Hostname: page-component-7bb8b95d7b-pwrkn Total loading time: 0 Render date: 2024-09-15T03:11:20.322Z Has data issue: false hasContentIssue false
  • English
  • Français

On Hermite-Fejér type interpolation

Published online by Cambridge University Press:  17 April 2009

H.-B. Knoop  and
B. Stockenberg
H.-B. Knoop
Affiliation:
Fachbereich II – Mathematikder Universität DuisburgLotharstr. 65D–4100 DuisburgWest Germany.
B. Stockenberg
Affiliation:
Fachbereich II – Mathematikder Universität DuisburgLotharstr. 65D–4100 DuisburgWest Germany.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For the Hermite-Fejér interpolation operator of higher order constructed on the roots , 1 ≤ k ≤ m, of the Jacobi-polynomial it is shown that is positive for all m ∈ N, if (α, β) ∈ [−¾, −¼]2. Further there is given an bound, which implies for arbitrary f ∈ C(I) and (α, β) ∈ [−¾, −¼]2.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 28 , Issue 1 , August 1983 , pp. 39 - 51
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]De Vore, Ronald A., The approximation of continuous functions by positive linear operators (Lecture Notes in Mathematics, 293. Springer-Verlag, Berlin, Heidelberg, New York, 1972).CrossRefGoogle Scholar
[2]Florica, Olariu, “Asupra ordinului de aproximatie prin polinoame de interpolare de tip Hermite-Fejér cu noduri cvadruple” [On the order of approximation by interpolating polynomials of Hermite-Fejér type with quadruple nodes], An. Univ. Timişoara Ser. Şti. Mat.-Fiz. 3 (1965), 227–234.Google Scholar
[3]Gonska, Heinz Herbert, “Quantitative Aussagen zur Approximation durch positive lineare Operatoren” (Dissertation, Universität Duisburg, 1979).Google Scholar
[4]Gonska, Heinz H., “A note on pointwise approximation by Hermite-Fejér type interpolation polynomials”, Functions, series operators (Proc. Conf. Budapest, Hungary, 1980. North-Holland, Groningen, to appear).Google Scholar
[5]Goodenough, S.J. and Mills, T.M., “The asymptotic behaviour of certain interpolation polynomials”, J. Approx. Theory 28 (1980), 309–316.CrossRefGoogle Scholar
[6]Haussmann, Werner und Knoop, Hans-Bernd, “Konvergenzordnung einer Folge positiver linearer Operatoren”, Rev. Anal. Numér. Théor. Approx. 4 (1975), 123–130.Google Scholar
[7]Knoop, H.-B., “Eine Folge positiver Interpolationsoperatoren”, Acta Math. Acad. Sci. Hungar. 27 (1976), 263–265.CrossRefGoogle Scholar
[8]Kryloff, N.M. and Stayermann, E., “Sur quelques formules d'interpolation convergentes pour toute fonction continue”, Bull. Acad. de l'Oucraine 1 (1923), 13–16.Google Scholar
[9]Laden, H.N., “An application of the classical orthogonal polynomials to the theory of interpolation”, Duke Math. J. 8 (1941), 591–610.CrossRefGoogle Scholar
[10]Mills, T.M., “On interpolation polynomials of the Hermite-Fejér type”, Colloq. Math. 35 (1976), 159–163.CrossRefGoogle Scholar
[11]Prasad, J., “On the rate of convergence of interpolation polynomials of Hermite-Fejér type”, Bull. Austral. Math. Soc. 19 (1978), 29–37.CrossRefGoogle Scholar
[12]Sharma, A. and Tzimbalario, J., “Quasi-Hermite-Fejer type interpolation of higher order”, J. Approx. Theory 13 (1975), 431–442.CrossRefGoogle Scholar
[13]Stancu, D.D., “Asupra unei demonstratii a teoremei lui Weierstrass” [On a proof of the theorem of Weierstrass], Bul. Inst. Politehn. Iaşi (N.S.) 5 (9)(1959), 47–50.Google Scholar
[14]Szász, Paul, “On a sum concerning the zeros of the Jacobi polynomials with application to the theory of generalized quasi-step parabolas”, Monatsh. Math. 68 (1964), 167–174.CrossRefGoogle Scholar
[15]Szegö, Gäbor, Orthogonal polynomials (American Mathematical Society Colloquium Publications, 23. American Mathematical Society, Providence, Rhode Island, 1975).Google Scholar