There is an approximation of a multivariate function by a Chebyshev polynomial of degree n. One needs to understand how the approximation error behaves depending on the degree of the polynomial or express the degree of approximation in terms of the error. There is an understanding of how to estimate the approximation error using Jackson's inequality, Markov's or Bernstein's inequality for one variable. How to estimate these errors in a multivariate case? Does Jackson's inequality applicable to non-trigonometric series? What are the formulas/theorems for estimating the error in the approximation of functions of several variables?
