Nonlinear Padé approximants for Legendre series

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Abstract
We introduce a new kind of Padé approximants for Legendre series based on the solution of a nonlinear system of equations. These Padé approximants have many properties in common with the well‐known Padé approximants for Taylor series. In the examples studied here, the poles and zeroes of the Padé approximants lie on the cut and separate each other‐a property which one expects to hold in general for Padé approximants. A proof of convergence follows the same lines as for Taylor series. Moreover, it turned out that these nonlinear approximants converge more rapidly than the linear approximants introduced in an earlier work. They may become a powerful tool in the summation of Legendre series.