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Fourier analysis

Schedule

Monday, 14 October 2019 to Friday, 29 May 2020
Total hours: 40
Hours of lectures: 40

Examination procedure

  • oral exam

Prerequisites

Lebesgue integral, measure theory, fundamental notions of functional analysis(Banach spaces and operators, weak topologies, theorems of Hahn-Banach, Banach-Alaoglu, uniform boundedness principle).

Undergraduate students of 4th and 5th year.

Syllabus

Fourier series. Fundamental properties, convergence problems, pointwise and in Lp norm. Summability methods. Relations between Abel summability and boundary behaviour of harmonic functions on the disc in C. Notion of conjugate harmonic of a harmonic function. 

The Riesz-Thorin interpolation theorem. Inequalities of Young and Hausdorff-Young.

Convergence of Fourier series in Lp norm. 

Fourier transform of a function defined on Rn. The Schwartz space and its invariance under Fourier transform. Trasforms of measures and tempered distributions. Harmonic functions on the upper half-space in Rn+1, Hilbert and Riesz transforms.

Fourier multipliers forseries and transforms. The deLeeuw theorem and the Poisson summation formula. 

The Hardy-Littlewood maximal function. Weak Lp spaces and the Marcinkiewicz interpolation theorem. The Calderón-Zygmund theory of singular integral operators. Lboundedness of the Hilbert and Riesz transforms. The Mihlin-Hörmander theorem for Fourier multipliers.

Series of Rademacher functions andLittlewood-Paley theory, with applications. Norm convergence of multiple Fourier series. Fefferman's theorem on spherical convergence.

 

Educational goals:

Provide fundamental notions and modern methods of Fourier analysis.

Bibliographical references

Notes will be made available during the course.