A theorem on weighted 𝐿¹-approximation

Schmidt, Darrell

doi:10.1090/S0002-9939-1987-0897074-4

A theorem on weighted $L^ 1$-approximation
HTML articles powered by AMS MathViewer

by Darrell Schmidt

Proc. Amer. Math. Soc. 101 (1987), 81-84

DOI: https://doi.org/10.1090/S0002-9939-1987-0897074-4

PDF | Request permission

Abstract:

It is proven that the $A$-property is necessary for a finitedimensional subspace to be Chebyshev in $C(K)$ with respect to the weighted ${L^1}$-norm $||f|{|_w} = \int _K {w|f|d\mu }$ for all weight functions $w$ in certain classes of functions.

References

András Kroó, On an $L_1$-approximation problem, Proc. Amer. Math. Soc. 94 (1985), no. 3, 406–410. MR 787882, DOI 10.1090/S0002-9939-1985-0787882-0
András Kroó, A general approach to the study of Chebyshev subspaces in $L_1$-approximation of continuous functions, J. Approx. Theory 51 (1987), no. 2, 98–111. MR 909801, DOI 10.1016/0021-9045(87)90024-4
András Kroó, Best $L_1$-approximation with varying weights, Proc. Amer. Math. Soc. 99 (1987), no. 1, 66–70. MR 866431, DOI 10.1090/S0002-9939-1987-0866431-4
András Kroó, Chebyshev rank in $L_1$-approximation, Trans. Amer. Math. Soc. 296 (1986), no. 1, 301–313. MR 837813, DOI 10.1090/S0002-9947-1986-0837813-5
R. R. Phelps, Čebyšev subspaces of finite dimension in $L_{1}$, Proc. Amer. Math. Soc. 17 (1966), 646–652. MR 194882, DOI 10.1090/S0002-9939-1966-0194882-9
A. Pinkus, Unicity subspaces in $L^1$-approximation, J. Approx. Theory 48 (1986), no. 2, 226–250. MR 862238, DOI 10.1016/0021-9045(86)90007-9
Manfred Sommer, Uniqueness of best $L_1$-approximations of continuous functions, Delay equations, approximation and application (Mannheim, 1984) Internat. Schriftenreihe Numer. Math., vol. 74, Birkhäuser, Basel, 1985, pp. 264–281. MR 899100
H. Strauss, $I_{1}$-Approximation mit Splinefunktionen, Numerische Methoden der Approximationstheorie, Band 2 (Tagung, Math. Forschungsinst., Oberwolfach, 1973) Internat. Schriftenreihe Numer. Math., Band 26, Birkhäuser, Basel, 1975, pp. 151–162 (German). MR 0382951
Hans Strauß, Eindeutigkeit in der $L_{1}$-Approximation, Math. Z. 176 (1981), no. 1, 63–74 (German). MR 606172, DOI 10.1007/BF01258905

Best

-approximation

41