Dirac Comb is Distribution
Theorem
Let $\phi \in \map \DD \R$ be a test function.
Suppose $\map {\operatorname {III} } 0$ is a Dirac comb such that:
- $\ds \map {\map {\operatorname {III} } 0} \phi := \sum_{n \mathop \in \Z} \map \phi n$
Then $\map {\operatorname {III} } 0$ is a distribution.
Proof
By definition of test function, $\phi$ is supported on a compact subset of $\R$.
Hence:
- $\exists N \in \N : \forall x \in \R \setminus \closedint {-N} N : \map \phi x = 0$
Therefore:
- $\ds \sum_{n \mathop \in \Z} \map \phi n = \sum_{n \mathop = - N}^N \map \phi n$
This is a finite sequence of real numbers.
Thus, the sequence converges, and $\map {\operatorname {III} } 0$ is a mapping such that $\map {\operatorname {III} } 0 : \map \DD \R \to \R$.
Linearity
Follows from Summation is Linear.
$\Box$
Convergence in $\map \DD \R$
Let $\sequence {\phi_n}$ be a convergent sequence in $\map \DD \R$ with the limit $\mathbf 0$.
By definition of convergence, all $\sequence {\phi_n}$ are supported on a compact subset of $\R$, say, $\closedint {-K} K$ with $K \in \N$.
By definition of convergence, $\sequence {\phi_n}$ converges to $\mathbf 0$ uniformly.
Hence:
- $\ds \forall k \in \R : \size k < K : \lim_{n \mathop \to \infty} \map {\phi_n} k = 0$
Therefore:
\(\ds \map {\map {\operatorname {III} } 0} {\phi_n}\) | \(=\) | \(\ds \sum_{k \mathop \in \Z} \map {\phi_n} k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = -K}^K \map {\phi_n} k\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \lim_{n \mathop \to \infty} \map {\map {\operatorname {III} } 0} {\phi_n}\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \sum_{k \mathop = -K}^K \map {\phi_n} k\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = -K}^K 0\) | Sum Rule for Real Sequences | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
$\Box$
By definition, the Dirac comb is a distribution.
$\blacksquare$
Also see
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.1$: A glimpse of distribution theory. Test functions, distributions, and examples