×

A two-dimensional analog of the effect of approximation improvement near the ends of an interval. (English. Russian original) Zbl 0862.41025

Vestn. St. Petersbg. Univ., Math. 27, No. 3, 44-49 (1994); translation from Vestn. St-Peterbg. Univ., Ser. I 1994, No. 3, 54-59 (1994).
Let \(\Lambda \subset \mathbb{R}^2\) be compact and \(f\in C (\Lambda)\). Theorem. There is a sequence \(\{Q_{n_1,n_2} (x_1,x_2) = \sum^{n_1}_{k=0} a_k(x_1) x_2^k + \sum^{n_2}_{k =0} b_k (x_2) x_1^k\}\) of continuous on \(\Lambda\) functions such that \(|f(x)- Q_{n_1, n_2} (x) |\leq A \omega (f;\min \{{1\over |n|}, \gamma (x)\})\) where \(A\) and \(\gamma \in C(A)\) are independent of \(f\) and \(\gamma |_{\partial \Lambda}= 0\). In the case of \(\Lambda = \{(x_1,x_2): x_1^2+ x^2_2\leq 1\}\) it is possible to take \(\gamma (x) = [\text{dist} (x, \partial D)]^{2/3}.\)

MSC:

41A63 Multidimensional problems
PDFBibTeX XMLCite