Classical and Modern Fourier AnalysisAn ideal refresher or introduction to contemporary Fourier Analysis, this book starts from the beginning and assumes no specific background. Readers gain a solid foundation in basic concepts and rigorous mathematics through detailed, user-friendly explanations and worked-out examples, acquire deeper understanding by working through a variety of exercises, and broaden their applied perspective by reading about recent developments and advances in the subject. Features over 550 exercises with hints (ranging from simple calculations to challenging problems), illustrations, and a detailed proof of the Carleson-Hunt theorem on almost everywhere convergence of Fourier series and integrals ofL p functions --one of the most difficult and celebrated theorems in Fourier Analysis. A complete Appendix contains a variety of miscellaneous formulae.L p Spaces and Interpolation. Maximal Functions, Fourier transforms, and Distributions. Fourier Analysis on the Torus. Singular Integrals of Convolution Type. Littlewood-Paley Theory and Multipliers. Smoothness and Function Spaces.BMO and Carleson Measures. Singular Integrals of Nonconvolution Type. Weighted Inequalities. Boundedness and Convergence of Fourier Integrals. For mathematicians interested in harmonic analysis. |
Contents
Appendix A Gamma and Beta Functions | 1 |
Appendix B Bessel Functions | 7 |
Rademacher Functions | 15 |
Copyright | |
12 other sections not shown
Common terms and phrases
approximate identity Avg ƒ B₁ ball compact support conclude continuous function converges Corollary cube Q defined Definition denote equal estimate Example Exercise exists a constant finite fixed follows Fourier coefficients Fourier series Fourier transform function f given by convolution Hardy spaces Hardy-Littlewood maximal Hilbert transform Hint Hölder's inequality homogeneous implies integrable function interpolation kernel L¹(R L¹(T L²(R Lebesgue Lemma Let f Let ƒ linear operator Littlewood-Paley Lº(R LP boundedness LP norm LP spaces LP(R LP(X maps LP maximal function maximal operator mean value zero measurable functions measure space mezn multiindices multiple observe obtain proof of Theorem properties Proposition Prove required conclusion satisfies Schwartz function sequence simple functions singular integrals smooth function subset Suppose supremum tempered distribution uniformly weak type