ALTERNATION THEOREM FOR $C(I,X)$ AND APPLICATION TO BEST LOCAL APPROXIMATION
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Abstract
Let $X$ be a Banach space with the approximation property, and $C(I,X)$ the space of continuous functions defined on $I = [0,1)$ with values in $X$. Let $u_i \in C(I,X)$, $i=1,2,\cdots, n$ and $M=span\{u_1, \cdots, u_n\}$. The object of this paper is to prove that if $\{u_1, \cdots, u_n\}$ satisfies certain conditions, then for $f \in C(I,X)$ and $g \in M$ we have $||f-g|| = \inf\{||f-h|| : h\in M\}$ if and only if $f-g$ has at least $n$-zeros. An application to best local approximation in $C(I,X)$ is given.
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References
A. Al-Zamel, and R. Khalil, "Unicity spaces in vector valued function spaces." Submitted. (1989).
J. Diestel, and J. Uhl, "Vector measures," Math. Surneys, no. 15 (1977).
W. Cheney, "Introduction to approximation theory," McGraw-Hill comp. New York. (1966).
C. Chui, O. Shisha, and P. Smith, "Best local approximation ," J. of approx. 15 (1975), 371-381.
W. Light, and E. Cheney, "Approximation theory in tensor product spaces," Lecture · notes in Math 1169. (1985).
A. Kroo, "Best L1-approximation of vector valued functions," Acta Math. Acad. Sci. Hungar., 39 (1982), 303-313.