A Refinement of Kolmogorov's Inequality.

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 ft, the j 1th derivative of ft are studied for an arbitrary real t and a prescribed j 0,...,n-1. A given value of the jth derivative of ft determines admissible values for the j 1th derivative of ft. These values are exactly determined in terms of the Euler spline E sub nt. Special differentiation formulas of cardinal interpolation type are developed to solve the problem.