Author | Edwards, R. E. author |
---|---|
Title | Fourier Series [electronic resource] : A Modern Introduction Volume 1 / by R. E. Edwards |
Imprint | New York, NY : Springer New York, 1979 |
Edition | Second Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4612-6208-4 |
Descript | XII, 228 p. online resource |
Contentss -- 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 Fejรฉr Kernels. Cesร ro Summability -- 5.1 The Dirichlet and Fejรฉr Kernels -- 5.2 The Localization Principle -- 5.3 Remarks concerning Summability -- Exercises -- 6 Cesร ro 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. Parsevalโs 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 Bochnerโs Theorem -- 9.1 Mise-en-Scรจne -- 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 Jordanโs Test -- 10.2 Remarks on Other Criteria for Convergence; Diniโs 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 -- Appendix A Metric Spaces and Baireโs Theorem -- A.1 Some Definitions -- A.2 Baireโs Category Theorem -- A.3 Corollary -- A.4 Lower Semicontinuous Functions -- A.5 A Lemma -- Appendix B Concerning Topological Linear Spaces -- B.1 Preliminary Definitions -- B.2 Uniform Boundedness Principles -- B.3 Open Mapping and Closed Graph Theorems -- B.4 The Weak Compacity Principle -- B.5 The Hahn-Banach Theorem -- Appendix D A WEAK FORM OF RUNGEโS THEOREM -- Research Publications -- Symbols