= Explicit Chebyshev construction for Bernstein lethargy
{title2=$E_n(f_\epsilon)\ge\epsilon_n$}
For a strictly decreasing positive sequence $\epsilon_n\to0$, put $a_0=\epsilon_0-\epsilon_1$ and $a_k=\epsilon_{3^{k-1}}-\epsilon_{3^k}$ for $k\ge1$. These are positive with sum $\epsilon_0$. The <positive lacunary Chebyshev series> $f_\epsilon=\sum_k a_kT_{3^k}$ has error $\epsilon_0$ at degree zero and error $\epsilon_{3^{K-1}}\ge\epsilon_n$ whenever $3^{K-1}\le n<3^K$. This proves the lower-bound form of <Bernstein's lethargy theorem> with an explicit continuous function.
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