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Convergence in variation and rates of approximation for Bernstein-type polynomials and singular convolution integrals. (English) Zbl 1049.41015

The article gives a long survey of known and new results in the approximation of functions of bounded variation by the Bernstein polynomials, by the Szász-Mirakjan operators, and by convolution operators associated with approximate identity kernels in the periodic case. The results include the variation detraction property of the operators, characterization of the class of functions for which the operators converge in variation to, and estimates on the degree of approximation in the total-variation seminorm. The second part of the paper is devoted to the investigation of the multivariate case of convolution operators associated with approximate identity kernels, as operators on the space of functions of bounded variation in the sense of Tonelli, in \(\mathbb{R}^n\). The authors extend many of the one dimensional results into this setting. I recommend that the paper be read by every person working on positive approximation operators.

MSC:

41A35 Approximation by operators (in particular, by integral operators)
41A25 Rate of convergence, degree of approximation
41A36 Approximation by positive operators
41A10 Approximation by polynomials
42A10 Trigonometric approximation
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
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