= Positive lacunary Chebyshev series
{title2=$f=\sum_{k\ge0}a_kT_{3^k},\quad E_n(f)=\sum_{3^k>n}a_k$}
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 $n=0$.
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