Positive lacunary Chebyshev series ($f={\sum }_{k\ge 0}{a}_k{T}_{3^k},E_n(f)={\sum }_{3^k>n}{a}_k$)

For positive summable coefficients, the Weierstrass M-test gives a continuous uniform sum. The partial sum over $3^k\le n$ is the unique best uniform approximation of degree at most $n$. If $3^K$ is the first omitted frequency, every tail term equals $(-1{)}^j$ at $x_j=\cos (j\pi /3^K)$ because its frequency ratio is odd. The entire tail therefore attains alternating extrema equal in magnitude to its coefficient sum at more than $n+1$ points. The Chebyshev alternation theorem proves the assertion, including the empty partial sum at $n=0$.