The periodic analog of Rolle's theorem for differential operators and approximation by L-splines

Abstract

Rolle's theorem, which expresses the relationship between the number of zeros of a differentiable function and the number :of:z~ros in its derivative, plays an important role in solution of problems in diverse areas of mathematics. It is of interest to';~"~tend Rolle's theorem to the case in which the derivative is replaced by a linear differential operator. To a considerable extent, interest in this problem results from the fact that for classes of smooth function defined by differential operators for which the analog of Rolle's theorem holds, many problems are solved in the theory of approximations. These include problems on diameters, quadrature formulas, problems in the theory of splines, and numerous others (see, for example, [1][2][3][4][5][6][7], and their bibliographies).In the first part of this paper, we prove the periodic analog of Rolle's theorem for linear differential operators satisfying certain additional factorizability constraints. We then use this result in the second part of the paper to find exact values for the upper bounds of the slopes of periodic interpolation L-splines. Let T denote the segment [0,27r], with its endpoints identified: for E = T and E = [a, b](-oo < a < b < + oo),