Frames and Bases in Hilbert Spaces
3-4 hours of lectures per week.
Lecturer
Henrik Stetkær
Content
An important tool in the study of a Hilbert space $\mathcal{H}$ is the concept of a basis. Given a basis every element of $\mathcal{H}$ can be written in exactly one way an infinite linear combination of the elements of the basis. The conditions on a basis are unfortunately quite restrictive - linear dependence is not allowed, and we often even require that the elements of the basis are orthogonal to one another. This makes it difficult, sometimes even impossible, to find bases that satisfy prescribed extra conditions. That is the reason why it is desirable to have a more flexible tool.
A frame is such a tool. A frame of $\mathcal{H}$ is a subset $\{ f_n\} _{n \in \Z }$ of $\mathcal{H}$ such that every element in $\mathcal{H}$ can be written as an infinite linear combination of the $f_n$, $n \in \Z $, but where the system $\{ f_n\} _{n \in \Z }$ need not be linearly independent.
In the course we will study the modern theory of bases and frames in Hilbert spaces. The emphasis will be on the mathematical aspects from the point of view of functional analysis. It should be mentioned that there are serious applications of the theory to signal analysis and image processing. We shall not discuss these applications.
Much of the theory is new, actually dating from the last 15 years, even though it has its roots in the classical theory of non-harmonic Fourier series: Here $\mathcal{H} = L^2(-\pi ,\pi )$ and $f_n(x) = e^{i\lambda _nx}$, where $\{ \lambda _n\} $ is a sequence of real or complex numbers; the harmonic case being the special case of $\lambda _n = n$. Newer examples of frames are the wavelet bases that by now are of great practical importance.
Prerequisites
The course Analysis 2
Literature
Ole Christensen,
An Introduction to Frames and Riesz Bases, Birkhauser, Boston, Basel, Berlin 2003.
Evaluation
Students who do not intend to take a degree in Mathematics or Statistics from the University of Aarhus, but wish to earn credits for a 2.dels course from the Department of Mathematics, should indicate at the beginning of the course that they wish to be examined.
The form of examination for these students will be active participation together with oral or written contributions.
Credits
10 ECTS
Quarter
Spring 2004