I have a continuous function $f: [0, T]\times \mathbb{R} \mapsto \mathbb{R} $ with the following properties:
1) $f_t, $ $f_x, $ and $f_{xx} $ are continuous. (These symbols denote partial derivatives).
My goal is to approximate the function $f $ given, on a compact set, using a sequence of functions $\{f_n(t, x): n \in \mathbb{N}\} $ where $$ f_n(t, x) = \sum_{k=0}^n g_{n,k}(t)h_{n,k}(t) $$ and where each of the $g_{n,k}() $ is continuous and has continuous first derivative, while each of the $h_{n,k}() $ is twice continuously differentiable.
I believe that the since polynomials are in this class of $f_n(t, x), $ Weierstrass approximation theorem guarantees that on $[0, T]\times [-N, N] $ we can approximate $f $ by $f_n $ uniformly. However, I would like to show the following:
$$ \lim_n \sup_{0\le t\le T}\sup_{|x| \le N} \bigl(|f(t,x) - f_n(t, x)|\, + \bigl|\tfrac{\partial}{\partial t}f(t, x) - \tfrac{\partial}{\partial t}f_n(t, x)\bigl|\bigl) = 0; $$ $$ \lim_n \sup_{0\le t\le T}\sup_{|x| \le N}\bigl|\tfrac{\partial}{\partial x}f(t, x) - \tfrac{\partial}{\partial x}f_n(t, x)\bigl| = 0; \hskip 5pt \text{ and } $$ $$ \lim_n \sup_{0\le t\le T}\sup_{|x| \le N}\bigl|\tfrac{\partial^2}{\partial x^2}f(t, x) - \tfrac{\partial^2}{\partial x^2}f_n(t, x)\bigl| = 0. $$
My intuition is that the "smoothness" of the $g_{n,k} $ and $h_{n,k} $ plays a role, but how does one prove the uniform convergence on compact sets of the partial derivatives listed above. I do not seem to be able to make these convergences satisfactorily formal.
Any help would be greatly appreciated. Thank you.
Maurice.