Favard's interpolation problem in one or more variables
The author discusses the interpolation of function values on discrete sets of data points, and the extent to which the derivatives of an interpolant can be bounded above in absolute value by a constant multiple of its necessary size, as determined by the divided differences of the data. The starting point of such investigations is a construction of \textit{J. Favard} [Sur l'interpolation. J. Math. Pures Appl., IX. Ser. 19, 281-306 (1940; Zbl 0026.01201)] that gives a complete answer to this question in the univariate case. First, the optimality of Favard's result is investigated in the article. Some improvement of the Favard's results is obtained with just a few minor changes to Favard's proof. It is shown that the \(\left(n-1\right)\)-th derivative of Favard's interpolant can be likewise bounded by divided differences, and that one can bound at best two consecutive derivatives of any interpolant by the corrseponding divided differences. Then the author considers the multidimensional problems. An extension of the Favard's result to the interpolation on lattices is discussed (the author has it proved in [\textit{Th. Kunkle}, J. Approximation Theory 71, No.~1, 94-103 (1992; Zbl 0767.41007)]). The main multidimensional results concern the interpolation on tensor product grids. Some useful conditions for the interpolants are obtained. The results obtained may be very interesting for the specialists in approximation theory as well as in numerical analysis.
scientific article; zbMATH DE number 1885731
scientific article