Let $K(x,y): \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ given by $K(x,y) = e^{-< x,y>^2}$ where $<\cdot,\cdot>$ denote the canonical inner product. Define integral operator $T:C(\mathbb{R}^n) \to C(\mathbb{R}^n)$ by $(Tf)(x) = \int_{\mathbb{R}^n}K(x,y)f(y) dy$ whenever this integration makes sense. (Here, $C(\mathbb{R}^n)$ is a space of continuous functions.)

Now consider a positive real analytic function $f(x)$ which decays exponentially in $< x,x>$. For such $f$, $Tf$ clearly makes sense. The question in my mind is, would $Tf$ be real analytic?

I think since the convergence is not only uniform, but also monotone, it's plausible to expect $Tf$ being analytic. But I have no clue how am I going to show this.

Another guess that I have is $f(x)$ smooth with some decaying condition also would imply analyticity of $Tf$, though I wouldn't need that much.

I'll appreciate any kind of comments on the problem. Thanks!

Junehyuk Jung

Added:

an alternate question will be this:

Let $0 \leq f_1 (x) \leq f_2 (x) \leq \cdots$ be the sequence of real analytic function defined on $\mathbb{R}^n$, which converges to some bounded function $f(x)$. Will $f(x)$ be real analytic?