= Haar approximation error for a function of bounded variation
{c}
{title2=$\|H_j(f)-f\|_1\le2^{-j}\operatorname{TV}(f)$}
Finite <total variation of a function> on $\mathbb R$ gives the following error bound for averaging on dyadic cells of length $\delta=2^{-j}$:
$$
\|H_j(f)-f\|_1\le\delta\operatorname{TV}(f).
$$
The cell error is at most its length times the oscillation on that cell, and the sum of the oscillations is at most the total variation. This proves that the difference is integrable even if $f$ is not globally in $L^1$. For an even function decreasing to zero on the positive half-line, $\operatorname{TV}(f)\le2f(0)$ and the bound becomes $f(0)2^{1-j}$.
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