Coeecients of Polynomials of Restricted Growth on the Real Line
نویسنده
چکیده
Let : (?1; 1) ! (0; 1) be a given continuous even function and let m be a positive integer. We show that, with some additional restrictions on , there exist decreasing sequences x 1 ; : : :; x m and y 1 ; : : :; y m?1 of symmetrically located points on (?1; 1) and corresponding polynomials P and Q of degrees m ? 1 and m, respectively, satisfying jP(x)j (x) m ; jQ(x)j (x) m ; ? 1 < x < 1; where equality holds with alternating signs at the corresponding sequence of points (and also at 1 for Q). Moreover, for any polynomial p of degree at most m, a) if jp(x j)j (x j) m for j = 1; : : :; m, then jp (k) (0)j jP (k) (0)j whenever k and m have opposite parity and 0 k < m. b) if jp(y j)j (y j) m for j = 1; : : :; m?1 and if lim sup y!1 jp(y)j==(y) m 1, then jp (k) (0)j jQ (k) (0)j whenever k and m have the same parity and 0 k m. We give two computational methods for determining these sequences of points and thus P and Q.
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