The paper has a methodical content and is addressed to young researchers. Its main goal is to prove how the property of monotonicity can be transferred from the sequence of univariate Bernstein polynomials to those of bivariate Bernstein polynomials.

Let \mathbb{N} be the set of positive integers, m,n\in\mathbb{N}, I=[0,1], I^2=[0,1]\times[0,1], \mathbb {R}^{I^2}=\{f|f:I^2\to\mathbb{R}\}, C(I^2)=\{f\in\mathbb {R}^{I^2}|f continuous on I^2\}. Denote by B_{m,n}:C(I^2)\to C(I^2) the Bernstein bivariate operator. This operator associates to each function f\in C(I^2) the bivariate Bernstein polynomial B_{m,n}(f;x,y). It is well known that the sequence \{B_{m,n}(f; x,y)\}_{m,n\in\mathbb { N}} converges to f, uniformly on I^2 for each f\in C(I^2).

In the present paper one investigates the monotonicity of the sequence \{B_{m,n}(f; x,y)\}_{m,n\in\mathbb{N}}. One proves that if f\in C(I^2) is convex of (1,1)-order on I^2 the sequence \{B_{m,n}(f; x,y)\}_{m,n\in\mathbb {N}} is monotonous decreasing and B_{{m,n}}(f; x,y)\geq f(x,y), (\forall)\, (x,y)\in I^2.

Additional Information


Bărbosu, Dan