Denote by $Sym_n$ the set of symmetric matrices on $\Bbb R^n$ and let $f: \Bbb R \to \Bbb R$ be a real analytic function.
Then it is well known how to associate in a real analytic manner to an element $S \in Sym_n$ a matrix $f(S) \in Sym_n$. For example one can use Cauchy's formula in a suitable manner, as pointed out on the wikipedia page https://en.wikipedia.org/wiki/Matrix_function#Cauchy_integral.
My question is: how to carry out the construction if $f:\Bbb R \to \Bbb R$ is only smooth, i.e., how to define the matrix $f(S)$ in this case?
I suspect that this should be possible, if not otherwise, then by uniformly approximating $f$ on compact intervals by (real analytic) polynomials, but I am hoping for a direct treatment without approximation.
My hope is that polished references can be quoted on this by people in the know-how, and I would much appreciate any help.
A posteriori, $f(S)$ can be constucted as follows.
Diagonalise $S$, that is choose $M$ so that $M^{-1}SM$ is diag$(x_1,\dots,x_n)=A$. Then $f(S)=Mf(A)M^{-1}$ and $f(A)=$diag$(f(x_1),\dots,f(x_n))$.
So you could use this definition in the general case.