ON SHARP KOLMOGOROV-TYPE INEQUALITIES TAKING INTO ACCOUNT THE NUMBER OF SIGN CHANGES OF DERIVATIVES

We obtain new sharp Kolmogorov-type inequalities, in particular the following sharp inequality for 2 pi-periodic functions x is an element of L(infinity)(T)(T): parallel to x((k))parallel to(1) <= (nu(x')/2)((1-1/p)alpha)parallel to phi(r -k)parallel to(1)/parallel to phi(r)parallel to(alpha)(p)parallel to x parallel to(alpha)(p)parallel to x((r))parallel to(1-alpha)(infinity), where k, is an element of N, k < r, r >= 3, p is an element of [1, infinity], alpha = (r - k)/(r - 1+ 1/p), phi(r) is the perfect Euler spline of order r, and nu(x') is the number of sign changes of x' on a period.