Integral representations of the error and asymptotic error bounds for generalized Padé type approximants (Q674430)

Let \(f\) be an analytic function defined on a domain \(A \subset {\mathbb C}\) by its expansion in series of functions \(\{g_i(t) \}\), whose generating function is \(G(x,t)= \sum_{i \geq 0} x^i g_i(t)\). Given \(n\) linear functionals \(L_i\) acting on the variable \(x\), and a non-negative integer \(k\), the generalized Padé approximant of order \((k/n)\) is the expression of the form \[ (k/n)_f^G(t)=\sum_{i=0}^k b_ig_i(t) + \sum_{i=0}^n a_i L_i(G(x,t)) , \] for which the expansion in the series of \(\{ g_i(t) \}\) coincides with that of \(f(t)\) up to the order at least \(k+n+1\). For the multipoint approximants (\(L_i(g)=g(x_i)\)), an integral representation of the error is obtained and upper bounds for the sequences \((0/n)_f^G\) are found. This allows to establish convergence results for different generating functions \(G(x,t)\), assuming an asymptotic distribution of interpolating points. Finally, this is extended to all columns \((k/n)_f^G\) (\(n \to \infty\) and \(k\) fixed) of the Padé table.