ALTERNATION THEOREM FOR $C(I,X)$ AND APPLICATION TO BEST LOCAL APPROXIMATION
Keyword(s):
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.
1989 ◽
Vol 32
(3)
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pp. 483-494
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Keyword(s):
2016 ◽
Vol 19
(04)
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pp. 1650024
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1989 ◽
Vol 31
(2)
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pp. 131-135
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1968 ◽
Vol 32
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pp. 287-295
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1999 ◽
Vol 92
(2)
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pp. 107-118
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1985 ◽
Vol 101
(3-4)
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pp. 203-206
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1989 ◽
Vol 32
(1)
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pp. 98-104
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