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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.

## 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

This image slightly hand-edited for clarity.

## Subfields

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

• 41A05: Interpolation, See also 42A15 and 65D05
• 41A10: Approximation by polynomials, For approximation by trigonometric polynomials, See 42A10
• 41A15: Spline approximation
• 41A17: Inequalities in approximation (Bernstein, Jackson, Nikolskii type inequalities)
• 41A20: Approximation by rational functions
• 41A21: Pade approximation
• 41A25: Rate of convergence, degree of approximation
• 41A27: Inverse theorems
• 41A28: Simultaneous approximation
• 41A29: Approximation with constraints
• 41A30: Approximation by other special function classes
• 41A35: Approximation by operators (in particular, by integral operators)
• 41A36: Approximation by positive operators
• 41A40: Saturation
• 41A44: Best constants
• 41A45: Approximation by arbitrary linear expressions
• 41A46: Approximation by arbitrary nonlinear expressions; widths and entropy
• 41A50: Best approximation, Chebyshev systems
• 41A52: Uniqueness of best approximation
• 41A55: Approximate quadratures
• 41A58: Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
• 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.), See also 30E15
• 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section)
• 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
• 41A80: Remainders in approximation formulas
• 41A99: Miscellaneous topics

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.

## Selected Topics at this site

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