ABSTRACT
Replacing fk) in (6.7) by Pn, we obtain a sequence,]", of polynomials such that II]" - fll k-t ~ 0 and ~k) = Pn , so lI~k)lIo ~ IIf(k)lIo· Consequently lilk~ Ifl k. Now apply (2.7) to j". Since limn-o+ooll ~J;, - ~fllo = 0, Lemma I implies that
~f E D k and (2.7) holds iff E D k • To complete the proof of Theorem I, it remains only to show that there are
polynomials an and bn satisfying (6.3)-(6.5) in addition to the hypotheses of Theorem I. Let b" be a sequence of polynomials constructed as in the last paragraph for k = m, and let bn(x) = b,,(x) + (b(l) - b,,(I»x. Then
(6.8) (6.9)
and b,,(i) = b(i), so b,,(0) ~ 0 and b,,(I) < O. The construction of an is similar, but certain refinements are required in order
to satisfy (6.5) for k = m and an(x) > 0 for 0 < x < I. The only coefficient dij' I <; j " i < m, that involves a(m) is
d m2 = mb(m-l) + 2-la(m).
