After successful completion of the course, students are able to remember and understand the basic concept of frames. Based on examples from practical application they will recognize the connection between (Gabor)framed, Fouriertransformation and redundant representations of functions and understand some advantages of redundant frame representations.
Although the first papers dealing with frames date back to the 1950s, the concept of frames and redundancy is relatively unknown in fields outside signal processing, which is interesting, because they provide the theoretical background for many algorithms that we use every day, e.g., when using phones or when we analyse music.
In a nutshell: Frames are (redundant) generating sets with special properties that allow the representation of elements of a (possibly infinite dimensional) vector space as a linear combination of frame elements. In that sense, frames are a generalization of orthogonal bases, but in contrast to bases a representation using frames is not unique anymore. This redundancy, however, allows more flexibility in constructing frames with special properties, e.g., such that the representation becomes more robust with respect to errors and noise, or that the representation using frames allows for an easier manipulation of elements of the vector space.
In the lecture we want to introduce frames in a finite dimensional as well as a infinite dimensional setting, we will introduce some of their properties and the theory behind frames. We also describe some examples where frames play an important role, e.g., Spectrogramms to analyse music and speech or frame-multipliers to manipulate signals (catchphrase: Photoshop for signals).