Approximation in the mean

Approximation of a given function $f(t)$, integrable on an interval $[a,b]$, by a function $\phi(t)$, where the quantity

$$\mu(f,\phi)=\int\limits_a^b|f(t)-\phi(t)|dt$$

is taken as the measure of approximation.

The more general case, when

$$\mu(f,\phi)=\int\limits_a^b|f(t)-\phi(t)|^qd\sigma(t)\quad(q>0),$$

where $\sigma(t)$ is a non-decreasing function different from a constant on $[a,b]$, is called mean-power approximation (with exponent $q$) with respect to the distribution $d\sigma(t)$. If $\sigma(t)$ is absolutely continuous and $\phi(t)=\sigma(t)$, then one obtains mean-power approximation with weight $\phi(t)$, and if $\sigma(t)$ is a step function with jumps $c_k$ at points $t_k$ in $[a,b]$, one has weighted mean-power approximation with respect to the system of points $\{t_k\}$ with measure of approximation

$$\mu(f,\phi)=\sum_kc_k|f(t_k)-\phi(t_k)|^q.$$

These concepts are extended in a natural way to the case of functions of several variables.

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. Linear theory , Addison-Wesley (1964)

References

[a1]	A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian)
[a2]	T.J. Rivlin, "An introduction to the approximation of functions" , Dover, reprint (1981)
[a3]	E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982) pp. 203ff