11 Spans of Translates. Closed Ideals. Closed Subalgebras. Banach Algebras. - 11. 1 Closed Invariant Subspaces and Closed Ideals. - 11. 2 The Structure of Closed Ideals and Related Topics. - 11. 3 Closed Subalgebras. - 11. 4 Banach Algebras and Their Applications. - Exercises. - 12 Distributions and Measures. - 12. 1 Concerning C?. - 12. 2 Definition and Examples of Distributions and Measures. - 12. 3 Convergence of Distributions. - 12. 4 Differentiation of Distributions. - 12. 5 Fourier Coefficients and Fourier Series of Distributions. - 12. 6 Convolutions of Distributions. - 12. 7 More about M and Lp. - 12. 8 Hilbert's Distribution and Conjugate Series. - 12. 9 The Theorem of Marcel Riesz. - 12. 10 Mean Convergence of Fourier Series in LP (1 < p < ?). - 12. 11 Pseudomeasures and Their Applications. - 12. 12 Capacities and Beurling's Problem. - 12. 13 The Dual Form of Bochner's Theorem. - Exercises. - 13 Interpolation Theorems. - 13. 1 Measure Spaces. - 13. 2 Operators of Type (p q). - 13. 3 The Three Lines Theorem. - 13. 4 The Riesz-Thorin Theorem. - 13. 5 The Theorem of Hausdorff-Young. - 13. 6 An Inequality of W. H. Young. - 13. 7 Operators of Weak Type. - 13. 8 The Marcinkiewicz Interpolation Theorem. - 13. 9 Application to Conjugate Functions. - 13. 10 Concerning ?*f and s*f. - 13. 11 Theorems of Hardy and Littlewood Marcinkiewicz and Zygmund. - Exercises. - 14 Changing Signs of Fourier Coefficients. - 14. 1 Harmonic Analysis on the Cantor Group. - 14. 2 Rademacher Series Convergent in L2(?). - 14. 3 Applications to Fourier Series. - 14. 4 Comments on the Hausdorff-Young Theorem and Its Dual. - 14. 5 A Look at Some Dual Results and Generalizations. - Exercises. - 15 Lacunary Fourier Series. - 15. 1 Introduction of Sidon Sets. - 15. 2 Construction and Examples of Sidon Sets. - 15. 3 Further Inequalities Involving Sidon Sets. -15. 4 Counterexamples concerning the Parseval Formula and Hausdorff-Young Inequalities. - 15. 5 Sets of Type (p q) and of Type ?(p). - 15. 6 Pointwise Convergence and Related Matters. - 15. 7 Dual Aspects: Helson Sets. - 15. 8 Other Species of Lacunarity. - Exercises. - 16 Multipliers. - 16. 1 Preliminaries. - 16. 2 Operators Commuting with Translations and Convolutions; m-operators. - 16. 3 Representation Theorems for m-operators. - 16. 4 Multipliers of Type (LP Lq). - 16. 15 A Theorem of KaczmarzStein. - 16. 6 Banach Algebras Applied to Multipliers. - 16. 7 Further Developments. - 16. 8 Direct Sum Decompositions and Idempotent Multipliers. - 16. 9 Absolute Multipliers. - 16. 10 Multipliers of Weak Type (p p). - Exercises. - Research Publications. - Corrigenda to 2nd (Revised) Edition of Volume 1. - Symbols. Language: English