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On a generalization of Bernstein-Chlodovsky polynomials for two variables. (English) Zbl 1116.41026

Summary: It is studied the following generalization of Bernstein-Chlodovsky polynomials \[ B_{n,m}(f;x,y)=\sum^n_{k=0}\sum^m_{j=0}f\left(\frac kn b_n,\frac jm b_m\right) \varphi_n^k\left(\frac{x}{b_n}\right) \varphi_j^m\left(\frac{y}{b_n}\right) \] where \(0\leq x\leq b_n\), \(0\leq y \leq b_m\) and \((b_n)\) is the sequence of real numbers and increasing which satisfies \(\lim_{n\to\infty}b_n=\infty\), \(\lim_{n \to\infty}=0\) and \(\varphi_n^k(t)={n\choose k}t^k(1-t)^{n-k}\). It may be also seen that \(B_{n,m}(f;x,y)\) is linear positive operator [D. D. Stancu, Stud. Univ. Babes-Bolyai, Math.-Phys. 14, No. 2, 31–44 (1969; Zbl 0205.36602)]. A theorem for convergence of \(B_{n,m}(f;x,y)\) to \(f(x,y)\) as \(n,m\to\infty\) in the space of continuous function on semi axes satisfying \(|f(x,y)|\leq C_f(1+ x^2+y^2)\) is established. And it is discussed the rate of approximation. Also if \(f\) has continuous partial derivatives we give a theorem for approximating properties of this operator \(B_{n,m}(f;x,y)\) and we give some examples.

MSC:

41A36 Approximation by positive operators
41A27 Inverse theorems in approximation theory

Citations:

Zbl 0205.36602
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