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| On the Lorentz Conjectures Under the L_1-Norm |
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Citation： |
Ye Maodong.On the Lorentz Conjectures Under the L_1-Norm[J].Chinese Annals of Mathematics B,1990,11(3):359~362 |
| Page view： 938
Net amount： 698 |
Authors： |
Ye Maodong; |
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| Abstract： |
Let f(x)\in C[-1,1],p_n^*(x) be the best approximation polynomial of degree n to f(x).G. Lorentz conjectured that if for all n, p^*_2n(x)=p^*_{2n+1}(x),then f is even; and if P^*_{2n+1}(x)=p^*_{2n+2}(x),p^*_0(x)\equiv 0, then f is odd.
In this paper, it is proved that, under the L_1-norm, the Lorentz conjecture is valid conditionally, i. e. if (i) (1-x^2)f(x) can be extended to an absolutely convergent Tchebyshev series; (ii) for every n, f(x)-p^*_{2n+1}(x) has exactly 2n+2 zeros (or, in the second situation,f(x)-p^*_{2n+2}(x) has exactly 2n+3 zeros), then Lorentz conjecture is valid. |
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