Behavior of a functional in learning theory

Abstract

Let H be a Hilbert space, A ∈ L(H), y ∈ R(A), and y ∈R(A). We study the behavior of the distance square between y and A(B_τ), defined as a functional F(τ), as the radius τ of the ball B_τ of H tends to ∞. This problem is important in estimating the approximation error in learning theory. Our main result is to estimate the asymptotic behavior of F(τ) without the compactness assumption on the operator A. We also consider the Peetre K-functional and its convergence rates.