appear in Volume 1, a Roman numeral "I" has been prefixed as a reminder to the reader; thus, for example, "I,B.2.1 " refers to Appendix B.2.1 in Volume 1. An understanding of the main topics discussed in this book does not, I hope, hinge upon repeated consultation of the items listed in the bibli ography. Readers with a limited aim should find strictly necessary only an occasional reference to a few of the book listed. The remaining items, and especially the numerous research papers mentioned, are listed as an aid to those readers who wish to pursue the subject beyond the limits reached in…mehr
appear in Volume 1, a Roman numeral "I" has been prefixed as a reminder to the reader; thus, for example, "I,B.2.1 " refers to Appendix B.2.1 in Volume 1. An understanding of the main topics discussed in this book does not, I hope, hinge upon repeated consultation of the items listed in the bibli ography. Readers with a limited aim should find strictly necessary only an occasional reference to a few of the book listed. The remaining items, and especially the numerous research papers mentioned, are listed as an aid to those readers who wish to pursue the subject beyond the limits reached in this book; such readers must be prepared to make the very considerable effort called for in making an acquaintance with current research literature. A few of the research papers listed cover devel opments that came to my notice too late for mention in the main text. For this reason, any attempted summary in the main text of the current standing of a research problem should be supplemented by anexamin ation of the bibliography and by scrutiny of the usual review literature.
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.
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.
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