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| A Kind of Best Approximation Using Function Pairs |
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Citation： |
Shi Yingguang.A Kind of Best Approximation Using Function Pairs[J].Chinese Annals of Mathematics B,1982,3(5):639~644 |
| Page view： 770
Net amount： 793 |
Authors： |
Shi Yingguang; |
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| Abstract： |
This paper discusses the minimization problem: For a Given n-dimensional subspace K and a function f of C(X), find a function pair (p_1,p_2)\in K x K, p_1 \geq f \geq p_2 such that $||p_1-p_2||=[\mathop {\mathop {\inf }\limits_{({q_1},{q_2}) \in K \times K} }\limits_{{p_1} \ge f \ge {q_2}} ||{q_1} - {q_2}||\]$
We call such a pair(p_1, p_2) (if any) a best approximation pair to f from K.
This paper has proved a characterization theorem for best L_1 approximation pairs which says that (p_1,p_2) is a best L_1 approximation pair if and only if p_1 and p_2 are respectively an upper sided and a lower sided best L_1 approximation to f.
With the provisos that K is a Haar subspace and thet 1\in K it turns out that the above conclusion of the characterization theorem for best L_\infty approximation pairs is also true. However we have further established, without the provisos at all, a “complete” characterization theorem for best L_\infty approximation pairs.
Furthermore, sufficient conditions for uniqueness of best L_\infty approximation pairs are also given. |
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