Numerical Analysis by Scott L. Ridgway

Author:Scott, L. Ridgway [Scott, L. Ridgway]
Language: eng
Format: epub
ISBN: 978-1-4008-3896-7
Publisher: Princeton University Press
Published: 2011-06-13T16:00:00+00:00

Chapter Ten

Polynomial Interpolation

The web site http://www.blackphoto.com/glossary/i.asp describes interpolation as “a technique used by digital cameras, scanners and printers to increase the size of an image in pixels by averaging the colour and brightness values of surrounding pixels.”

The approximation of general functions by simple classes of functions has many applications as well as theoretical implications. The uniform approximation of a general continuous function on an interval by polynomials (a theorem of Weierstrass1 is a fundamental result that casts light on the nature of both polynomials and continuous functions. In the era of modern computers, approximation via interpolation has emerged as a general paradigm for computing elementary functions as part of typical system software on current computers [107]. Probably one of the earliest applications of interpolation was simply to link scattered data to provide some sense of what a continuum representation might look like. The phrase “connecting the dots” has become a common metaphor for problem solving, but this is precisely what polynomial interpolation does.

One feature of the subject is that it introduces infinite-dimensional vector spaces in a natural way. Dealing with such spaces in a complete (pun intended) way is beyond the scope of this book, but we hope that the ideas stimulate interest in further study of functional analysis. We start by considering approximation by polynomials in one dimension. Some of the technology we develop applies to other classes of approximating spaces, as well as multivariate approximation.

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