A Refinement of Kolmogorov's Inequality.

摘要

For any n-times differentiable function f with uniform bounds on f and the nth derivative of f, the pair of values the (jth derivative of f(t), the (j + 1)th derivative of f(t)) are studied for an arbitrary real t and a prescribed j = 0,...,n-1. A given value of the jth derivative of f(t) determines admissible values for the (j + 1)th derivative of f(t). These values are exactly determined in terms of the Euler spline E sub n(t). Special differentiation formulas of cardinal interpolation type are developed to solve the problem.