Introduction to harmonic analysis

2022/2023
Programme:
Mathematics, Second Cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M1
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

Fourier series, summation methods, Riesz-Thorin interpolation theorem,
Harmonic functions, Poisson integrals, Hardy spaces, harmonic conjugate, Hilbert transform,
Schwartz class, Fourier transform, distributions and tempered distributions,
weak Lp spaces and the Marcinkiewicz interpolation theorem, the Paley-Wiener theorem and the uncertainty principle,
Hardy-Littlewood maximal function,
Calderón-Zygmund singular integral operators,
linear partial differential operators with constant coefficients, fundamental solution, Sobolev spaces.

Readings

L. Grafakos: Classical Fourier Analysis, Second Edition, Graduate Texts in Mathematics 249, Springer, 2008.
E. M. Stein, G. L. Weiss: Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971.
A. Torchinsky: Real-Variable Methods in Harmonic Analysis, Academic Press, 1986.
Y. Katznelson: An introduction to harmonic analysis, Dover, New York,1976.
L. Hörmander: The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Berlin Heidelberg New York 1990.

Objectives and competences

Acquiring knowledge of fundamental notions and tools of euclidean harmonic analysis, placing them into the context of partial differential equations.

Intended learning outcomes

Knowledge and understanding: Mastering basic concepts of euclidean harmonic analysis.
Application: PDE, mathematical physics, natural sciences, medicine.
Reflection: The course subject is one of the cornerstones of modern mathematical analysis.
Transferable skills: Recognition of problems in the realm of harmonic analysis, formulation and solving problems with methods of classical Fourier analysis.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

Homework assignments
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

DRAGIČEVIĆ, Oliver, VOLBERG, Alexander. Linear dimension-free estimates in the embedding theorem for Schrödinger operators. Journal of the London Mathematical Society, ISSN 0024-6107, 2012, vol. 85, p. 1, str. 191-222. [COBISS-SI-ID 16214873]
DRAGIČEVIĆ, Oliver, VOLBERG, Alexander. Bilinear embedding for real elliptic differential operators in divergence form with potentials. Journal of functional analysis, ISSN 0022-1236, 2011, vol. 261, iss. 10, str. 2816-2828. [COBISS-SI-ID 16051545]
DRAGIČEVIĆ, Oliver. Weighted estimates for powers of the Ahlfors-Beurling operator. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2011, vol. 139, no. 6, str. 2113-2120. [COBISS-SI-ID 15876697]