Suppose we have function $f$ such that;
$$f(t)=a(t)\cdot x(t)$$
And I want an approximation like below;
$$f(t) = a_0\cdot x(t) + a_1\cdot x'(t) + a_2\cdot x''(t) + \cdots + a_n\cdot x^{(n)}$$
$n$ is integer that can be finite or infinite it does not matter for me.
If both equations are true, what is the relation with $a_n$ and $a(t)$ such that $a_n$ is not dependent on $t$.
I tried to google it but i couldn't express myself :(
Edit: $a(t)$ and $x(t)$ are continuous, infinitely diffetentable and $t$ is positive real number.
Edit2: The reason why I want to find it out is that to solve the equation below.
$$x'(t) = a(t)\cdot x(t) + b(t)$$
If, somehow, I can approximate $a(t)\cdot x(t)$ , the system can be solved as non homogenous linear differential equation.
So much depends upon the nature of $a(t)$ and $x(t)$ that I do not think a meaningful answer is possible. When asking a very general question such as this, try a simple example.
Let $a(t)=t$ and, since we presumably would like lots of derivatives, let $x(t)=\sin(t)$. So we want some fixed values $a_0,a_1,a_2,\cdots$ such that we can say
\begin{align} t\sin(t)&\approx a_0\sin(t)+a_1\cos(t)-a_2\sin(t)-a_3\cos(t)+\cdots\\ &=\sin(t)\sum_{k=0}(-1)^ka_{2k}+\cos(t)\sum_{k=0}(-1)^ka_{2k+1} \end{align}
So certainly the sums must converge and we find we must approximate an unbounded function by a bounded function. Possibly this could be done over some limited domain for this particular example, but no mention was made of such limits in the question.
Complications arising from simple examples should convince one of the necessity of further circumscribing the problem.