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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-1992-1034660-1
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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  • [1] D. L. Berman, On the impossibility of constructing a linear operator furnishing an approximation within the order of the best approximation, Dokl. Akad. Nauk SSSR 120 (1958), 1175-1177. (Russian) MR 0098941 (20:5387)
  • [2] B. L. Chalmers, The $ n$-dimensional Hàlder equality condition, (submitted).
  • [3] C. Franchetti and E. W. Cheney, Minimal projections in $ {L^1}$-space, Duke Math. J. 43 (1976), 501-510. MR 0423061 (54:11044)
  • [4] Charles R. Hobby and John R. Rice, A moment problem in $ {L^1}$ approximation, Proc. Amer. Math. Soc. 16 (1965), 665-670. MR 0178292 (31:2550)
  • [5] Robert C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265-292. MR 0021241 (9:42c)
  • [6] B. R. Kripke and T. J. Rivlin, Approximation in the metric of $ {L^1}(X,\mu )$, Trans. Amer. Math. Soc. 119 (1965), 101-122. MR 0180848 (31:5078)
  • [7] Pol V. Lambert, On the minimum norm property of the Fourier projection in $ {L^1}$ spaces, Bull. Soc. Math. Belg. 21 (1969), 370-391. MR 0273293 (42:8173)
  • [8] P. D. Morris and E. W. Cheney, On the existence and characterization of minimal projections, J. Reine Angew. Math. 270 (1974), 61-76. MR 0358188 (50:10653)
  • [9] M. M. Rao, Measure theory and integration, Wiley, New York, 1987. MR 891879 (89k:28001)
  • [10] I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Springer-Verlag, Berlin, 1970. MR 0270044 (42:4937)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1034660-1
Article copyright: © Copyright 1992 American Mathematical Society

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