Abstract and Applied Analysis
Volume 5 (2000), Issue 4, Pages 207-226
doi:10.1155/S1085337501000227
Integration with respect to a vector measure and function
approximation
Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, C amino de Vera, Valencia 14 46022, Spain
Received 13 May 2000
Copyright © 2000 L. M. García-Raffi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.