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Principles of Fourier Analysis
Convergence and Fourier’s Conjecture: The Proofs
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Publication date:
May 18 2001
Publisher:
CRC Press
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Book Chapter
Publication date:
May 18 2001
Pages
: 191-208
DOI:
10.1201/9781420036909-17
SO-VID:
255abd91-6813-4489-be95-d8f84a829e41
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Book chapters
pp. 17
The Starting Point
pp. 21
Basic Terminology, Notation, and Conventions
pp. 29
Basic Analysis I: Continuity and Smoothness
pp. 51
Basic Analysis II: Integration and Infinite Series
pp. 63
Symmetry and Periodicity
pp. 71
Elementary Complex Analysis
pp. 87
Functions of Several Variables
pp. 109
Heuristic Derivation of the Fourier Series Formulas
pp. 115
The Trigonometric Fourier Series
pp. 135
Fourier Series over Finite Intervals (Sine and Cosine Series)
pp. 143
Inner Products, Norms, and Orthogonality
pp. 157
The Complex Exponential Fourier Series
pp. 167
Convergence and Fourier’s Conjecture
pp. 191
Convergence and Fourier’s Conjecture: The Proofs
pp. 209
Derivatives and Integrals of Fourier Series
pp. 227
Applications
pp. 259
Heuristic Derivation of the Classical Fourier Transform
pp. 265
Integrals on Infinite Intervals
pp. 287
The Fourier Integral Transforms
pp. 305
Classical Fourier Transforms and Classically Transformable Functions
pp. 325
Some Elementary Identities: Translation, Scaling, and Conjugation
pp. 345
Differentiation and Fourier Transforms
pp. 365
Gaussians and Other Very Rapidly Decreasing Functions
pp. 385
Convolution and Transforms of Products
pp. 409
Correlation, Square-Integrable Functions, and the Fundamental Identity of Fourier Analysis
pp. 429
Identity Sequences
pp. 449
Generalizing the Classical Theory: A Naive Approach
pp. 477
Fourier Analysis in the Analysis of Systems
pp. 493
Gaussians as Test Functions, and Proofs of Some Important Theorems
pp. 513
A Starting Point for the Generalized Theory
pp. 517
Gaussian Test Functions
pp. 539
Generalized Functions
pp. 569
Sequences and Series of Generalized Functions
pp. 589
Basic Transforms of Generalized Fourier Analysis
pp. 631
Generalized Products, Convolutions, and Definite Integrals
pp. 651
Periodic Functions and Regular Arrays
pp. 675
General Solutions to Simple Equations and the Pole Functions
pp. 711
Periodic, Regular Arrays
pp. 723
Sampling and the Discrete Fourier Transform
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