Let $$f_n (x) := \frac{1}{n} \sum_{i=1}^n \exp(-a_i \cdot x^2),$$ with $x \in \mathbb{R}$, and $(a_i)_{i \in \mathbb{N}}$ monotone sequence with $a_i >0 \; \forall i \in \mathbb{N}$.
I have to determine whether $f_n (x)$ converges for $x \in \mathbb{R}$, and eventually if it converges uniformly.
My attempt:
as $f_n(x)$ is:
- bounded for each $x \in \mathbb{R}$: we have $ 0 \leq f_n(x) \leq 1 \; \forall n \in \mathbb{N}$ and $\forall x \in \mathbb{R}$,
- monotone: we have $f_n(x) \geq f_{n+1}(x) \; \forall n \in \mathbb{N}$ and $\forall x \in \mathbb{R}$,
then it converges pointwise.
To understand whether it converges uniformly, as I could not determine explicitely the limit f(x), I tried to use Cauchy criteria (Cauchy Criterion for Uniform Convergence of Functions).
In the trivial case in which $a_i=a>0$ (the $a_i$'s are all the same) then it is clearly uniformly convergent, as $f_n(x) = f(x) = \exp(-a \cdot x^2)$, but in the general case I could neither show uniform convergence nor an objection to it.
Any idea? Any thought would be greatly appreciated.
Best Answer
I suppose you are assuming that $a_i$'s are monotonically increasing. [The case when they are decreasing is easier and I will leave that to you].
Case 1): $(a_i)$ increases to $\infty$. In this case $e^{-a_ix^{2}} \to 0$ as $i \to \infty$ for each $x \neq 0$. Hence, $f_n(x)$ also tends to $0$ for $x \neq 0$. But $f_n(0)=1$ for all $n$. If $f_n$ converges uniformly then the limit function $f$ is continuous. But $f(x)=0$ for $ \neq 0$ and $f(0)=1$. This proves that the convergence is not uniform.
Suppose $a_i$ increases to a finite limit $a$. Then $f_n(x) \to e^{-ax^{2}}$ uniformly. Here are some hints for the proof: Split $f_n(x)$ into the sum of $\frac 1 n\sum\limits_{i=1}^{k}e^{-a_ix^{2}}$ and $\frac 1 n\sum\limits_{i=k+1}^{n}e^{-a_ix^{2}}$. It is easy to see that the first part tends to $0$ uniformly for fixed $k$. We can choose $k$ so large that $a-\epsilon <a_i <a+\epsilon$ for $i >k$. Now use Squeeze Theorem to finish the proof.