Approximation of some classes of functions of many variables by harmonic splines

By C./ we denote the space of continuous functions uW! R with the norm kukC./ D maxfju.x/jW x 2 g: Spline functions play an important role in the theory of approximation of functions of one and many variables. Piecewise linear functions (broken-line functions in the case of functions of one variable) serve as an example of 1 Dnepropetrovsk National University, Dnepropetrovsk, Ukraine. 2 Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, Donetsk, Ukraine. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 8, pp. 1011–1024, August, 2012. Original article submitted April 14, 2011.