A Convergence Criterion
4. Double Fourier Series for a Function with Different Periods in x and y
5. The Fourier Integral as a Limiting Case of the Fourier Series
6. Improper Integrals Depending on a Parameter
7. Two Lemmas
8. Proof of the Fourier Integral Theorem
9. Different Forms of the Fourier Integral Theorem
* 10. The Fourier Transform
*II. The Spectral Function
PROBLEMS
8 - BESSEL FUNCTIONS AND FOLIRIER-BESSEL SERIES
I. Bessel’s Equation
2. Bessel Functions of the First Kind of Nonnegative Order
3. The Gamma Function
4. Bessel Functions of the First Kind of Negative Order
5. The General Solution of Bessel’s Equation
6. Bessel Functions of the Second Kind
7. Relations between Bessel Functions of Different Orders
8. Bessel Functions of the First Kind of Half-Integral Order
9. Asymptotic Formulas for the Bessel Functions
10. Zeros of Bessel Functions and Related Functions
II. Parametric Form of Bessel’s Equation
12. Orthogonality of the Functions Jp(λx)
13. Evaluation of the Integral
*14. Bounds for the Integral
15. Definition of Fourier-Bessel Series
16. Criteria for the Convergence of Fourier-Bessel Series
*17. Bessel’s Inequality and Its Consequences
*18. The Order of Magnitude of the Coefficients which Guarantees Uniform Convergence of a Fourier-Bessel Series
*19. The Order of Magnitude of the Fourier-Bessel Coefficients of a Twice Differentiable Function
*20. The Order of Magnitude of the Fourier-Bessel Coefficients of a Function Which is Differentiable Several Times
*21. Term by Term Differentiation of Fourier-Bessel Series
22. Fourier-Bessel Series of the Second Type
*23. Extension of the Results of Sees. 17 – 21 to Fourier-Bessel Series of the Second Type
24. Fourier-Bessel Expansions of Functions Defined on the Interval [0, l]
PROBLEMS
9 - THE EIGENFUNCTION METHOD AND ITS APPLICATIONS TO MATHEMATICAL PHYSICS
Part I. THEORY
Part II. APPLICATIONS
PROBLEMS
ANSWERS TO PROBLEMS
BIBLIOGRAPHY
INDEX
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