ON THE LORENTZ CONJECTURES UNDER THE L_1-NORM
Aodong Depertment · 1990
Let f (x) ∈ C [-1, 1], p_n~* (x) be the best approximation polynomial of degree n to f (x). G. Iorentz 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_o~* (z) = 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 Tehebyshev sories; (ii) for every n, f (x) - p_(2n+1)~* (x) has exactly 2n + 2 zeros (or, in the arcond situation, f (x) - p_(2n+2)~* (x) has exaetly 2n+3 zeros), then Lorentz conjecture is valid.