Weighted approximation by Bernstein-Chlodowsky polynomials
The authors study weighted polynomial approximation of continuous functions on the real line by the so-called Bernstein-Chlodowsky polynomials \(\widetilde B_n\). If \(B_n\) denotes the Bernstein polynomial of degree \(n\), the polynomials \(\widetilde B_n\) are given by \(\widetilde B_n(f,x):=B_n(f_n,{x\over {b_n}})\), where \((b_n)\) is a sequence of positive numbers such that \(b_n\to\infty\), \(b_n/n\to 0\) for \(n\to\infty\) and where \(f_n(t):=f(b_n t)\). The main result states that \[ {1\over\sqrt{b_n}}\sup_{0\leq x\leq b_n}{|\widetilde B_n(f,x)-f(x)|\over 1+x^2}\to 0\quad (n\to\infty) \] for every continuous function \(f\) on \([0,\infty)\) which satisfies \(| f(x)|\leq M_f(1+x^2)\) for some constant \(M_f\).
scientific article; zbMATH DE number 1275953