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Aug 29, 2022 at 14:22 comment added Peter O. Indeed, $|B_{2n}(f) - B_n(f)| \le 2M/n$ whenever $|B_n(f) - f| \le M/n$ (Ditzian and May, A Saturation Result for Combinations of Bernstein Polynomials, Tôhoku Math. J. 28 [1976]), answering question 2.
Aug 29, 2022 at 14:08 comment added Peter O. Thus, the answers to 1, 3, and 4 are "no". I thought I found a rate of $O(1/n^2)$, but I had written the numerical code for it wrong.
Aug 29, 2022 at 14:07 vote accept Peter O.
Aug 29, 2022 at 11:02 history answered Giorgio Metafune CC BY-SA 4.0