ABSTRACT
Lemma 4.4 Let 19 E [O, T ] he f i e d . Th,em exists a constant c(19 j > 0 such that for a ~ b i t 7 - q ~ k k No there exists an index k* f No with k* 1 k and \iF pk*(cosfiIl > e(Bj. PROOF. The case k = 0 is trivial since Po = 1. So let k > 0 and assume at first 29 = 0 or 29 = T . By observing IPk(cos29)j =
|Pk(±1)| = 1, we obtain the estimate√ 2k + 1 4π
|Pk(cosϑ)| = √
2k + 1 4π
≥ 1 2 √
π > 0. (4.15)
By choosing k∗ = k and c(ϑ) ∈ (0, 1 2 √
π ) arbitrary, assertion (4.14) is
fulfilled. Now fix ϑ ∈ (0, π). Employing the approximation formula from (3.2) we conclude
√ 2k + 1 4π
|Pk(cosϑ)| = 1 π √ sinϑ
∣∣∣∣∣cos ((
k + 1 2
) ϑ− π
)∣∣∣∣∣+O (k−1) . The asymptotic part O (k−1) vanishes for k → ∞. The constant
sinϑ is strictly positive. So let us assume
cos
(( k +
1 2
) ϑ− π
) k→∞−→ 0.
