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.

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