Number Theory, Fourier Analysis and Geometric Discrepancy

Erscheinungsjahr 2014
ISBN: 978-1-107-61985-2
Verlag: Cambridge University Press

Buch, Englisch, Band 81, 248 Seiten, Format (B × H): 152 mm x 229 mm, Gewicht: 371 g

Reihe: London Mathematical Society Student Texts

The study of geometric discrepancy, which provides a framework for quantifying the quality of a distribution of a finite set of points, has experienced significant growth in recent decades. This book provides a self-contained course in number theory, Fourier analysis and geometric discrepancy theory, and the relations between them, at the advanced undergraduate or beginning graduate level. It starts as a traditional course in elementary number theory, and introduces the reader to subsequent material on uniform distribution of infinite sequences, and discrepancy of finite sequences. Both modern and classical aspects of the theory are discussed, such as Weyl's criterion, Benford's law, the Koksma–Hlawka inequality, lattice point problems, and irregularities of distribution for convex bodies. Fourier analysis also features prominently, for which the theory is developed in parallel, including topics such as convergence of Fourier series, one-sided trigonometric approximation, the Poisson summation formula, exponential sums, decay of Fourier transforms, and Bessel functions.

Autoren/Hrsg.

Travaglini, Giancarlo

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie
• Mathematik | Informatik Mathematik Mathematische Analysis Harmonische Analysis, Fourier-Mathematik

Inhaltsverzeichnis

Part I. Elementary Number Theory: 1. Prelude; 2. Arithmetic functions and integer points; 3. Congruences; 4. Quadratic reciprocity and Fourier series; 5. Sums of squares; Part II. Fourier Analysis and Geometric Discrepancy: 6. Uniform distribution and completeness of the trigonometric system; 7. Discrepancy and trigonometric approximation; 8. Integer points and Poisson summation formula; 9. Integer points and exponential sums; 10. Geometric discrepancy and decay of Fourier transforms; 11. Discrepancy in high dimension and Bessel functions; References; Index.

Über Autor(innen)

Travaglini, Giancarlo
Giancarlo Travaglini is Professor of Mathematics at the University of Milano-Bicocca. He is the author of numerous research papers and several books in the areas of abstract harmonic analysis, Fourier analysis and discrepancy theory. He is a member of the Unione Matematica Italiana and the American Mathematical Society. His current mathematical interests include Fourier analysis, discrepancy theory, radon transforms and didactics of mathematics.