A note on integral modification of the Meyer-König and Zeller operators. (English) Zbl 1023.41014

Gupta, Vijay; Kumar, Niraj

Int. J. Math. Math. Sci. 2003, No. 31, 2003-2009 (2003).

The Meyer-König and Zeller operators \(M_n (n\in\mathbb{N})\) are defined by \[ M_n(f,x)= \sum^\infty_{k=0} p_{n,k}(x)f ({k\over n+k}),\;x\in [0,1] \tag{1} \] and \[ p_{n,k}(x)= {n+k-1\choose k} x^k(1-x)^n.\tag{2} \] The integral modifications of operators (1) were given by S. S. Guo (see S. S. Guo [Approximation Theory Appl. 4, 9-18; Zbl 0686.41016)]) as follows \[ \widehat M_n(f,x) =\sum^\infty_{k=1} p_{n,k+1}(x) {(n+k-2) (n+k-3)\over n-2}\int^1_0 p_{n-2,k-1} (t)f(t)dt\tag{3} \] where \(p_{n,k}(x)\) is defined in (2). In their note, the authors give an improved estimate for the rate of convergence of functions of bounded variation for the operators (3). This result is the following theorem. Let \(f\) be a function of bounded variation on \([0,1]\). Then, for every \(x\in (0,1)\) and \(n\) sufficiently large, \[ \begin{split}\biggl|\widehat M_n(f,x)- {1\over 2}\bigl\{f(x^+) +f(x^-)\bigr\} \biggr|\leq{7\over nx} \sum^n_{k=1} V^{x+(1-x)/ \sqrt k}_{x-x/ \sqrt k}(g_x)+ \left(3+{1 \over \sqrt 8e}\right) {1\over\sqrt nx^{3/2}}\\ \bigl|f(x^+)-f(x^-)\bigr |, \end{split} \] where \(V^b_a(g_x)\) is the total variation of \(g_x\) on \([a,b]\) and \[ g_x(t)= \begin{cases} f(t)-f(x^-), \quad & 0\leq <x\\ 0,\quad & t=x\\ f(t)-f(x^+), \quad & x<t\leq 1.\end{cases}. \]

Reviewer: Zoltán Finta (Cluj-Napoca)

MSC:

41A30	Approximation by other special function classes
41A36	Approximation by positive operators
41A25	Rate of convergence, degree of approximation

Keywords:

Meyer-König operators; function of bounded variation; rate of convergence; Zeller operators