Young's Inequality for Convolutions/Corollary 2
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Corollary to Young's Inequality for Convolutions
Let $f, g: \R^n \to \R$ be Lebesgue integrable functions.
Then their convolution $f * g$ is also Lebesgue integrable, and:
- $\left\Vert{f * g}\right\Vert \le \left\Vert{f}\right\Vert \, \left\Vert{g}\right\Vert$
Thus, convolution may be seen as a binary operation:
- $*: \mathcal L^1 \times \mathcal L^1 \to \mathcal L^1$
on the space of integrable functions $\mathcal L^1$.
Proof
Apply Young's Inequality for Convolutions with $p = q = r = 1$.
$\blacksquare$