The present book is a collection of variations on a theme which can be summed up as follows: It is impossible for a non-zero function and its Fourier transform to be simultaneously very small. In other words, the approximate equalities x :::::: y and x :::::: fj cannot hold, at the same time and with a high degree of accuracy, unless the functions x and yare identical. Any information gained about x (in the form of a good approximation y) has to be paid for by a corresponding loss of control on x, and vice versa. Such is, roughly speaking, the import of the Uncertainty Principle (or UP for short) referred to in the title ofthis book. That principle has an unmistakable kinship with its namesake in physics - Heisenberg's famous Uncertainty Principle - and may indeed be regarded as providing one of mathematical interpretations for the latter. But we mention these links with Quantum Mechanics and other connections with physics and engineering only for their inspirational value, and hasten to reassure the reader that at no point in this book will he be led beyond the world of purely mathematical facts. Actually, the portion of this world charted in our book is sufficiently vast, even though we confine ourselves to trigonometric Fourier series and integrals (so that "The U. P. in Fourier Analysis" might be a slightly more appropriate title than the one we chose).

Inhalt

One. The Uncertainty Principle Without Complex Variables.- 1. Functions and Charges with Semibounded Spectra.- The Uncertainty Principle for Charges with Semibounded Spectra. The F. and M. Riesz Theorem.- Sharpness of the F. and M. Riesz Theorem. The Rudin-Carleson Theorem.- A Quantitative Refinement of the F. and M. Riesz Theorem.- The De Leeuw-Katznelson Theorem.- One More Quantitative Refinement of the F. and M. Riesz Theorem.- Some Multidimensional Generalizations and Analogs of the F. and M. Riesz Theorem. The Approach of Aleksandrov and Shapiro.- Perturbed Plus-Charges.- Notes.- 2. Some Topics Related to the Harmonic Analysis of Charges.- r-Charges.- Cantor Measures.- The Riesz Products.- The Ivashev-Musatov Theorem.- The Lemma on the Optimal Rime. The Role of Regularity Properties of the Majorant in the Ivashev-Musatov Theorem.- Deep Zeros and Sparse Spectra. A "Real" Proof of the Mandelbrojt Theorem.- Notes.- 3. Hilbert Space Methods.- Mutually Annihilating Pairs of Subspaces.- Annihilation of Supports and Spectra of Finite Volume. The Amrein-Berthier and Slepian-Pollak Theorems.- Functions with Sparse Spectra. The Mikheyev Theorem.- Supports Strongly Annihilating any Bounded Spectrum. The Logvinenko-Sereda Theorem.- Notes.- Two. Complex Methods.- 1. The Uncertainty Principle from the Complex Point of View. First Examples.- Introductory Remarks.- The Limit Speed of Decay of the Fourier Transform of a Rapidly Decreasing Function. The Dzhrbashyan Theorem.- A Complex Proof of the Mandelbrojt Theorem.- The UP for Plus-Functions from a New Point of View.- Hardy Classes in the Upper Half-Plane.- The Lindel Theorem. Entire Functions of the Cartwright Class.- Notes.- 2. The Logarithmic Integral Diverges.- Divergence of a Logarithmic Integral and Some Forms of the UP for Functions and Charges with Rapidly Decreasing Amplitudes. The Beurling Theorem.- Charges with a Spectral Gap.- One-Sided Decrease of Amplitudes. The Volberg Theorems.- One-Sided Decrease and One-sided Growth of Amplitudes. The Borichev Approach.- Notes.- 3. The Logarithmic Integral Converges.- Outer Functions. Sharpness of Some Forms of the UP.- The Khinchin-Ostrowski Theorem.- Unilateral Decrease of Amplitudes and the Size of Support. The Hruscev Theorem.- The Hruscev Theorem: the End of the Proof.- The First Beurling-Malliavin Theorem.- Functions with Finite Dirichlet Integral in a Half-Plane and Their Boundary Values.- Logarithmic Potential and Distributions of Finite Energy.- The Spectral Gap Problem Revisited.- The Sapogov Problem: Characteristic Functions with a Spectral Gap.- Notes.- 4. Missing Frequencies and the Diameter of the Support. The Second Beurling-Malliavin Theorem and the Fabry Theorem.- Statement of the Problem. The Diameter of a Divisor.- Systems of Long and Short Intervals. A Covering Lemma.- Three Definitions of the Integral Density of a Measure.- The Estimate ?3(?) ? R(?).- The Estimate ?1(?) ? R(?).- The Fabry Theorem: Technical Preliminaries.- The Fabry and Carlson-Landau Theorems. The First Proof.- Sharpness of the Carlson-Landau Theorem.- The Fabry Theorem. The Second Proof.- . The Fabry Theorem. The Third Proof.- . The Amrein-Berthier Theorem Revisited. The Nazarov Approach.- . Concluding Remarks. The Fabry Phenomenon.- Notes.- 5. Local and Non-local Convolution Operators.- Symbols of Local Operators.- Semirational Symbols and Complete Antilocality.- The Cauchy Problem for the Laplace Equation.- Complete Antilocality of One-Dimensional M. Riesz Potentials.- Notes.- References.- Author Index.