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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 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

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

Other web sites with this focus

• Approximation Theory Network (AT-NET)
• SIAM, the Society for Industrial and Applied Mathematics.
• Here are the AMS and Goettingen resource pages for area 41.

Selected Topics at this site

• Overview of options and pitfalls of (1-dimensional) interpolation.
• When is right to do polynomial interpolation?
• Does polynomial interpolation of decreasing data yield a decreasing function? (no)
• Interpolating a function on R^2 from values at discrete points.
• Interpolating a function on R^2 from values at discrete points.
• Pointer for 2D interpolation.
• Interpolation for functions of several variables.
• Tetrahedron interpolation using barycentric coordinates
• Isn't linear interpolation easiest? (not in multivariable settings)
• Pointer to software for interpolating over a sphere
• Using even distributions of points on a sphere for interpolation
• How does a Taylor series behave on the circle of convergence?
• Why use Pade Approximants?
• Under what circumstances do the Pade approximations converge to the original function?
• Is a Pade approximation the right one to use?
• Approximating functions with quadratics
• Basics: how to calculate a spline-curve?
• Elementary construction of cubic splines
• What is the basic idea behind splines?
• When does it make sense to use Bézier curves instead of interpolating polynomials (say)?
• The Budan-Fourier theorem to determine the maximum possible number of real roots of an equation on a given interval.
• What is Watson's Lemma (asymptotic expansions)
• Approximation of Lipschitzian functions by C^1 functions having the same Lipschitz constant