A Panorama of Harmonic Analysis (Carus Mathematical Monographs, Series Number 27)

Annotation Tracing a path from the earliest beginnings of Fourier series through to the latest research A Panorama of Harmonic Analysis discusses Fourier series of one and several variables, the Fourier transform, spherical harmonics, fractional integrals, and singular integrals on Euclidean space. The climax is a consideration of ideas from the point of view of spaces of homogeneous type, which culminates in a discussion of wavelets. This book is intended for graduate students and advanced undergraduates, and mathematicians of whatever background who want a clear and concise overview of the subject of commutative harmonic analysis Title ......Page 3 Contents ......Page 9 Preface ......Page 13 0.1 Pre-Basics ......Page 17 0.2 A Whirlwind Review of Measure Theory ......Page 18 0.3 The Elements of Banach Space Theory ......Page 29 0.4 Hilbert Space ......Page 37 0.5 Two Fundamental Principles of Functional Analysis ......Page 42 1.0 The Pre-History of Fourier Analysis ......Page 47 1.1 The Rudiments of Fourier Series ......Page 58 1.2 Summability of Fourier Series ......Page 65 1.3 A Quick Introduction to Summability Methods ......Page 71 1.4 Key Properties of Summability Kernels ......Page 78 1.5 Pointwise Convergence for Fourier Series ......Page 87 1.6 Norm Convergence of Partial Sums and the Hilbert Transform ......Page 92 2.1 Basic Properties of the Fourier Transform ......Page 111 2.2 Invariance and Symmetry Properties of the Fourier Transform ......Page 114 2.3 Convolution and Fourier Inversion ......Page 119 2.4 The Uncertainty Principle ......Page 133 3.1 Various Methods of Partial Summation ......Page 137 3.2 Examples of Different Types of Summation ......Page 142 3.3 Fourier Multipliers and the Summation of Series ......Page 146 3.4 Applications of the Fourier Multiplier Theorems to Summation of Multiple Trigonometric Series ......Page 156 3.5 The Multiplier Problem for the Ball ......Page 164 4.1 A New Look at Fourier Analysis in the Plane ......Page 187 4.2 Further Results on Spherical Harmonics ......Page 199 5.1 Fractional Integrals and Other Elementary Operators ......Page 215 5.2 Prolegomena to Singular Integral Theory ......Page 219 5.3 An Aside on Integral Operators ......Page 224 5.4 A Look at Hardy Spaces in the Complex Plane ......Page 225 5.5 The Real-Variable Theory of Hardy Spaces ......Page 235 5.6 The Maximal-Function Characterization of Hardy Spaces ......Page 241 5.7 The Atomic Theory of Hardy Spaces ......Page 243 5.8 Ode to BMO ......Page 245 6.1 Spaces of Homogeneous Type ......Page 251 6.2 Integral Operators on a Space of Homogeneous Type ......Page 257 6.3 A New Look at Hardy Spaces ......Page 278 6.4 The T(1) Theorem ......Page 282 7.1 Localization in the Time and Space Variables ......Page 289 7.2 Building a Custom Fourier Analysis ......Page 292 7.3 The Haar Basis ......Page 295 7.4 Some Illustrative Examples ......Page 301 7.5 Construction of a Wavelet Basis ......Page 313 8.1 Fourier Analysis: An Historical Overview ......Page 329 Appendix I, The Existence of Testing Functions and Their Density in Lp ......Page 331 Appendix II, Schwartz Functions and the Fourier Transform ......Page 333 Appendix III, The Interpolation Theorems of Marcinkiewicz and Riesz-Thorin ......Page 334 Appendix IV, Hausdorff Measure and Surface Measure ......Page 336 Appendix VI, The Banach-Alaoglu Theorem ......Page 339 Appendix VIII, The Stone-Weierstrass Theorem ......Page 340 Appendix IX, Landau's O and i Notation ......Page 341 Table of Notation ......Page 343 Bibliography ......Page 355 Index ......Page 363 Steven Krantz......Page 374 Cover......Page 376 Back cover......Page 377 A Panorama of Harmonic Analysis treats the subject of harmonic analysis, from its earliest beginnings to the latest research. Following both an historical and a conceptual genesis, the book discusses Fourier series of one and several variables, the Fourier transform, spherical harmonics, fractional integrals, and singular integrals on Euclidean space. The climax of the book is a consideration of the earlier ideas from the point of view of spaces of homogeneous type. The book culminates with a discussion of wavelets-one of the newest ideas in the subject. A Panorama of Harmonic Analysis is intended for graduate students, advanced undergraduates, mathematicians, and anyone wanting to get a quick overview of the subject of cummutative harmonic analysis. Applications are to mathematical physics, engineering and other parts of hard science. Required background is calculus, set theory, integration theory, and the theory of sequences and series. This Book Treats The Subject Of Harmonic Analysis, From Its Earliest Beginnings To The Latest Research. Following Both An Historical And A Conceptual Genesis, The Book Discusses Fourier Series Of One And Several Variables, The Fourier Transform, Spherical Harmonics, Fractional Integrals, And Singular Integrals On Euclidean Space.

the Climax Of The Book Is A Consideration Of The Earlier Ideas From The Point Of View Of Spaces Of Homogeneous Type. The Book Culminates With A Discussion Of Wavelets -- One Of The Newest Ideas In The Subject.

a Panorama Of Harmonic Analysis Is Intended For Graduate Students, Advanced Undergraduates, Mathematicians, And Anyone Wanting To Get A Quick Overview Of The Subject Of Commutative Harmonic Analysis. Applications Are To Mathematical Physics, Engineering, And Other Parts Of Hard Science. Required Background Is Calculus, Set Theory, Integration Theory, And The Theory Of Sequences And Series.

We take it for granted that the reader of this book has some acquaintance with elementary real analysis.