Definition 30.5.1 ($\ell^{1}(\integer)$).label Let $\ell^{1}(\integer)$ be the $\ell^{1}$ sequence space on $\integer$. For each $f, g \in \ell^{1}(\integer)$, let
\[(f * g)(n) = \sum_{k \in \integer}f(k)g(n - k)\]
then:
- (1)
$\ell^{1}(\integer)$ is a commutative Banach algebra.
- (2)
The multiplicative unit of $\ell^{1}(\integer)$ is $\delta_{0} = \one_{\bracs{n = 0}}$.
The space $\ell^{1}(\integer)$ is the convolution algebra on $\integer$.
Proof. For each $f, g \in \ell^{1}(\integer)$, by Fubini’s Theorem,
\begin{align*}\normn{f * g}_{\ell^1(\integer)}&= \sum_{n \in \integer}\abs{\sum_{k \in \integer}f(k)g(n - k)}\\&\le \sum_{n, k \in \integer}|f(k)| \cdot |g(n-k)| \le \sum_{k \in \integer}|f(k)| \cdot \sum_{n \in \integer}|g(n - k)| \\&= \norm{f}_{\ell^1(\integer)}\cdot \norm{g}_{\ell^1(\integer)}\end{align*}
$\square$
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