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? Is 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?
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- $\begingroup$ Is this helpful? mathoverflow.net/questions/130684/… $\endgroup$Margaret Friedland– Margaret Friedland2024-02-21 02:34:39 +00:00Commented Feb 21, 2024 at 2:34
- $\begingroup$ @MargaretFriedland Thank you for the help. I found another formulation:, multivariate Jackson theorem takes the form . If I understand correctly, this is an estimate of the best polynomial approximation from the set of all polynomials of degree n. And your version gives an estimate for a given polynomial. There is also a difference on the right side of the inequality in how partial derivatives are taken. $\endgroup$Don– Don2024-02-21 15:05:49 +00:00Commented Feb 21, 2024 at 15:05
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