- Define
\[ f \ast g ( x ) : = \int _ { - \infty } ^ { \infty } f ( y ) g ( x - y ) \ : d y . \]
Prove that if \(f , g \in L ^ { 2 } ( \mathbb { R } )\) , then \(f * g\) is a continuous function on \(\mathbb { R }\)
Solution
1. \(f\ast g\) is defined everywhere and bounded by \(\norm f_2\norm g_2\).
By the Cauchy–Schwarz inequality, \(\abs{\int f(y)g(x-y)\,dy}\le\norm f_2\norm g_2\).
2. Q.E.D.
By the Cauchy–Schwarz inequality, \(\abs{(f\ast g)(x+h)-(f\ast g)(x)}\le\norm f_2\norm{g(\cdot+h)-g}_2\), which tends to \(0\) as \(h\to0\) by continuity of translation in \(L^2\) [@Fol13].