Linear span of weighted powers

Question

I am reading Functional Analysis by Peter Lax, and I do not understand the passage where it says that $w(t)e^{i\zeta t}$ belongs to $C$, where:

• $\zeta$ is a complex variable, and
• $C$ is the set of continuous functions on $\mathbb R$ that vanish at $\infty$, as is stated above (2).

Thank you anyone for help.

Answer 1

Define $x(t) = w(t) e^{i \zeta t}$. We want to show that $x \in C$, i.e, $\lim_{|t| \rightarrow \infty} x(t) = 0$ according to (2), or, equivalently, $|x(t)| \rightarrow 0$ as $|t| \rightarrow \infty$.

Note that $e^{i \zeta t} = e^{i (Re \zeta + i Im \zeta) t} = e^{-(Im \zeta) t} e^{i (Re \zeta) t}$, and note also that the text requires $|Im \zeta| < c$, so that $c + Im \zeta > 0$ and $c - Im \zeta > 0$.

According to (1), we have that $w$ satisfies $0 < w(t) < ae^{-c |t|}$ for some $a > 0$ and $c > 0$.

We analyze the two cases when $|t| \rightarrow \infty$:

When $t \rightarrow \infty$, we have $$ |x(t)| = |w(t) e^{i \zeta t}| = |w(t)| |e^{i \zeta t}| = |w(t)| |e^{-(Im \zeta) t}| |e^{i (Re \zeta) t}| = w(t) e^{-(Im \zeta) t} < ae^{-c |t|} e^{-(Im \zeta) t} = ae^{-c t} e^{-(Im \zeta) t} = a e^{-(c + Im \zeta) t} \rightarrow 0 $$ since $t > 0$ and $-(c + Im \zeta) < 0$.

Similarly, when $t \rightarrow -\infty$, we have $$ |x(t)| = |w(t) e^{i \zeta t}| = |w(t)| |e^{i \zeta t}| = |w(t)| |e^{-(Im \zeta) t}| |e^{i (Re \zeta) t}| = w(t) e^{-(Im \zeta) t} < ae^{-c |t|} e^{-(Im \zeta) t} = ae^{c t} e^{-(Im \zeta) t} = a e^{(c - Im \zeta) t} \rightarrow 0 $$ since $t < 0$ and $c - Im \zeta > 0$.