Weak Chebyshev subspaces and 𝐴-subspaces of 𝐶[𝑎,𝑏]

Li, Wu

doi:10.1090/S0002-9947-1990-1010886-6

Weak Chebyshev subspaces and $A$-subspaces of $C[a,b]$
HTML articles powered by AMS MathViewer

by Wu Li PDF

Trans. Amer. Math. Soc. 322 (1990), 583-591 Request permission

Abstract:

In this paper we show some very interesting properties of weak Chebyshev subspaces and use them to simplify Pinkus’s characterization of $A$subspaces of $C[a,b]$. As a consequence we obtain that if the metric projection ${P_G}$ from $C[a,b]$ onto a finite-dimensional subspace $G$ has a continuous selection and elements of $G$ have no common zeros on $(a,b)$, then $G$ is an $A$-subspace.

References

M. P. Carroll and D. Braess, On uniqueness of $L_{1}$-approximation for certain families of spline functions, J. Approximation Theory 12 (1974), 362–364. MR 358150, DOI 10.1016/0021-9045(74)90078-1
Frank Deutsch, Günther Nürnberger, and Ivan Singer, Weak Chebyshev subspaces and alternation, Pacific J. Math. 89 (1980), no. 1, 9–31. MR 596912, DOI 10.2140/pjm.1980.89.9
P. V. Galkin, The uniqueness of the element of best mean approximation of a continuous function by splines with fixed nodes, Mat. Zametki 15 (1974), 3–14 (Russian). MR 338623
S. Ja. Havinson, On uniqueness of functions of best approximation in the metric of the space $L_{1}$, Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 243–270 (Russian). MR 0101443
Dunham Jackson, Note on a class of polynomials of approximation, Trans. Amer. Math. Soc. 22 (1921), no. 3, 320–326. MR 1501177, DOI 10.1090/S0002-9947-1921-1501177-1
R. C. Jones and L. A. Karlovitz, Equioscillation under nonuniqueness in the approximation of continuous functions, J. Approximation Theory 3 (1970), 138–145. MR 264303, DOI 10.1016/0021-9045(70)90021-3

The

problem in abstract linear normed space

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, 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
A. Kroó, D. Schmidt, and M. Sommer, Some properties of $A$-spaces and their relationship to $L^1$-approximation, Constr. Approx. 7 (1991), no. 3, 329–339. MR 1120407, DOI 10.1007/BF01888161
Wu Li, Continuous selections for metric projections in $C(X)$. II. Alternating signs, Acta Math. Sinica 31 (1988), no. 1, 11–20 (Chinese). MR 951468

Continuous selections for metric projections and interpolating subspaces

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
A. Pinkus and B. Wajnryb, Necessary conditions for uniqueness in $L^1$-approximation, J. Approx. Theory 53 (1988), no. 1, 54–66. MR 937143, DOI 10.1016/0021-9045(88)90076-7
Darrell Schmidt, A theorem on weighted $L^1$-approximation, Proc. Amer. Math. Soc. 101 (1987), no. 1, 81–84. MR 897074, DOI 10.1090/S0002-9939-1987-0897074-4
Manfred Sommer, Weak Chebyshev spaces and best $L_{1}$-approximation, J. Approx. Theory 39 (1983), no. 1, 54–71. MR 713361, DOI 10.1016/0021-9045(83)90068-0

Uniqueness of best

approximation of continuous functions

Bernd Stockenberg, Subspaces of weak and oriented Tchebyshev-spaces, Manuscripta Math. 20 (1977), no. 4, 401–407. MR 442555, DOI 10.1007/BF01171129
Hans Strauss, Best $L_{1}$-approximation, J. Approx. Theory 41 (1984), no. 4, 297–308. MR 753026, DOI 10.1016/0021-9045(84)90087-X