Is there any upper bound on $L_2$-norm of a convergent power series (in R), in terms of coefficients? I have $f(x) = a_0+a_1\frac{x}{1!}+a_2\frac{x^2}{2!}+a_3\frac{x^3}{3!}+...$.
I need something like: $$ \|f\|^2_{L_2({\mathbb R})}\leq r(a_0, a_1, a_2,\cdots) $$ where $r(a_0, a_1, a_2,\cdots) = \sum_{ij}c_{ij}a^\ast_i a_j$ is some infinite positive definite quadratic form.