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 $f$?
Thanks in advance.
1) No, in general, convergence does not hold because of the presence of spurious poles. A typical result, see [1], is
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*}
[1] H. Wallin, The convergence of Padé approximants and the size of the power series coefficients. Applicable Anal. 4 (1974), no. 3, 235-251.
