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Miklós Schweitzer 1986, Problem 8

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September 12, 2016
Miklos Schweitzercollege contestsalgebrapolynomialreal analysis

Problem Statement

Let a0=0a_0=0, a1,…,aka_1, \ldots, a_k and b1,…,bkb_1, \ldots, b_k be arbitrary real numbers. (i) Show that for all sufficiently large nn there exist polynomials pnp_n of degree at most nn for which pn(i)(−1)=ai,     pn(i)(1)=bi,     i=0,1,…,kp_n^{(i)} (-1)=a_i,\,\,\,\,\, p_n^{(i)} (1)=b_i,\,\,\,\,\, i=0, 1, \ldots, k and max⁡∣x∣≤1∣pn(x)∣≤cn2          (∗)\max_{|x|\leq 1} |p_n (x)|\leq \frac{c}{n^2}\,\,\,\,\,\,\,\,\,\, (*) where the constant cc depends only on the numbers ai,bia_i, b_i.
(ii) Prove that, in general, (*) cannot be replaced by the relation lim⁡n→∞n2⋅max⁡∣x∣≤1∣pn(x)∣=0\lim_{n\to\infty} n^2\cdot \max_{|x|\leq 1} |p_n (x)| = 0 [J. Szabados]
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