Approximation order

The order of the error of approximation as a variable quantity, depending on a continuous or discrete argument $\tau$, relative to another variable $\phi(\tau)$ whose behaviour, as a rule, is assumed to be known. In general, $\tau$ 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 $\tau$ may, moreover, have an infinite or finite limit point. The function $\phi(\tau)$ 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 $\phi(\tau)$.

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 $O(h^m)$, where $h$ is the step of the method, is the exponent $m$.

References

[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)

References

[a1]	M.W. Müller, "Approximationstheorie" , Akad. Verlagsgesellschaft (1978)
[a2]	G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966) pp. Chapt. 2