Date of Award

Spring 1-1-2011

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Applied Mathematics

First Advisor

Gregory Beylkin

Second Advisor

Lucas Monzon

Third Advisor

James Curry

Abstract

We introduce a new computationally efficient algorithm for constructing near optimal rational approximations of large data sets. In contrast to wavelet-type approximations often used for the same purpose, these new approximations are effectively shift invariant. On the other hand, when dealing with large data sets the complexity of our current non-linear algorithms for computing near optimal rational approximations prevents their direct use. By using an intermediate representation of the data via B-splines, followed by a rational approximation of the B-splines themselves, we obtain a suboptimal rational approximation of data segments. Then, using reduction and merging algorithms for data segments, we arrive at an efficient procedure for computing near optimal rational approximations for large data sets. A motivating example is the compression of audio signals and we provide several examples of compressed representations produced by the algorithm.

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