Abstract and Applied Analysis
Volume 5 (2000), Issue 4, Pages 207-226

Integration with respect to a vector measure and function approximation

L. M. García-Raffi, D. Ginestar, and E. A. Sánchez-Pérez

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.


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 is a Lebesgue measurable set, fL2(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 fL2(Ω,μ0). (2) The integral distance between f and f0 on the sets {Ai} is small.