Multivariate \(L_ p\)-error estimates for positive linear operators via the first-order \(\tau\)-modulus (Q1120767)

Let I be an m-dimensional compact interval and let M(I) be the set of all real-valued, bounded, measurable functions on I. Let \(L: M(I)\to M(I)\) be a multivariate positive operator. The author gives estimates for \(\| f-Lf\|_ p\) (the \(L_ p\)-norm) using the so-called averaged modulus of smoothness or \(\tau\)-modulus of first order. The result is quite similar to the univariate one, but the techniques used (interpolation results of Riesz-Thorin type for the \(\tau\)-modulus) are completely different.