Xii CONTENTS 3 FOURIER ANALYSIS IN U SPACES 169 3.1 Motivation and Heuristics 169 3.2 The M. Riesz-Thorin Interpolation Theorem 169 3.2.0.1 Generalized Young's Inequality 174 3.2.0.2 The Hausdorff-Young Inequality 174 3.2.1 Stein's Complex Interpolation Theorem 175 3.3 The Conjugate Function or Discrete Hilbert Transform 176 3.3.1 LP Theory of the Conjugate Function 177 3.3.2 L1 Theory of the Conjugate Function 179 3.3.2.1 Identification as a Singular Integral 183 3.4 The Hilbert Transform on R 184 3.4.1 L2 Theory of the Hilbert Transform 185 3.4.2 U Theory of the Hilbert Transform, 1 p oo 186 3 A.2.1 Applications to Convergence of Fourier Integrals 187 3.4.3 L1 Theory of the Hilbert Transform and Extensions 188 3.4.3.1 Kolmogorov's Inequality for the Hilbert Transform 192 3.4.4 Application to Singular Integrals with Odd Kernels 194 3.5 Hardy-Littlewood Maximal Function 197 3.5.1 Application to the Lebesgue Differentiation Theorem 200 3.5.2 Application to Radial Convolution Operators 202 3.5.3 Maximal Inequalities for Spherical Averages 203 3.6 The Marcinkiewicz Interpolation Theorem 206 3.7 Calderon-Zygmund Decomposition 209 3.8 A Class of Singular Integrals 210 3.9 Properties of Harmonic Functions 212 3.9.1 General Properties 212 3.9.2 Representation Theorems in the Disk 214 3.9.3 Representation Theorems in the Upper Half-Plane 216 3.9.4 Herglotz/Bochner Theorems and Positive Definite Functions 219 4 POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES 222 4.1 Motivation and Heuristics 222 4.2 The Poisson Summation Formula in E1 223 4.2.1 Periodization of a Function 223 4.2.2 Statement and Proof 225 4.2.3 Shannon Sampling 228 4.3 Multiple Fourier Series 230 4.3.1 Basic Ll Theory 231 4.3.1.1 Pointwise Convergence for Smooth Functions 233 4.3.1.2 Representation of Spherical Partial Sums 233
Previous Page Next Page