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Approximation by linear combination of Szász-Mirakian operators. (English) Zbl 0941.41008

The paper considers approximation of functions by linear combinations of generalized Bernstein polynomials.
Using \[ S_n(f;x)=\sum_{\nu=0}^\infty e^{-nx} {(nx)^\nu\over \nu !} f\left({ \nu \over n}\right) \] and \[ S_n(f,k,x) =\sum_{j=0}^k\Biggl(\prod_{\substack{ i=0\\ i\not= j}} ^k {d_j\over d_j-d_i}\Biggr) S_{d_jn}(f; x), \] where \(d_0,\ldots , d_k\) are arbitrary fixed positive integers, bounded functions and their derivatives can be approximated well. The approach improves the estimates of S. P. Singh [Bull. Aust. Math. Soc., II. Ser. 24, 221-225 (1981; Zbl 0511.41027)] and X. Sun [J. Approximation Theory 55, No. 3, 279-288 (1988; Zbl 0697.41009)].

MSC:

41A25 Rate of convergence, degree of approximation
41A28 Simultaneous approximation
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