An introduction to frames and wavelet frames
Time and place
12:30 PM on Tuesday, March 6th, 2012;
Azita Mayeli (Queensborough Community College)
Abstract
In the study of signal and image analysis, one wishes to cut up data into dif- ferent frequency components, and then study each component separately. In pure mathematics, this discussion can be translated as following: For a given vector in a Hilbert space, how can one expand the vector into linear combination of some \simple building blocks", and hereby get information about the vector through the knowledge of the coeficients of the expansion? Of course, an orthonormal basis provides an answer to this question. However, the conditions on these bases are limiting and the linear independency and the orthogonality are required. Therefore we look for more flexible tools over orthonormal bases. A frame for a Hilbert space is a set of non-independent elements which, nevertheless, can be used to write every vector in the space as a completely explicit linear combination of frame elements. In this talk, I shall briefly introduce frames for general Hilbert spaces followed by some simple as well as less simple examples, and I will also mention some of signicant advantages of working with frames. Further, I will introduce an important class of frames, namely, wavelet frames in L2(R). These frames are feasible tools in practice due to their time-frequency localization property. I will conclude my talk with the generalization of these concepts in some complicated setting, namely, when R is replaced by the three-dimensional Heisenberg group. If time permits, I will also mention some of the applications of the results in the study of Besov spaces which are useful tools in the study of ordinary and partial differential equations.
