Approximately Linear Functionals on Hilbert Spaces

Minimizing Functionals on Normed - linear Spaces

This paper extends results of [1], [2], of Goldstein, and [3] of Vainberg concerning steepest descent and related topics. An example Is given taken from a simple rendezvous problem in control theory. The problem is one of minimizing a norm on an affine subspace. The problem here is solved in the primal. A solution in the dual is given by Neustadt [4]. I. GENERATION OF MINIMIZING SEQUENCES Let E...

Linear Functionals on Certain Banach Spaces

2 . 1 The linear functionals on Hilbert space

Proof. (Projection Theorem) Showing that the existence of minimizer implies that V is closed is left as an exercise. So we assume that V is closed. For f ∈ H, let α := infv∈V ‖v − f‖. It is a little easier to work with this in an equivalent form, α2 = infv∈V ‖v − f‖2. Thus, for every ε > 0 there is a vε ∈ V such that α2 ≤ ‖vε − f‖2 < α2 + ε. By choosing ε = 1/n, where n is a positive integer, w...

On adaptive inverse estimation of linear functionals in Hilbert scales

We address the problem of estimating the value of a linear functional h f , xi from random noisy observations of y 1⁄4 Ax in Hilbert scales. Both the white noise and density observation models are considered. We propose an estimation procedure that adapts to unknown smoothness of x, of f , and of the noise covariance operator. It is shown that accuracy of this adaptive estimator is worse only b...

ON APPROXIMATION OF LINEAR FUNCTIONALS ON Lp SPACES

In a recent paper certain approximations to continuous nonlinear functionals de-ned on an L p space (1 < p < 1) are shown to exist. These approximations may be realized by sigmoidal neural networks employing a linear input layer that implements nite sums of integrals of a certain type. In another recent paper similar approximation results are obtained using elements of a general class of contin...