The linearity coefficient of metric projections onto one-dimensional Chebyshev subspaces of the space C

The paper deals with the operator of metric projection onto an arbitrary one-dimensional Chebyshev subspace 〈φ〉 of the space C[K] of real-valued functions defined and continuous on a Hausdorff compact set K. The linearity coefficient of the operator is calculated in terms of the parameters of the generating function φ. As a consequence, a new estimate of the Lipschitz constant of the operator is obtained.     

Convergence of metric projections

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 10, pp. 1411–1413, October, 1989.     

On a subspace metric based on matrix rank

The metric between subspaces M,N⊆C_n,1, defined by δ(M,N)=rk(P_M-P_N), where rk(·) denotes rank of a matrix argument and P_M and P_N are the orthogonal projectors onto the subspaces M and N, respectively, is investigated. Such a metric takes integer values only and is not induced by any vector norm. By exploiting partitioned representations of the projectors, several features of the metric δ(M,N) are identified. It turns out that the metric enjoys several properties possessed also by other measures used to characterize subspaces, such as distance (also called gap), Frobenius distance, direct distance, angle, or minimal angle.     

The metric projection onto the soul

We study geometric properties of the metric projection of an open manifold with nonnegative sectional curvature onto a soul . is shown to be up to codimension 3. In arbitrary codimensions, small metric balls around a soul turn out to be convex, so that the unit normal bundle of also admits a metric of nonnegative curvature. Next we examine how the horizontal curvatures at infinity determine the geometry of , and study the structure of Sharafutdinov lines. We conclude with regularity properties of the cut and conjugate loci of .

     

Weak continuity of metric projections

A necessary and sufficient condition is found for weak continuity of a metric projection onto a finite-dimensional subspace inl _p (1<p≠2). A metric projection onto a boundedly compact set inl _p is sequentially weakly upper semicontinueus. An example is given on a convex, compact set inl ₂ onto which the metric projection is not weakly continuous. Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 345–356, September, 1977.     

The contraction linear metric projections inL p spaces

In this paper, we study the contraction linearity for metric projection in Lp spaces. A geometrical property of a subspace Y of Lp is given on which Pr is a contraction projection.     

Structure of the metric projection to a one-dimensional subspace in Lϕ

V. Kapsukas Vilnius State University. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 3, pp. 464–473, July–September, 1989.     

Controllability onto a subspace of regular differential-algebraic systems with numerous delays

We study the H-controllability and complete H-controllability problems for differential-algebraic systems with numerous delays in the control. The original system can be reduced by special transformations to a delay-free system, which is analyzed with the use of results obtained earlier by the authors.     

Chebyshev approximation by an N-dimensional subspace on a set of N+1 points

In this note we discuss Chebyshev approximation by an n-dimensional subspace on a set of n+1 points and prove the two conjectures proposed by C. B. Dunham in [2]. The Project Supported by National Natural Science Foundation of China