Sharp estimates for functions of class W̊2 r(- 1,1)

A study was conducted to investigate sharp estimates for functions of class Sobolev spaces W̊r 2(-1, 1). The class Sobolev spaces consisted of all functions f : [-1, 1] ? R having absolutely continuous derivatives up to order r - 1. Calculation of exact constants in Kolmogorov-type inequalities for intermediate derivatives in various cases was considered in monographs. The zero boundary conditions on f are automatically satisfied at the left end, while at the right end (x = +1), they were equivalent to the family of orthogonality conditions.