Interpolation, Least Squares Approximations, Numerical Integration and Roots of Orthogonal Polynomials
By the time Euler was at the peak of his mathematical powers in the 18th century, the process of constructing polynomials that passed through a given set of points was well-understood. We’ll start with a look at this theory, and see why, when the data is experimental (i.e.), interpolation does not lead to viable models.
Gauss, and later Legendre, independently developed least squares as a tool to "see through" errors in datasets. When applied in the context of more general inner product spaces, however, least squares can serve as a lens to view the development of vast expanses of modern analysis. In this talk, we'll follow a small strand of this history and see how to develop useful tools for numerical integration that turn out to form part of the underlying justification for such modern techniques as the fast Fourier transform.
A Ramble Around the History of Pi
As of fall 2010, more than 5 trillion digits of pi have been calculated. While there is almost no conceivable practical use for this, the intellectual curiosity driving human interest in this modest constant has driven innovation in theoretical and computational mathematics for at least 5000 years.
This talk is meant to be an entertaining perusal of some of the high (and maybe even low) points of that history.
(More to come, or contact me.)
MAA Online (www.maa.org) is the principal website of the Mathematical Association of America. In addition to information about the MAA and its programs, this is the place to access Math in the News, regular columns and upcoming events sponsored by the MAA.
Some additional MAA links that may not be obvious:
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