Fourier series: a modern introduction/ R.E. Edwards

1 Trigonometric Series and Fourier Series.-
1.1 The Genesis of Trigonometric Series and Fourier Series.-
1.2 Pointwise Representation of Functions by Trigonometric Series.-
1.3 New Ideas about Representation.- Exercises.-
2 Group Structure and Fourier Series.-
2.1 Periodic Functions.-
2.2 Translates of Functions. Characters and Exponentials. The Invariant Integral.-
2.3 Fourier Coefficients and Their Elementary Properties.-
2.4 The Uniqueness Theorem and the Density of Trigonometric Polynomials.-
2.5 Remarks on the Dual Problems.- Exercises.-
3 Convolutions of Functions.-
3.1 Definition and First Properties of Convolution.-
3.2 Approximate Identities for Convolution.-
3.3 The Group Algebra Concept.-
3.4 The Dual Concepts.- Exercises.-
4 Homomorphisms of Convolution Algebras.-
4.1 Complex Homomorphisms and Fourier Coefficients.-
4.2 Homomorphisms of the Group Algebra.- Exercises.-
5 The Dirichlet and Fejer Kernels. Cesaro Summability.-
5.1 The Dirichlet and Fejer Kernels.-
5.2 The Localization Principle.-
5.3 Remarks concerning Summability.- Exercises.-
6 Cesaro Summability of Fourier Series and its Consequences.-
6.1 Uniform and Mean Summability.-
6.2 Applications and Corollaries of.1.1 90.-
6.3 More about Pointwise Summability.-
6.4 Pointwise Summability Almost Everywhere.-
6.5 Approximation by Trigonometric Polynomials.-
6.6 General Comments on Summability of Fourier Series.-
6.7 Remarks on the Dual Aspects.- Exercises.-
7 Some Special Series and Their Applications.-
7.1 Some Preliminaries.-
7.2 Pointwise Convergence of the Series (C) and (S).-
7.3 The Series (C) and (S) as Fourier Series.-
7.4 Application to A(Z).-
7.5 Application to Factorization Problems.- Exercises.-
8 Fourier Series in L2.-
8.1 A Minimal Property.-
8.2 Mean Convergence of Fourier Series in L2. Parseval's Formula.-
8.3 The Riesz-Fischer Theorem.-
8.4 Factorization Problems Again.-
8.5 More about Mean Moduli of Continuity.-
8.6 Concerning Subsequences of sNf.-
8.7 A(Z) Once Again.- Exercises.-
9 Positive Definite Functions and Bochner's Theorem.-
9.1 Mise-en-Scene.-
9.2 Toward the Bochner Theorem.-
9.3 An Alternative Proof of the Parseval Formula.-
9.4 Other Versions of the Bochner Theorem.- Exercises.-
10 Pointwise Convergence of Fourier Series.-
10.1 Functions of Bounded Variation and Jordan's Test.-
10.2 Remarks on Other Criteria for Convergence; Dini's Test.-
10.3 The Divergence of Fourier Series.-
10.4 The Order of Magnitude of sNf. Pointwise Convergence Almost Everywhere.-
10.5 More about the Parseval Formula.-
10.6 Functions with Absolutely Convergent Fourier Series.-
Exercises.-