Quadrature formulas constructed by using certain linear positive operators

4 years ago

Abstract

Let [a,b] be a compact interval of the real line IR. It is known that the classical theorem of Bohman-Korovkin states that in order that a sequence of positive linear operators (L_m), mapping into itself the space C[a,b] of continuous real-valued functions on [a,b], equipped with the uniform norm, to have the property that, for any f∈C[a,b], if m → ∞ we have lim L_mf = f, uniformly on [a,b], it is necessary and sufficient that such a convergence occur for a triplet of “test functions” from C[a,b], forming a so called Korovkin system. For C[a,b] the three monomials e₀,e₁,e₂ where e_j(x) := x^j (j = 0,1,2), represent such a system.

Authors

Dimitrie D. Stancu
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania

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D.D. Stancu, Quadrature formulas constructed by using certain linear positive operators, Numerical Integration (Proc. Conf. Mth. Res. Inst. Oberwolfach, 1981; ed. G. Hämmerlin; ISNM 57), Birkhäuser, Basel, 1982, 241–251, https://doi.org/10.1007/978-3-0348-6308-7_23

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1982 – Stancu – Quadrature formulas constructe dby using certain linear positive operators

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Springer

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Birkhäuser, Basel

DOI

https://doi.org/10.1007/978-3-0348-6308-7_23

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[1] BRASS, H., Eine Verallgemeinerung der Bernsteinschen Operatoren. Abhandl. Math. Sem. Hamburg, 36 (1971),111–122. CrossRef Google Scholar
[2] CHENEY, E.W., A. Sharma, Bernstein power series. Canad. J. Math., 16 (1964), 241–252. CrossRef Google Scholar
[3] MOLDOVAN, G., Discrete convolutions and linear positive operators. Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 15 (1972), 31–44. Google Scholar
[4] POPOVICIU, T., Sur le reste dans certaines formules linéaires d’approximation de l’analyse. Mathematica (Cluj), 1(24) (1959), 95–142. Google Scholar
[5] STANCU, D.D., Approximation of functions by a new class of linear polynomial operators. Rev. Roum. Math. Pures Appl. 13 (1968), 1173–1194. Google Scholar
[6] STANCU, D.D., On a generalization of the Bernstein polynomials. Studia Univ. Babes-Bolyai, Ser. Math.-Phys., 14 (1969), 31–45 (Rumanian). Google Scholar
[7] STANCU, D.D., Use of linear interpolation for constructing a class of Bernstein polynomials. Studii Cercet. Matem. (Bucharest), 28 (1976), 369–379 (Rumanian). Google Scholar
[8] STANCU, D.D., Approximation of functions by means of a new generalized Bernstein operator. (To appear).