SUBJECT

Fourier integral

lecture

master

Faculty of Science

Mathematics MSc

2+1

Semesters 1-4

Autumn/Spring semester

Fourier transform of functions in L_1. Riemann Lemma. Convolution in L_1. Inversion formula. Wiener’s theorem on the closure of translates of L_1 functions. Applications to Wiener’s general Tauberian theorem and special Tauberian theorems.

Fourier transform of complex measures. Characterizing continuous measures by its Fourier transform. Construction of singular measures.

Fourier transform of functions in L_2. Parseval formula. Convolution in L_2. Inversion formula. Application to non-parametric density estimation in statistics.

Young-Hausdorff inequality. Extension to L_p. Riesz-Thorin theorem. Marczinkiewicz interpolation theorem.

Application to uniform distribution. Weyl criterion, its quantitative form by Erdős-Turán. Lower estimation of the discrepancy for disks.

Characterization of the Fourier transform of functions with bounded support. Paley-Wiener theorem.

Phragmén-Lindelöf type theorems.

• E.C. Titchmarsh: Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford, 1937.
• A. Zygmund: Trigonometric Series, University Press, Cambridge, 1968, 2 volumes
• R. Paley and N. Wiener: Fourier Transforms in the Complex Domain, American Mathematical Society, New York, 1934.
• J. Beck and W.L. Chen: Irregularities of Distribution, University Press, Cambridge, 1987.