Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T15:04:23.939Z Has data issue: false hasContentIssue false
  • English
  • Français

On almost-Hermite-Fejér-interpolation: pointwise estimates

Published online by Cambridge University Press:  17 April 2009

Heinz H. Gonska
Heinz H. Gonska
Affiliation:
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12181, USA Department of Mathematics, University of Duisburg, D-4100 Duisburg 1, West 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.

We give a brief survey of the results obtained by numerous authors in so-called almost-Hermite-Fejér-interpolation and deal mainly with new quantitative assertions.

These are based upon more general theorems for certain continuous linear operators which yield estimates involving different types of moduli of continuity.

Our paper shows that in the case of almost-Hermite-Fejér-interpolation the underlying general technique can be used to treat three essentially different cases: sequences of positive operators, which converge uniformly for every continuous function on [−1, 1], sequences of non-positive operators doing the same, and sequences of operators which converge on proper subspaces of C[−1, 1] only.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 25 , Issue 3 , June 1982 , pp. 405 - 423
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Berman, D.L., “Convergence of the possible extensions of the Hermite-Fejér interpolation process”, Soviet Math. (Iz. VUZ) 19 (1975), no. 8, 79–82.Google Scholar
[2]Berman, D.L., “Investigation of the convergence of all possible extensions of the Hermite-Fejer interpolation process construct constructed on the Chebyshev nodes of the second kind”, Soviet Math. (Is. VUZ) 21 (1977), no. 1, 106–108.Google Scholar
[3]Fejér, Leopold, “Ueber Interpolation”, Nachr. Ges. Wissensch. Göttingen (1916), 66–91.Google Scholar
[4]Gonska, Heinz Herbert, “Quantitative Aussagen zur Approximation durch positive lineare Operatoren” (Dissertation, Universitát Duisburg, 1979).Google Scholar
[5]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
[6]Gonska, Heinz H., “On quasi-Hermite-Fejér interpolation: pointwise estimates”, submitted.Google Scholar
[7]Gonska, Heinz H. and Meier, Jutta, “On approximation by Bernstein type operators: best constants”, submitted.Google Scholar
[8]Goodenough, S.T. and Mills, T.M., “A new estimate for the approximation of functions by Hermite-Fejér interpolation polynomials”, J. Approx. Theory 31 (1981), 253–260.CrossRefGoogle Scholar
[9]Knoop, Hans-Bernd, “Hermite-Fejér-Interpolation mit Randbedingungen” (Habilitationsschrift, Universität Duisburg, 1981).Google Scholar
[10]Kumar, V. and Mathur, K.K., “On the rapidity of convergence of a quasi Hermite-Fejér interpolation polynomial”, Studia Sci. Math. Hunyar. 9 (1974), 313–319 (1975).Google Scholar
[11]Mills, T.M., “Some techniques in approximation theory”, Math. Sci. 5 (1980), 105–120.Google Scholar
[12]Минькова, P.M. [Min'kova, R.M.], “Сходимость производных линейных операторов” [Convergence of the derivatives of linear operators], Izv. Vysš. Učebn. Zaved Matematika (1976), no. 8, 52–59.Google ScholarPubMed
[13]Popoviciu, Tiberiu, “Asupra demonstratiei teoremei lui Weierstrass cu ajutorul polinoamelor de interpolare”, Lucraˇrile sesiunii generate ştiinţifice, 2–12 iunie 1950, 1664–1667 (Academiei Republicii Populare Române, Române, 1951).Google Scholar
[14]Prasad, J. and Saxena, R.B., “Degree of convergence of quasi Hermite-Fejér interpolation”, Publ. Inst. Math. (Beograd) (N.S.) 19 (33) (1975), 123–130.Google Scholar
[15]Prasad, J. and Varma, A.K., “A study of some interpolatory processes based on the roots of Legendre polynomials”, J. Approx. Theory 31 (1981), 244–252.CrossRefGoogle Scholar
[16]Saxena, R.B., “On the convergence and divergence behavior of Hermite-Fejer and extended Hermite-Fejer interpolations”, Univ. e Politeo. Torino Rend. Sem. Mat. 27 (1967/1968), 223–235.Google Scholar
[17]Schönhage, Arnold, Approximationstheorie (Walter de Gruyter, Berlin, New York, 1971).CrossRefGoogle Scholar
[18]Schönhage, A., “Zur Konvergenz der Stufenpolynome über den Nullstellen der Legendre-Polynome”, Linear operators and approximation, 448–451 (Proc. Conf. Oberwolfach Mathematical Research Institute, Black Forest, 1971. International Series of Numerical Mathematics, 25. Birkhaüser Verlag, Basel und Stuttgart, 1972).CrossRefGoogle Scholar
[19]Szász, Paul, “The extended Hermite-Fejér interpolation formula with application to the theory of generalized almost-step parabolas”, Publ. Math. Debrecen 11 (1964), 85–100.CrossRefGoogle Scholar
[20]Vértesi, P., “Hermite-Fejér type interpolations. III”, Acta Math. Acad. Sci. Hungar. 34 (1979), 67–84.CrossRefGoogle Scholar