Approximation of functions, measure of
A quantitative expression for the error of an approximation. When the discussion is about the approximation of a function 
by a function 
, the measure of approximation 
is usually defined by the metric in a function space containing both 
and 
. For example, if 
and 
are continuous functions on a segment 
, the uniform metric of 
is commonly used, i.e. one puts
 |
If continuity of the approximated function is not guaranteed or if the conditions of the problem imply that it is important that 
and 
are close on 
in an average sense, the integral metric of a space 
may be used, putting
 |
where 
is a weight function. The case 
is most often used and is most convenient from a practical point of view (cf. Mean-square approximation of a function).
The measure of approximation may take into account only values of 
and 
in discrete points 
, 
, of 
, e.g.
 |
 |
where 
are certain positive coefficients.
One defines in an analogous way the measure of approximation of functions in two or more variables.
The measure of approximation of a function 
by a family 
of functions is usually taken to be the best approximation:
 |
The quantity
 |
is usually taken as the measure of approximation of a class 
of functions 
by functions 
from a certain fixed set 
. It characterizes the maximal deviation of functions in 
from functions in 
that are closest to them.
In general, when approximation in an arbitrary metric space 
is considered, the measure of approximation 
of an element 
by an element 
(a set 
) is the distance 
(or 
) between 
and 
(or 
) in the metric of 
.
References
| [1] | V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian) |
| [2] | S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian) |
| [3] | J.R. Rice, "The approximation of functions" , 1–2 , Addison-Wesley (1964–1968) |
Comments
The measure of approximation is also called the error measure.
References
| [a1] | G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966) |
| [a2] | A. Pinkus, " -widths in approximation theory" , Springer (1985) (Translated from Russian) |
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