The linear conditional expectation in Hilbert space

Wahba (1978) and Weinert et al. (1980), using different models, show that an optimal smoothing spline can be thought of as the conditional expectation of a stochastic process observed with noise. This observation leads to efficient computational algorithms. By going back to the Hilbert space formulation of the spline minimization problem, we provide a framework for linking the two different stochastic models. The last part of the paper reviews some new efficient algorithms for spline smoothing.

Download Full-text

The Approach to Equilibrium of a Class of Quantum Dynamical Semigroups

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025798000302 ◽

1998 ◽

Vol 01 (04) ◽

pp. 561-572 ◽

Cited By ~ 17

Author(s):

Franco Fagnola ◽

Rolando Rebolledo

Keyword(s):

Hilbert Space ◽

Asymptotic Behavior ◽

Conditional Expectation ◽

Sufficient Conditions ◽

Approach To Equilibrium ◽

Bounded Operators ◽

Quantum Dynamical Semigroup ◽

Quantum Dynamical ◽

Dynamical Semigroup ◽

Necessary And Sufficient

This paper deals with the asymptotic behavior of a quantum dynamical semigroup [Formula: see text] acting on the algebra of all linear bounded operators on a given Hilbert space. In practice, all these semigroups have a generator which can be written in a well-known form named after Lindblad and Davies. If the semigroup has a faithful normal stationary state ρ, necessary and sufficient conditions are derived for the w*-convergence of [Formula: see text] to [Formula: see text], where [Formula: see text] is the conditional expectation of an element X onto the subalgebra of fixed points. Our main results are expressed in terms of the Lindblad–Davies generator .

Download Full-text

On the equivalence of two stochastic approaches to spline smoothing

Journal of Applied Probability ◽

10.2307/3214367 ◽

1986 ◽

Vol 23 (A) ◽

pp. 391-405 ◽

Cited By ~ 7

Author(s):

Craig F. Ansley ◽

Robert Kohn

Keyword(s):

Stochastic Process ◽

Hilbert Space ◽

Minimization Problem ◽

Stochastic Models ◽

Conditional Expectation ◽

Smoothing Spline ◽

Computational Algorithms ◽

Spline Smoothing ◽

Optimal Smoothing ◽

Space Formulation

Wahba (1978) and Weinert et al. (1980), using different models, show that an optimal smoothing spline can be thought of as the conditional expectation of a stochastic process observed with noise. This observation leads to efficient computational algorithms. By going back to the Hilbert space formulation of the spline minimization problem, we provide a framework for linking the two different stochastic models. The last part of the paper reviews some new efficient algorithms for spline smoothing.

Download Full-text

Conditional Expectations for Unbounded Operator Algebras

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2007/80152 ◽

2007 ◽

Vol 2007 ◽

pp. 1-22 ◽

Cited By ~ 1

Author(s):

Atsushi Inoue ◽

Hidekazu Ogi ◽

Mayumi Takakura

Keyword(s):

Hilbert Space ◽

Conditional Expectation ◽

Unbounded Operator ◽

Operator Algebras ◽

Linear Map ◽

Conditional Expectations ◽

Positive Linear Map

Two conditional expectations in unbounded operator algebras (O∗-algebras) are discussed. One is a vector conditional expectation defined by a linear map of anO∗-algebra into the Hilbert space on which theO∗-algebra acts. This has the usual properties of conditional expectations. This was defined by Gudder and Hudson. Another is an unbounded conditional expectation which is a positive linear mapℰof anO∗-algebraℳonto a givenO∗-subalgebra𝒩ofℳ. Here the domainD(ℰ)ofℰdoes not equal toℳin general, and so such a conditional expectation is called unbounded.

Download Full-text