Generalized spline algorithms and conditions of their linearity and centrality (Q2848847)

The authors focus on the construction of a linear central algorithm, for various equations in Hilbert and Fréchet spaces, namely an algorithm based on a linear, spline and optimal estimator. The framework is the ill-posed linear setting, with the error measured using a metric. The notion of a solution operator of an operator equation is used as a linear operator mapping, a linear space over the scalar field of real or complex numbers into a local convex metric linear space over the scalar field of real or complex numbers with a given metric. Elements from an absolutely convex set in the domain of the operator are called the problem elements and the image of the operator on these elements is called the solution elements which have to be computed with a minimal error. The case considered in this paper, taking into account a solution operator mapping a metrizable locally convex space in the same space extends the already known case when the solution operator acts on a linear space containing a decreasing sequence of problem elements sets. Thus, the notion of generalized interpolating spline introduced in the paper, extends the already known similar notion for the case when we have not only one set of problem elements sets on the linear space, but a decreasing sequence of them. Based on this sequence, a generalization of the Minkowski functional which generates metrizable locally convex topology is given. Moreover, the generalized interpolating spline proposed in this paper, provides a minimum for both metric and generalized Minkowski functional. Further, the generalized central algorithm is given for a solution operator mapping a Fréchet space into the same space. Also, the authors consider an operator equation with a selfadjoint and positive defined operator in a Hilbert space and give a linear and generalized central algorithm together with some examples of differential operators. An operator equation in a Hilbert space with a selfadjoint, positive, one-to-one compact operator, possessing dense image is considered and a linear generalized spline and central algorithm is proposed together with some examples of inverses of strong elliptic operators and of some integral operators.