Padé approximants for entire functions with regularly decreasing Taylor coefficients

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For a class of entire functions the asymptotic behaviour of the Hadamard determinants as and 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 in certain classes of entire functions (with regular Taylor coefficients) the Padé approximants , which provide the locally best possible rational approximations, converge to the given function uniformly on the compact set with asymptotically best rate.