Harmonic Analysis (AN8)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN8
Semester	2
Course Title	Harmonic Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The plan of the course is the achievement by the graduate student of the theoretical background in the theory of differentiation bases on Euclidean spaces, their connection with maximal operators and the study of fractal sets. Also the study of general theory of maximal operators will be considered.
General Competences	The objective of the course is the graduate student’s ability achievement in analysis and synthesis of the basic background in the area of Harmonic Analysis connected with differentiation bases on Euclidian spaces and geometric measure theory on the plane. Also connections with maximal operators will be given.

Syllabus

Busemman – Feller differentiation bases on Euclidian spaces and associated maximal operators, covering lemmas and applications to the behavior of maximal operators, connection of differentiation bases of certain spaces with respective properties of the related maximal operators, the basis B2 of intervals on Euclidian spaces and its differentiation properties, covering properties of the basis B2, the basis B3 of rectangles on Euclidian spaces: The Perron tree, Fefferman’s lemma, Besicovitch and Kakeya sets, the Nikodym set, subbases of B3 and differentiation properties, Hausdorff measure and dimension on the plane, fractal sets and densities, regular and irregular sets, tangency and projection properties, the theory of maximal operators from the general point of view.

Teaching and Learning Methods - Evaluation

Delivery	Teaching with talks on the blackboard.
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 110 Exercises solutions 38.5 Course total 187.5
Student Performance Evaluation	Combination of writing and oral examination at the end of the semester.

Attached Bibliography

• M. De Guzman, Real Variable Methods in Fourier Analysis, North Holland - Mathematical Studies.