Harmonic Analysis

Harmonic Analysis is an important tool that plays a vital role in many areas of mathematics as well as applications. It studies functions by decomposing them into components that are special functions. A prime example is decomposing a periodic function into a linear combination of sines and cosines. The subject is vast, and this book covers only the selection of topics that was dealt with in the course given at the Courant Institute in 2000 and 2019. These include standard topics like Fourier series and Fourier transforms of functions, as well as issues of convergence of Abel, Feier, and Poisson sums. At a slightly more advanced level the book studies convolutions with singular integrals, fractional derivatives, Sobolev spaces, embedding theorems, Hardy spaces, and BMO. Applications to elliptic partial differential equations and prediction theory are explored. Some space is devoted to harmonic analysis on compact non-Abelian groups and their representations, including some details about two groups: the permutation group and SO(3). The text contains exercises at the end of most chapters and is suitable for advanced undergraduate students as well as first- or second-year graduate students specializing in the areas of analysis, PDE, probability or applied mathematics. Contents Preface Chapter 1. Fourier Series 1.1. Introduction 1.2. Convergence of Fourier series 1.3. Special case p=2 1.4. Higher dimensions 1.5. Maximal inequality 1.6. Exercises Chapter 2. Fourier Transforms on R^{d} 2.1. Smooth rapidly decaying functions 2.2. Exercises Chapter 3. Singular Integrals 3.1. Interpolation theorems 3.2. Weak type inequality 3.3. Exercises Chapter 4. Riesz Transforms on R^{d} 4.1. Singular integrals on R^{d} 4.2. Riesz kernels 4.3. Exercises Chapter 5. Sobolev Spaces 5.1. Generalized derivatives 5.2. Approximation theorems 5.3. Embedding theorems 5.4. Trace and extension theorems 5.5. Fractional derivatives 5.6. Generalized functions 5.7. Exercises Chapter 6. Hardy Spaces 6.1. Stationary Gaussian processes 6.2. Hardy spaces 6.3. Inner and outer functions 6.4. Connection to prediction theory 6.5. Exercises Chapter 7. Bounded Mean Oscillation 7.1. Functions of bounded mean oscillation 7.2. Duality of BMO and H1 7.3. Exercises Chapter 8. Elliptic PDEs Chapter 9. Banach Algebras and Wiener’s Theorem Chapter 10. Compact Groups 10.1. Haar measure 10.2. Representations of a group 10.3. Representations of a compact group Chapter 11. Representations of Two Compact Groups 11.1. Representations of the permutation group 11.2. Representations of SO(3) References Index "Harmonic Analysis is an important tool that plays a vital role in many areas of mathematics as well as applications. It studies functions by decomposing them into components that are special functions. A prime example is decomposing a periodic function into a linear combination of sines and cosines. The subject is vast, and this book covers only the selection of topics that was dealt with in the course given at the Courant Institute in 2000 and 2019. These include standard topics like Fourier series and Fourier transforms of functions, as well as issues of convergence of Abel, Feier, and Poisson sums. At a slightly more advanced level the book studies convolutions with singular integrals, fractional derivatives, Sobolev spaces, embedding theorems, Hardy spaces, and BMO. Applications to elliptic partial differential equations and prediction theory are explored. Some space is devoted to harmonic analysis on compact non-Abelian groups and their representations, including some details about two groups: the permutation group and SO(3)."--Page 4 of printed paper wrapper "Harmonic Analysis is an important tool that plays a vital role in many areas of mathematics as well as applications. It studies functions by decomposing them into components that are special functions. A prime example is decomposing a periodic function into a linear combination of sines and cosines. The subject is vast, and this book covers only the selection of topics that was dealt with in the course given at the Courant Institute in 2000 and 2019. These include standard topics like Fourier series and Fourier transforms of functions, as well as issues of convergence of Abel, Feier, and Poisson sums. At a slightly more advanced level the book studies convolutions with singular integrals, fractional derivatives, Sobolev spaces, embedding theorems, Hardy spaces, and BMO. Applications to elliptic partial differential equations and prediction theory are explored. Some space is devoted to harmonic analysis on compact non-Abelian groups and their representations, including some details about two groups: the permutation group and SO(3)." Sommario tratto dalla copertina