We construct one-step iterative process for an α- nonexpansive mapping and a mapping satisfying condition (C) in the framework of a convex metric space. We study △-convergence and strong convergence of the iterative process to the common fixed point of the mappings. Our results are new and are valid in hyperbolic spaces, CAT(0) spaces, Banach spaces and Hilbert spaces, simultaneously.

We identify multiresolution subspaces giving rise via Hankel transforms to Bessel functions. They emerge as orthogonal systems derived from geometric Hilbert-space considerations, the same way the wavelet functions from a multiresolution scaling wavelet construction arise from a scale of Hilbert spaces. We study the theory of representations of the C ∗-algebra Oν+1 arising from this multiresolu...

We show that, assuming that quantum mechanics holds locally, the finite speed of information is the principle that limits all possible correlations between distant parties to be quantum mechanical as well. Local quantum mechanics means that a Hilbert space is assigned to each party, and then all local positive-operator-valued measurements are (in principle) available; however, the joint system ...

The Bogomolny equations for Yang-Mills-Higgs monopoles follow from a system of linear equations which may be solved through a parametric Riemann-Hilbert problem. We extend this approach to noncommutative R and use it to (re)construct noncommutative Dirac, Wu-Yang, and BPS monopole configurations in a unified manner. In all cases we write down the underlying matrix-valued functions for multi-mon...

We consider the Schrödinger operator −∆ − V (x) on R, but with the difference from the usual case that V is a Hermitian matrix-valued potential. In other words, the Hilbert space is not L(R) but L(R;C). The values of functions in this space, ψ(x), are N−dimensional vectors. (What we say here easily generalizes to ‘operatorvalued’ potentials, i.e., C is replaced by a Hilbert space such as L(R), ...

We study decompositions of operator measures and more general sesquilinear form measures E into linear combinations of positive parts, and their diagonal vector expansions. The underlying philosophy is to represent E as a trace class valued measure of bounded variation on a new Hilbert space related to E. The choice of the auxiliary Hilbert space fixes a unique decomposition with certain proper...

A method is introduced to learn and represent similarity with linear operators in kernel induced Hilbert spaces. Transferring error bounds for vector valued large-margin classifiers to the setting of Hilbert-Schmidt operators leads to dimension free bounds on a risk functional for linear representations and motivates a regularized objective functional. Minimization of this objective is effected...

In this paper we are interested in the numerical approximation of the marginal distributions of the Hilbert space valued solution of a stochastic Volterra equation driven by an additive Gaussian noise. This equation can be written in the abstract Itô form as dX(t) + (∫ t 0 b(t− s)AX(s) ds ) dt = dWQ (t), t ∈ (0, T ]; X(0) = X0 ∈ H, where WQ is a Q-Wiener process on the Hilbert space H and where...

0. Introduction. It is the purpose of this note to show that the several minimum properties of odd degree polynomial spline functions [4, 18] all derive from the fact that spline functions are representers of appropriate bounded linear functionals in an appropriate Hilbert space. (These results were first announced in Notices, Amer. Math. Soc., 11 (1964) 681.) In particular, spline interpolatio...