 Uniform hard-threshold convergence of normalized Haar expansions

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 $|c_H|\le {\omega }_f(\ell )\sqrt{\ell }$ 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.