Integration with respect to a vector measure and function approximation

Abstract

The integration with respect to a vector measure may be applied in order to approximate a function in a Hilbert space by means of a finite orthogonal sequence $\left\{f_i\right\}$ attending to two different error criterions. In particular, if $\Omega \in ℝ$ is a Lebesgue measurable set, $f\in L_2\left(\Omega \right)$, and $\left\{A_i\right\}$ is a finite family of disjoint subsets of $\Omega$, we can obtain a measure ${\mu }_0$ and an approximation $f_0$ satisfying the following conditions: (1) $f_0$ is the projection of the function $f$ in the subspace generated by $\left\{f_i\right\}$ in the Hilbert space $f\in L_2\left(\Omega ,{\mu }_0\right)$. (2) The integral distance between $f$ and $f_0$ on the sets $\left\{A_i\right\}$ is small.