= Uniform hard-threshold convergence of normalized Haar expansions
{title2=$\|\sum_{|c_H|\geq\delta}c_HH-f\|_\infty\to0$}
Use the full orthonormal Haar system, including the constant function. Dyadic partial sums are cell averages and converge uniformly for continuous $f$. On a cell of length $\ell$, a normalized detail has $|c_H|\leq\omega_f(\ell)\sqrt\ell$ and pointwise size $|H|=\ell^{-1/2}$. Fix a coarse level where the modulus is small. Above a level chosen by $\delta/\sqrt\ell$ reaching that small modulus, no detail can survive the threshold. Below it, omitted details along any one dyadic chain sum to a geometric bound. Thus the thresholded expansion differs from a sufficiently fine cell average by a uniformly small amount.
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