Padé approximants for entire functions with regularly decreasing Taylor coefficients

For a class of entire functions the asymptotic behaviour of the Hadamard determinants $D_{n,m}$ as $0\leqslant m\leqslant m(n)\to\infty$ and $n\to\infty$ is described. This enables one to study the behaviour of parabolic sequences from Padé and Chebyshev tables for many individual entire functions. The central result of the paper is as follows: for some sequences $\{(n,m(n))\}$ in certain classes of entire functions (with regular Taylor coefficients) the Padé approximants $\{\pi_{n,m(n)}\}$, which provide the locally best possible rational approximations, converge to the given function uniformly on the compact set $D=\{z:|z|\leqslant 1\}$ with asymptotically best rate.