Characterization of Generalized Haar Spaces

✍ Scribed by M Bartelt; W Li

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 414 KB
• Volume: 92
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1996.3108

✦ Synopsis

We say that a subset G of C 0 (T, R k ) is rotation-invariant if [Qg: g # G]=G for any k_k orthogonal matrix Q. Let G be a rotation-invariant finite-dimensional subspace of C 0 (T, R k ) on a connected, locally compact, metric space T. We prove that G is a generalized Haar subspace if and only if P G ( f ) is strongly unique of order 2 whenever P G ( f ) is a singleton.

1998 Academic Press

1. Introduction

Let T be a locally compact Hausdorff space and G a finite-dimensional subspace of C 0 (T, R k ), the space of vector-valued functions f on T which vanish at infinity, i.e., the set

The metric projection P G from C 0 (T, R k ) to G is given by

for f # C 0 (T, R k ),

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