Introduction to Fourier Analysis on Euclidean Spaces (PMS-32), Volume 32 (Princeton Mathematical Series)

The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations, and to motivate the study of harmonic analysis on more general spaces having an analogous structure, e.g., symmetric spaces. Preface Contents Chapter I. The Fourier Transform 1. The basic L1.heory of the Fourier transform 2. The L2 theory and the Plancherel theorem 3. The class of tempered distributions 4. Further results Chapter II. Boundary Values of Harmonic Functions 1. Basic properties of harmonic functions 2. The characterization of Poisson integrals 3. The Hardy-Littlewood maximal function and nontangential convergence of harmonic functions 4. Subharmonic functions and majorization by harmonic functions 5. Further results Chapter III. The Theory of Hp Spaces on Tubes 1. Introductory remarks 2. The H2 theory 3. Tubes over cones 4. The Paley-Wiener theorem 5. The Hp theory 6. Further results Chapter IV. Symmetry Properties of the Fourier Transform 1. Decomposition of L2(E2) into subspaces invariant under the Fourier transform 2. Spherical harmonics 3. The action of the Fourier transform on the spaces 4. Some applications 5. Further results Chapter V. Interpolation of Operators 1. The M. Riesz convexity theorem and interpolation of operators defined on Lp spaces 2. The Marcinkiewicz interpolation theorem 3. L(p, q) spaces 4. Interpolation of analytic families of operators 5. Further results Chapter VI. Singular Integrals and Systems of Conjugate Harmonic Functions 1. The Hilbert transform 2. Singular integral operators with odd kernels 3. Singular integral operators with even kernels 4. Hp spaces of conjugate harmonic functions 5. Further results Chapter VII. Multiple Fourier Series 1. Elementary properties 2. The Poisson summation formula 3. Multiplier transformations 4. Summability below the critical index (negative results) 5. Summability below the critical index 6. Further results Bibliography Index