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 {fi} attending to two different error criterions. In particular, if Ω∈ℝ is a Lebesgue measurable set, f∈L2(Ω), and {Ai} is a finite family of disjoint subsets of Ω, we can obtain a measure μ0 and an approximation f0 satisfying the following conditions: (1) f0 is the projection of the function f in the subspace generated by {fi} in the Hilbert space f∈L2(Ω,μ0). (2) The integral distance between f and f0 on the sets {Ai} is small.