Duality for frames in Krein spaces

A J‐frame for a Krein space is in particular a frame for (in the Hilbert space sense). But it is also compatible with the indefinite inner‐product of , meaning that it determines a pair of maximal uniformly definite subspaces, an analogue to the maximal dual pair associated with an orthonormal basis in a Krein space. This work is devoted to study duality for J‐frames in Krein spaces. Also, tight and Parseval J‐frames are defined and characterized.     

Weighted Generalized Inverses,Oblique Projections,and Least-Squares Problems

A generalization with singular weights of Moore–Penrose generalized inverses of closed range operators in Hilbert spaces is studied using the notion of compatibility of subspaces and positive operators.     

Algebraic splines in locally convex spaces

In a vector space of continuous functions, a variational solution of a finite system of linear functional equations is found. The locally convex topology on the vector space and the properties of the objective functional required for obtaining the solution in the form of a decomposition in the basis dual to the family of functionals of the system are determined. The basis elements are calculated exactly and called basis algebraic splines; their linear span is called the space of algebraic splines in the corresponding locally convex space.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 339–353.Original Russian Text Copyright © 2005 by A. P. Kolesnikov.This revised version was published online in April 2005 with a corrected issue number.     

On singular critical points of positive operators in Krein spaces

We give an example of a positive operator in a Krein space with the following properties: the nonzero spectrum of consists of isolated simple eigenvalues, the norms of the orthogonal spectral projections in the Krein space onto the eigenspaces of are uniformly bounded and the point is a singular critical point of

     

Mixed interpolating-smoothing splines and the ν-spline

In their monograph, Bezhaev and Vasilenko have characterized the “mixed interpolating-smoothing spline” in the abstract setting of a Hilbert space. In this paper, we derive a similar characterization under slightly more general conditions. This is specialized to the finite-dimensional case, and applied to a few well-known problems, including the ν-spline (a piecewise polynomial spline in tension) and near-interpolation, as well as interpolation and smoothing. In particular, one of the main objectives in this paper is to show that the ν-spline is actually a mixed spline, an observation that we believe was not known prior to this work. We also show that the ν-spline is a limiting case of smoothing splines as certain weights increase to infinity, and a limiting case of near-interpolants as certain tolerances decrease to zero. We conclude with an iteration used to construct curvature-bounded ν-spline curves.     

Local spectral theory for normal operators in Krein spaces

Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the resolvent of the imaginary part (close to the real axis) is of finite order, then the normal operator possesses a local spectral function defined for Borel subsets of the spectrum which belong to positive (negative) type spectrum. Moreover, the restriction of the normal operator to the spectral subspace corresponding to such a Borel subset is a normal operator in some Hilbert space. In particular, if the spectrum consists entirely out of positive and negative type spectrum, then the operator is similar to a normal operator in some Hilbert space. We use this result to show the existence of operator roots of a class of quadratic operator polynomials with normal coefficients.     

Signatures for ‐hermitians and ‐unitaries on Krein spaces with Real structures

For J‐hermitian operators on a Krein space satisfying an adequate Fredholm property, a global Krein signature is shown to be a homotopy invariant. It is argued that this global signature is a generalization of the Noether index. When the Krein space has a supplementary Real structure, the sets of J‐hermitian Fredholm operators with Real symmetry can be retracted to certain of the classifying spaces of Atiyah and Singer. Secondary ‐invariants are introduced to label their connected components. Related invariants are also analyzed for J‐unitary operators.     

The Krein formula in almost Pontryagin spaces. A proof via orthogonal coupling

A new proof is provided for the Krein formula for generalized resolvents in the context of symmetric operators or relations with defect numbers $(1,1)$ in an almost Pontryagin space. The new proof is geometric and uses the orthogonal coupling of the almost Pontryagin spaces induced by the $Q$-function and the parameter function in the Krein formula.     

Abstract Hardy-Sobolev spaces and interpolation

The purpose of this work is to describe an abstract theory of Hardy-Sobolev spaces on doubling Riemannian manifolds via an atomic decomposition. We study the real interpolation of these spaces with Sobolev spaces and finally give applications to Riesz inequalities.     

Krein空间上J-正常算子的可定化性

对于Krein空间上J-正常算子 的各种可定化性进行了研究. 利用可定化J-正常算子的谱函数, 给出了临界线的概念, 得到了可定化的J-正常算子成为强可定化算子和一致可定化算子的充要条件.     

Interpolatory Hermite splines on rectangular domains

Based on Bittner and Urban’s construction of interpolatory multiwavelets [Kai Bittner, Karsten Urban, On interpolatory divergence-free wavelets, Math. Comput. 76 (258) (2007) 903-929], we use the truncation method to obtain interpolatory multiwavelets on rectangular domains in this paper. In addition, the characterization for the corresponding functional spaces is given.     

Simultaneously Definitizable and Spectralizable Operators in Krein Space

We study the spectral properties of a self–adjoint operator in Krein space such that one of its natural powers is J–non–negative and some other power is a spectral operator in the Dunford sense.     

Projections in Krein spaces

Let (H,J) be a Krein space with selfadjoint involution J. Starting with a canonical representation of a J-selfadjoint projection, J-projection in short, as the sum of a J-positive projection and a J-negative one we study in detail the structure of a regular subspace, that is, the range of a J-projection. We treat the problem when the sum of two regular subspaces is again regular. We also treat the problem when the closure of the range of the product of a J-contraction and a J-expansion becomes regular.     

Method of successive projections for finding a common point of sets in metric spaces

Many problems in applied mathematics can be abstracted into finding a common point of a finite collection of sets. If all the sets are closed and convex in a Hilbert space, the method of successive projections (MOSP) has been shown to converge to a solution point, i.e., a point in the intersection of the sets. These assumptions are however not suitable for a broad class of problems. In this paper, we generalize the MOSP to collections of approximately compact sets in metric spaces. We first define a sequence of successive projections (SOSP) in such a context and then proceed to establish conditions for the convergence of a SOSP to a solution point. Finally, we demonstrate an application of the method to digital signal restoration.     

Abstract functions with continuous differences and Namioka spaces

Let be a semigroup and a topological space. Let be an Abelian topological group. The right differences of a function are defined by for . Let be continuous at the identity of for all in a neighbourhood of . We give conditions on or range under which is continuous for any topological space . We also seek conditions on under which we conclude that is continuous at for arbitrary . This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.

     

The bidimensional interproximation problem and mixed splines

Interproximation methods for surfaces can be used to construct a smooth surface interpolating some data points and passing through specified regions. In this paper we study the use of mixed splines, that is smoothing splines with additional interpolation constraints, to solve the interproximation problem for surfaces in the case of scattered data. The solution is obtained by solving a linear system whose structure can be improved by using “bell-shaped” thin plate splines.