Lebl-Basic-Analysis-I · Problem 5.5.14

The Riemann Integral

Mathematics · improper-integrals-convolution

Problem

Let

f : \mathbb{R} \to \mathbb{R}

and

g : \mathbb{R} \to \mathbb{R}

be continuous functions, where

g(x) = 0

for all

x \notin [a, b]

for some interval

[a, b]

. a) Show that the convolution

(g * f)(x) \coloneqq \int_{-\infty}^{\infty} f(t) g(x - t) \, dt

is well-defined for all

x \in \mathbb{R}

. b) Suppose

\int_{-\infty}^{\infty} \left|f(x)\right| dx < \infty

. Prove that

\lim_{x \to -\infty} (g * f)(x) = 0, \quad \text{and} \quad \lim_{x \to \infty} (g * f)(x) = 0.