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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The determination of minimal projections and extensions in $ L\sp 1$


Authors: B. L. Chalmers and F. T. Metcalf
Journal: Trans. Amer. Math. Soc. 329 (1992), 289-305
MSC: Primary 41A35; Secondary 41A65
MathSciNet review: 1034660
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Abstract: Equations are derived which are shown to be necessary and sufficient for finite rank projections in $ {L^1}$ to be minimal. More generally, these equations are also necessary and sufficient to determine operators of minimal norm which extend a fixed linear action on a given finite-dimensional subspace of $ {L^1}$ and thus may be viewed as an extension of the Hahn-Banach theorem to higher dimensions in the $ {L^1}$ setting. These equations are solved in terms of an $ {L^1}$ best approximation problem and the required orthogonality conditions. Moreover, this solution has a simple geometric interpretation. Questions of uniqueness are considered and a number of examples are given to illustrate the usefulness of these equations in determining minimal projections and extensions, including the minimal $ {L^1}$ projection onto the quadratics.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1034660-1
PII: S 0002-9947(1992)1034660-1
Article copyright: © Copyright 1992 American Mathematical Society