Codimension-one minimal projections onto Haar subspaces |
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Authors: | Grzegorz Lewicki Michael Prophet |
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Affiliation: | a Department of Mathematics, Jagiellonian University, 30-059 Kraków, Reymonta 4, Poland;b Department of Mathematics, University of Northern Iowa, Cedar Falls, IA 50614-0506, USA |
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Abstract: | Let Hn be an n-dimensional Haar subspace of and let Hn−1 be a Haar subspace of Hn of dimension n−1. In this note we show (Theorem 6) that if the norm of a minimal projection from Hn onto Hn−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]. |
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Keywords: | Haar subspace Minimal projection Interpolating projection |
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