Bernstein's lethargy theorem

Bernstein's lethargy theorem says that best approximation errors in nested approximation spaces can tend to zero arbitrarily slowly. For algebraic polynomials on $[-1,1]$, given any decreasing positive sequence ${\delta }_n\to 0$, there is a continuous function $f$ whose best degree-$n$ uniform-approximation error is at least ${\delta }_n$ for every $n$.

For a strictly decreasing positive sequence ${ϵ}_n\to 0$, put $a_0={ϵ}_0-{ϵ}_1$ and $a_k={ϵ}_{3^{k-1}}-{ϵ}_{3^k}$ for $k\ge 1$. These are positive with sum ${ϵ}_0$. The positive lacunary Chebyshev series $f_{ϵ}={\sum }_k{a}_k{T}_{3^k}$ has error ${ϵ}_0$ at degree zero and error ${ϵ}_{3^{K-1}}\ge {ϵ}_n$ whenever $3^{K-1}\le n<3^K$. This proves the lower-bound form of Bernstein's lethargy theorem with an explicit continuous function.