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Approximation of unbounded functions by linear positive operators. (English) Zbl 0860.41024

The authors define a unified class of linear positive operators. Let \[ L_{n,c}(f,x)=\sum^\infty_{a=0} p_{n,k}(c,x)f\Biggl({k\over n}\Biggr), p_{n,k}(c,x)=e^{-nx}{(nx)^k\over k!}\quad\text{for }c=0\text{ and} \]
\[ p_{n,k}(c,x)= {k+n/c-1\choose k} {(cn)^k\over (1+cx)^{n/c+k}}\quad\text{for } c\neq 0. \] The cases \(c=-1\), \(c=0\), and \(c=1\) give the well-known operators of Bernstein, Szász-Miraktsan and Lupas, respectively. The first author [Proc. R. Ir. Acad., Sect. 89 A, No. 1, 75-77 (1989; Zbl 0661.41012); Rev. Roum. Math. Pures Appl. 37, No. 6, 491-498 (1992; Zbl 0776.41007), Acta Math. Hung. 61, No. 3-4, 281-288 (1993; Zbl 0794.41014)] and many others have studied the approximation properties of these operators. The authors of this paper obtain some approximation estimates for unbounded functions defined on \([0,\infty)\) and \(c\geq 0\). They establish the convergence of \(L_{n,c}(f,x)\) to \(\{f(x+)+ f(x-)\}/2\) under the assumption that \(f\) is exponentially bounded and both left and right limits exist. For the particular case \(c=0\) their result sharpens and improves the earlier estimates due to F. Cheng [J. Approximation Theory 40, 226-241 (1984; Zbl 0532.41026)] and X. Sun [J. Approximation Theory 55, No. 3, 279-288 (1988; Zbl 0697.41009)]. Finally, they give an estimate for continuous functions of exponential growth involving higher order modulus of continuity on any compact subinterval of \(g(0,\infty)\).

MSC:

41A36 Approximation by positive operators
41A25 Rate of convergence, degree of approximation
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References:

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