Pointwise and uniform convergence for particularly well-behaved functions

Question

I'm asking myself...

if $f_n(x):\mathbb{R}\to\mathbb{R}$ are infinitely differentiable functions, and each $f_n$ is such that $f_n(x)\underbrace{=}_{|x|\to\infty}\mathcal{O}(|x|^{-N})$ for any $N\in\mathbb{N}$, then is it true that:

$$\underbrace{\lim_{n\to\infty}f_n(x)=0}_{\text{only pointwise}}\Rightarrow \lim_{n\to\infty}\max_{x\in\mathbb{R}}\{f_n(x)\}=0$$

?

I can't set up a proof, but intuitively I can't find a counterexample.

Thanks in advance for any help.

Answer 1

As mentioned in the comment above, your assumption is not enough to guarantee uniform convergence to $0$. Let me rephrase your question.

Let $f_n \in C^\infty(\mathbb{R};\mathbb{R})$ such that, for every $n \in \mathbb{N}$ and every $N \in\mathbb{N}$, there exists $C_{n,N} > 0$ such that: $$ \forall x \in \mathbb{R}, \quad |f_n(x)| \leq C_{n,N} |x|^{-N}.$$ Does pointwise convergence of $f_n$ to $0$ imply uniform convergence of $f_n$ to $0$?

A key weakness of your assumption is that it does not prevent "mass escape" at infinity. This is a typical example for compact embedding results such as Fréchet-Kolmogorov theorem, which involves a condition called "equitightness".

As a counter-example, let $\phi \in C^\infty(\mathbb{R};\mathbb{R})$ be a non-zero function which is compactly supported in $[-1,1]$ and let $f_n(x) := \phi(x-n)$. Then for every $n$, $\max_{x\in \mathbb{R}} |f_n(x)| = \max_{[-1,1]} |\phi| > 0$ does not tend to zero. But you can check that $f_n \to 0$ pointwise (since, for every $x \in \mathbb{R}$, for every $n$ large enough, $f_n(x) = 0$. Also, these functions satisfy your decay assumption since they are compactly supported.