The approximation properties of generalized Bernstein polynomials of two variables
The authors generalize the 2-dimensional Bernstein polynomials along the lines of Stancu's generalization of of the ordinary Bernstein polynomials, namely, for parameters \(0\leq\alpha_k\leq\beta_k\), \(k=1,2\), and a continuous function \(f\) defined on the unit square, they denote \[ B^{\alpha_k,\beta_k}_{n,m}(f;x,y):=\sum_{i=0}^n\sum_{j=0}^mf\left(\frac{i+\alpha_1} {n+\beta_1},\frac{j+\alpha_2}{m+\beta_2}\right)\binom ni\binom mjx^i(1-x)^{n-i}y^j(1-y)^{m-j}. \] The authors prove that their generalized Bernstein polynomials converge uniformly in \([0,1]\times[0,1]\) to \(f\) and obtain some estimates on the degree of approximation involving the modulus of continuity of \(f\). The results are straightforward, moreover, considering that similar results are known for the ordinary 2-dimensional Bernstein polynomials, all that is needed is the trivial estimate of the distance between the point \(\left(\frac{i+\alpha_1} {n+\beta_1},\frac{j+\alpha_2}{m+\beta_2}\right)\) and the point \(\left(\frac in,\frac jm\right)\). The latter tends to zero uniformly in \(0\leq i\leq n\) and \(0\leq j\leq m\), as \(n,m\to\infty\). In fact, all one needs is to observe that \[ \frac{i+\alpha_1}{n+\beta_1}-\frac in=\frac{n\alpha_1+i\beta_1}{n(n+\beta_1)}\leq\frac{2\beta_1}{n+\beta_1}, \] and similarly for the other pair. The reviewer sees no point in this generalization and no benefit in the figures, tables, and obvious computer programs in the article.
scientific article; zbMATH DE number 2118022
scientific article