Convergence of Sequence of Test Functions in Test Function Space implies Convergence in Schwartz Space
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Theorem
Let $\map \DD \R$ be the test function space.
Let $\map \SS \R$ be the Schwartz space.
Let $\sequence {\phi_n}_{n \mathop \in \N}$ be a sequence of test functions in $\map \DD \R$.
Let $\mathbf 0 : \R \to 0$ be the zero mapping.
Suppose $\sequence {\phi_n}$ converges to $\mathbf 0$ in $\map \DD \R$:
- $\phi_n \stackrel \DD \longrightarrow \mathbf 0$
Then $\sequence {\phi_n}$ converges to $\mathbf 0$ in $\map \SS \R$:
- $\phi_n \stackrel \SS \longrightarrow \mathbf 0$
Proof
For all $n \in \N$ let $\phi_n$ be a test function.
By definition, $\phi_n$ has a compact support $I_n \subset \R$:
- $\forall x \notin I_n \implies \map {\phi_n} x = 0$
Let:
- $a \in \R : a > 0 : \forall n \in \N : I_n \subseteq \closedint {-a} a$
Then:
\(\ds \forall m,k \in \N : \sup_{x \mathop \in \R} \size {x^k \map {\phi^{\paren m}_n} x}\) | \(=\) | \(\ds \map {\sup_{\size x \mathop \le a} } {\size x^k \size {\map {\phi^{\paren m}_n} x} }\) | Absolute Value of Product | |||||||||||
\(\ds \) | \(\le\) | \(\ds a^k \sup_{\size x \mathop \le a} \size {\map {\phi^{\paren m}_n} x}\) |
Suppose:
- $\phi_n \stackrel \DD \longrightarrow \mathbf 0$
Then all derivatives of $\phi_n$ on $\closedint {-a} a$ converge uniformly to $\mathbf 0$.
Hence:
\(\ds \lim_{n \mathop \to \infty} \sup_{x \mathop \in \R} \size {x^k \map {\phi^{\paren m}_n} x}\) | \(\le\) | \(\ds a^k \lim_{n \mathop \to \infty} \sup_{\size x \mathop \le a} \size {\map {\phi^{\paren m}_n} x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a^k \sup_{\size x \mathop \le a} \size {\mathbf 0}\) | $\phi_n \stackrel \DD \longrightarrow \mathbf 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds a^k \cdot 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
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In other words:
- $\ds \forall m,k \in \N : \lim_{n \mathop \to \infty} \sup_{x \mathop \in \R} \size {x^k \map {\phi^{\paren m}_n} x} = 0$
By definition:
- $\phi_n \stackrel \SS \longrightarrow \mathbf 0$
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.5$: A glimpse of distribution theory. Fourier transform of (tempered) distributions