Approximation of functions from \(C^ r\) by splines of minimal defect (Q920311)

The author gives an alternative simple proof of the inequalities for the best approximation in the class \(C^{r,\omega}\) with a periodic modulus of continuity and a Jackson's type inequality for \(C^ r\)-functions in \(L^ p\)-norms. The best approximations are considered with respect to splines of order r and of defect 1. Further, there is proved a theorem on estimates in \(L^ 1(0,x)\) of nonincreasing rearrangement of derivatives of functions by those of the ideal Euler splines and the best approximation inequality for splines of order zero in an arbitrary symmetric space.