  We present the Wilsonian effective action as a solution of the exact RG
equation for the critical $O(N)$ vector model in the large $N$ limit. Below
four dimensions, the exact effective action can be expressed in a closed form
as a transcendental function of two leading scaling operators with infinitely
many derivatives. From the exact solution that describes the RG flow from a UV
theory to the fixed point theory in the IR, we obtain the mapping between UV
operators and IR scaling operators. It is shown that IR scaling operators are
given by sums of infinitely many UV operators with infinitely many derivatives.
