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 Kaczmarz—Stein.- 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.
