Overlap multiplicity bound for piecewise difference operators ($\Vert R_{\alpha }\Vert \le \sqrt{2m}/\alpha$)

Suppose a piecewise finite difference uses two translated evaluations divided by $\alpha$. If the pulled-back evaluation intervals cover almost every input point at most $m$ times, then $|a-b{|}^2\le 2(|a{|}^2+|b{|}^2)$ gives $\Vert R_{\alpha }f{\Vert }_2^2\le 2m\Vert f{\Vert }_2^2/{\alpha }^2$. This converts an interval-overlap count into an operator norm estimate without requiring pointwise values of an L2 space equivalence class outside the almost-everywhere evaluation rule.