Approximation order

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The order of the error of approximation as a variable quantity, depending on a continuous or discrete argument , relative to another variable whose behaviour, as a rule, is assumed to be known. In general, is a parameter that is a numerical characteristic of the approximating set (e.g. its dimension) or of the method of approximation (e.g. the interpolation step). The set of values of may, moreover, have an infinite or finite limit point. The function is most often a power, an exponential or a logarithmic function. The modulus of continuity (cf. Continuity, modulus of) of the approximated function (or that of some derivative of it) or a majorant of it may figure as .

The approximation order is characterized both by the properties of the approximation method, as well as by a definite property of the approximated object, e.g. the differential-difference properties of the approximated function (cf. Approximation of functions, direct and inverse theorems).

In numerical analysis, the approximation order of a numerical method having error , where is the step of the method, is the exponent .


[1] V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian)
[2] A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian)
[3] N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)



[a1] M.W. Müller, "Approximationstheorie" , Akad. Verlagsgesellschaft (1978)
[a2] G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966) pp. Chapt. 2
How to Cite This Entry:
Approximation order. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by N.P. KorneichukV.P. Motornyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article