Budakçı, Gülter; Oruç, Halil Bernstein-Schoenberg operator with knots at the \(p\)-integers. (English) Zbl 1255.41003 Math. Comput. Modelling 56, No. 3-4, 56-59 (2012). Summary: We consider a special knot sequence \(u_{i+1}=qu_{i}+1\) and define a one parameter family of Bernstein-Schoenberg operators. We prove that this operator converges to \(f\) uniformly for all \(f\) in \(C[0,1]\). This operator also inherits the geometric properties of the classical Bernstein-Schoenberg operator. Moreover we show that the error function \(E_{m,n}\) has a particular symmetry property, that is \(E_{m,n}(f;x;q)=E_{m,n}(f;1 - x,1/q)\) provided that \(f\) is symmetric on \([0,1]\). Cited in 1 Document MSC: 41A35 Approximation by operators (in particular, by integral operators) 41A15 Spline approximation Keywords:Bernstein-Schoenberg operator; B-spline; Marsden’s identity; \(q\)-integer PDF BibTeX XML Cite \textit{G. Budakçı} and \textit{H. Oruç}, Math. Comput. Modelling 56, No. 3--4, 56--59 (2012; Zbl 1255.41003) Full Text: DOI References: [1] Schoenberg, I.J., On spline functions, (), 255-291 · Zbl 0147.32101 [2] de Boor, C., On calculating with \(B\)-splines, Journal of approximation theory, 6, 1, 50-62, (1972) · Zbl 0239.41006 [3] Goodman, T.N.T., Total positivity and the shape of curves, (), 157-186 · Zbl 0894.68159 [4] Goodman, T.N.T.; Sharma, A., A property of bernstein – schoenberg spline operators, Proceedings of the Edinburgh mathematical society, 28, 333-340, (1985) · Zbl 0596.41023 [5] Marsden, M., An identity for spline functions with applications to variation-diminishing spline approximation, Journal of approximation theory, 3, 7-49, (1970) · Zbl 0192.42103 [6] Marsden, M.; Schoenberg, I.J., On variation diminishing spline approximation methods, Mathematica, 8, 31, 61-82, (1966) · Zbl 0171.31001 [7] Phillips, G.M., Interpolation and approximation by polynomials, (2003), Springer-Verlag New York · Zbl 1023.41002 [8] Koçak, Z.; Phillips, G.M., \(B\)-splines with geometric knot spacing, BIT numerical mathematics, 34, 388-399, (1994) · Zbl 0815.41010 [9] Phillips, G.M., Bernstein polynomials based on the \(q\)-integers, Annals of numerical mathematics, 4, 511-518, (1997) · Zbl 0881.41008 [10] Phillips, G.M., Survey of results on the \(q\)-Bernstein polynomials, IMA journal of numerical analysis, 30, 1, 277-288, (2010) · Zbl 1191.41002 [11] Oruç, H.; Phillips, G.M., \(q\)-Bernstein polynomials and Bèzier curves, Journal of computational and applied mathematics, 151, 1-12, (2003) · Zbl 1014.65015 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.