# Rate of convergence of Padé approximants

Let $$f$$ be an entire function of order $$1$$. Two questions:

1) Can one assert that the diagonal Padé approximants converge to $$f$$ (pointwise or uniformly over compacts of $$\mathbb C$$)?

2) if yes, can one estimate $$|P_n(x)f(x)-Q_n(x)|$$ in function of $$n$$ and $$x$$ (and $$f$$ of course), where $$(P_n,Q_n)$$ is the $$[n,n]$$-Padé approximants of $$f$$?

Theorem. Let $$(n_{\nu})_{1}^{\infty}$$ be a sequence of positive integers satisfying $$n_{\nu}>2 n_{\nu-1}$$ for all $$\nu$$. Then there exists an entire function $$f(z)=\sum_{i=0}^{\infty} a_{i} z^{i}$$ with $$a_{i}$$ tending to 0 arbitrarily fast, so that the $$\left(n_{v}, n_{v}\right)$$ Padé approximant to $$f$$ has a pole at any prescribed point $$b_{\nu} \neq 0$$ of the complex plane, for $$\nu=1,2, \ldots$$
2) The exponential function is an entire function for which local uniform convergence of the diagonal Padé approximants $$R_{n,n}$$ is known (this is due to Padé). The rate of convergence at a point $$z=x+iy\in\mathbb{C}$$ is given by $$\begin{equation*} |e^z-R_{n,n}(z)|=\frac{(n!)^{2}}{(2n)!(2n+1)!}|z|^{2n+1} e^{x}(1+o(1)). \end{equation*}$$