Result on the approximation of multivariable functions-generalization of a theorem of Walsh

In a theorem of Walsh, the approximation of a given function by a rational function has been considered. Walsh stated that the best approximate of a given function is one which interpolates the given function at several properly chosen points. Transfer functions of multidimensional digital filters with separable denominator are used for the approximation of given multivariate functions. It is shown that the result of Walsh can be generalized in a straightforward manner. By an example, it is illustrated how the new result can be applied, for example, to the order-reduction of a higher order system. The usefulness and the limitation of the result are discussed