 Solution

ID: past-exam-of-the-functional-analysis-and-banach-and-c-algebras-course-of-the-university-of-oxford/2007/a/1/ii/solution

The kernel is globally Lipschitz with constant

1/ δ^{2}

: this holds on each of its linear pieces, and summing across the finitely many intervening breakpoints proves it across pieces as well. Consequently

∣ (f * g) (t_{1}) - (f * g) (t_{2}) ∣ \leq \int ∣ f (s) ∣ \frac{∣ t _{1} - t _{2} ∣}{δ ^{2}} d s = ∥ f ∥_{1} \frac{∣ t _{1} - t _{2} ∣}{δ ^{2}} .

 New to topics? Read the docs here!