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41: Approximations and expansions


Introduction

Approximations and expansions primarily concern the approximation of classes of real functions by functions of special types. This includes approximations by linear functions, polynomials (not just the Taylor polynomials), rational functions, and so on; approximations by trigonometric polynomials is separated into Fourier analysis (below). Topics include criteria for goodness of fit, error bounds, stability upon change of approximating family, and preservation of functional characteristics (e.g. differentiability) under approximation. Effective techniques for specific kinds of approximation are also prized. This is also the area covering interpolation and splines.

History

Applications and related fields

For all approximation theory in the complex domain, See 30Exx, 30E05, and 30E10; for all trigonometric approximation and interpolation, see 42Axx, 42A10, and 42A15; for numerical approximation, See 65Dxx [Schematic of subareas and related areas]

This image slightly hand-edited for clarity.

Subfields

There is only one division (41A) but it is subdivided:

Browse all (old) classifications for this area at the AMS.


Textbooks, reference works, and tutorials

de Boor, Carl: "A practical guide to splines", Applied Mathematical Sciences, 27. Springer-Verlag, New York-Berlin, 1978. 392 pp. ISBN 0-387-90356-9 MR80a:65027

For numerical issues regarding interpolation consult the appropriate portion of the Numerical Analysis FAQ

Software and tables

Interpolation software approximations software.

Other web sites with this focus

Selected Topics at this site


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Last modified 2000/01/14 by Dave Rusin. Mail: