(Created 2010-07-25.)

FUNCTIONAL ANALYSIS AND HARMONIC ANALYSIS | FMA260 |

**Aim**

Both functional analysis and harmonic analysis are fundamental tools in many mathematical applications ( e.g., in field theory, solid mechanics, control theory, signal processing) and in mathematical statistics and numerical analysis. The aim of the course is to convey knowledge about basic concepts and methods, to give ability to follow discussions where these are used and to give ability independently to solve mathematical problems which appear in the applications. One particular goal of the course is to develop a power of abstraction which makes it easier to see analogies between problems from apparently different fields.

*Knowledge and understanding*

For a passing grade the student must

be familiar with the meaning of the concepts of completeness and compactness, and be aware that the choice of norm has an impact on these properties of a space. In particular, the student should have a good understanding of the difference between the infinite and the finite dimensional case.

be familiar with Banach's fixed point theorem and be able to give examples of its applications.

be familiar with how the properties of a function are reflected in its Fourier transform.

be familiar with the most common classes of linear and bounded operators, and have an understanding of how spectral theory provides information on the properties of a linear operator.

be able to explain the basic theory in an oral examination.

*Skills and abilities*

For a passing grade the student must

in writing and orally, with proper terminology and clear logic be able to explain the solution to a mathematical problem within the course.

be able to translate concrete mathematical problem settings into the abstract concepts of the course.

**Contents***Functional analysis:* norms and approximation, the contraction theorem, compactness, function spaces, Hilbert spaces, orthogonality and orthogonal systems, linear operators, spectral theory.

*Harmonic analysis:* the Fourier transform, Fourier series and discrete Fourier transforms. Uncertainty relations, the sampling theorem, Fourier transforms and analytic functions, the Hilbert transform, wavelet transforms.

**Literature**

Renardy and Rogers. An Introduction to Partial Differential Equations, 2nd ed. Springer. ISBN 0-387-00444-0.

Some further material and completions.