Convergence in Variation and Rates of Approximation for Bernstein-Type Polynomials and Singular Convolution Integrals
This paper is concerned with the Variation detracting property and the rate of approximation of the Bernstein, Kantorovitch and Szasz-Mirak'jan Operators, as well as of general singular convolution operators. These problems are studied with respect to the Variation seminorm. In the instance of the Bernstein polynomials the Variation detracting property Vio,i][ßn/l < ^o,i)[/] holds for all functions / of bounded Variation. However, Um„_,oo V\o,i][Bnf / ] = 0 is vahd if and only if f is absolutely continuous. The chief aim is the approach for multivariate convolution Integrals, not Standard in approximation, employing the theory of functions of bounded Variation in the sense of Tonelli, perfected by the work T. Rado and C. Vinti. 1991 Mathematics Subject Classification: 41A35, 41A25, 41A10, 42A10, 26B30, 26A45.