The linearity coefficient of metric projections onto a Chebyshev subspace

The linearity coefficient λ(Y) of a metric projection P _Y onto a subspace Y in a Banach space X is determined. This coefficient turns out to be related to the Lipschitz norm of the operator P _Y . It is proved that, for any Chebyshev subspace Y in the space C or L ₁, either λ(Y) = 1 (which corresponds to the linearity of P _Y ) or λ(Y) ≤ 1/2.     

The norm in C of orthogonal projections onto subspaces of polygonal functions

Let P be an orthogonal projection (in the sense of L₂) onto the subspace of polygonal functions over a certain partition of the segment [0, 1]. Z. Ciesielski has established the following estimate for the norm of this operators, as acting from C into C, valid for an arbitrary partition: P _CC 3. In this note it is proved that this estimate is final; more precisely, it is shown that .Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 495–502, April, 1977.     

Norm-one projections onto subspaces ofl p

Summary We give an intrinsic characterization of norm-one complemented subspaces of finite codimension in l _p (1<p< ,p 2).Work prepared under the auspices of M.P.I. (Ministero della Pubblica Istruzione) and G.N.A.F.A. of C.N.R. (Consiglio Nazionale delle Ricerche).     

Codimension-one minimal projections onto Haar subspaces

Let H_n be an n-dimensional Haar subspace of and let H_n−1 be a Haar subspace of H_n of dimension n−1. In this note we show (Theorem 6) that if the norm of a minimal projection from H_n onto H_n−1 is greater than 1, then this projection is an interpolating projection. This is a surprising result in comparison with Cheney and Morris (J. Reine Angew. Math. 270 (1974) 61 (see also (Lecture Notes in Mathematics, Vol. 1449, Springer, Berlin, Heilderberg, New York, 1990, Corollary III.2.12, p. 104) which shows that there is no interpolating minimal projection from C[a,b] onto the space of polynomials of degree n, (n2). Moreover, this minimal projection is unique (Theorem 9). In particular, Theorem 6 holds for polynomial spaces, generalizing a result of Prophet [(J. Approx. Theory 85 (1996) 27), Theorem 2.1].     

Unique continuous selections for metric projections of C(X) onto finite-dimensional vector subspaces, II

Best approximation in C(X) by elements of a Chebyshev subspace is governed by Haar's theorem, the de la Vallée Poussin estimates, the alternation theorem, the Remez algorithm, and Mairhuber's theorem. J. Blatter (1990, J. Approx. Theory 61, 194–221) considered best approximation in C(X) by elements of a subspace whose metric projection has a unique continuous selection and extended Haar's theorem and Mairhuber's theorem to this situation. In the present paper we so extend the de la Vallée Poussin estimates, the alternation theorem, and the Remez algorithm.     

A note on norm-one projections onto subspaces of finite codimension ofl ∞

Research partially supported by Italian Ministero della Pubblica Istruzione (M.P.I.) and G.N.A.F.A. (C.N.R.)     

Structure of balls and uniqueness subspaces in a linear metric space

In linear metric space X we investigate the structure of balls in terms of their convex hulls. We generalize the theorem of [2] having to do, in particular, with the conditions under which each subspace L X is a set of uniqueness [3].Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 541–550, April, 1973.     

Chebyshev subspaces of vector-valued functions

It is shown that if on a compact space Q any polynomial 0} $$ " align="middle" border="0"> , in a system of continuous vector functions with real coefficients such that N=n·s and s=2p +1 has at most n–1 zeros, then Q is homeomorphic to a circle or a part of one.Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 347–352, March 1976.In conclusion, the author thanks S. B. Stechkin for stating the problem globally.