Approximation of functions, measure of

A quantitative expression for the error of an approximation. When the discussion is about the approximation of a function $f$ by a function $\phi$, the measure of approximation $\mu (f, \phi )$ is usually defined by the metric in a function space containing both $f$ and $\phi$. For example, if $f$ and $\phi$ are continuous functions on a segment $[a, b]$, the uniform metric of $C [a, b]$ is commonly used, i.e. one puts

$$\mu (f, \phi ) = \ \max _ {a \leq t \leq b } \ | f (t) - \phi (t) | .$$

If continuity of the approximated function is not guaranteed or if the conditions of the problem imply that it is important that $f$ and $\phi$ are close on $[a, b]$ in an average sense, the integral metric of a space $L _ {p} [a, b]$ may be used, putting

$$\mu (f, \phi ) = \ \int\limits _ { a } ^ { b } q (t) | f (t) - \phi (t) | ^ {p} \ dt,\ p > 0,$$

where $q (t)$ is a weight function. The case $p = 2$ 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 $f$ and $\phi$ in discrete points $t _ {k}$, $k = 1 \dots n$, of $[a, b]$, e.g.

$$\mu (f, \phi ) = \ \max _ {1 \leq k \leq n } \ | f (t _ {k} ) - \phi (t _ {k} ) | ,$$

$$\mu (f, \phi ) = \sum _ {k = 1 } ^ { n } q _ {k} | f (t _ {k} ) - \phi (t _ {k} ) | ^ {p} ,$$

where $q _ {k}$ 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 $f$ by a family $F$ of functions is usually taken to be the best approximation:

$$E (f, F) = \ \mu (f, F) = \ \inf _ {\phi \in F } \ \mu (f, \phi ).$$

The quantity

$$E ( \mathfrak M , F) = \ \mu ( \mathfrak M , F) = \ \sup _ {f \in \mathfrak M } \ \inf _ {\phi \in F } \ \mu (f, \phi )$$

is usually taken as the measure of approximation of a class $\mathfrak M$ of functions $f$ by functions $\phi$ from a certain fixed set $F$. It characterizes the maximal deviation of functions in $\mathfrak M$ from functions in $F$ that are closest to them.

In general, when approximation in an arbitrary metric space $X$ is considered, the measure of approximation $\mu (x, u)$ of an element $x$ by an element $u$( a set $F$) is the distance $\rho (x, u)$( or $\rho (x, F)$) between $x$ and $u$( or $F$) in the metric of $X$.

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, "$n$-widths in approximation theory" , Springer (1985) (Translated from Russian)