Integration with Respect to a Vector Measure and Function Approximation
نویسندگان
چکیده
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 ∈ R is a Lebesgue measurable set, f ∈ L2( ), and {Ai} is a finite family of disjoint subsets of , 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( ,μ0). (2) The integral distance between f and f0 on the sets {Ai} is small.
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