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Modified Szász operators. (English) Zbl 0717.41041

Mathematical analysis and its applications, Proc. Int. Conf., Safat/Kuwait 1985, KFAS Proc. Ser. 3, 29-41 (1988).
[For the entire collection see Zbl 0698.00024.]
S. P. Singh and O. P. Varshney [Rend. Mat. Appl., VII. Ser. 2, 565-571 (1982; Zbl 0512.41019)] defined a sequence of Szász-type operators which maps the space of bounded continuous functions into itself as \[ (S_{n,x}f)(t)=\sum^{\infty}_{k=0}p_{n,k}(t)f(x+(k/n)), \] where \(p_{n,k}(t)=e^{-nt}((nt)^ k/k!)\) and \(x\in [0,\infty)\) is fixed, and proved some approximation properties. We propose a sequence of modified Szász operators defined on the space of integrable functions on \([0,\infty)\) as \[ (M_{n,x}f)(t)\equiv M_{n,x}(f(y);t)=n\sum^{\infty}_{k=0}p_{n,k}(t)\int^{\infty}_{0 }p_{n,k}(y)f(x+y)dy, \] where t,x\(\in [0,\infty)\) and x is fixed. Clearly, \((M_{n,x}f)(t)\) is a linear positive operator.

MSC:

41A36 Approximation by positive operators
44A15 Special integral transforms (Legendre, Hilbert, etc.)