Codimension-one minimal projections onto Haar subspaces

Abstract

Let H n be an n-dimensional Haar subspace of X=C R [a,b] 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, ( n⩾2). 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].