Abstract
When considering approximation of continuous periodic functions f: R d → R by blending-type approximants which depend on directions ξ1,…,ξν ∈ R d directional moduli of smoothness (1) are appropriate measures of smoothness of /. In this paper, we introduce equivalent directional K- functionals. As an application, we obtain a result on the degree of approximation by certain trigonometric blending functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Cottin: Mixed K-functionals: A measure of smoothness for blending-type approximation, Math. Z. 204 (1990), 69–83.
C. Cottin: Approximation by bounded pseudo-polynomials, in ldFunction Spaces” (Proc. Conf. Poznan 1989, J. Musielak, H. Hudzik, R. Urbański, eds., Teubner, Leipzig, 1991, 152–161.
C. Cottin: Directional extensions of univariate operators and Boolean sum approximation. Preprint 1992.
R.A.De Vore: Degree of approximation, in “Approximation Theory II” (Proc. Conf. Austin 1976, G.G. Lorentz, C.K. Chui, L.L. Schumaker, eds., Academic Press, New York, 1976.
Z. Ditzian and V. Totik: “Moduli of smoothness”, Springer Series in Computational Math. 9, Springer, New York, 1987.
H.H. Gonska: Quantitative Approximation in C(X), “Habilitationsschrift”, University of Duisburg, 1985.
H.H. Gonska: Degree of simultaneous approximation of bivariate functions by Gordon operators, J. Approx. Theory 62 (1990), 170–191.
K. Jetter: The Bernoulli spline and approximation by trigonometric blending polynomials, Results in Mathematics 16 (1989), 243–252.
K. Jetter: On Jackson-Favard estimates for blending approximation, in “Approximation Theory VI” (Proc. Conf. College Station 1989, C.K. Chui, L.L. Schumaker, J.D. Ward, eds., Academic Press, New York, 1989, 241–344.
H. Johnen and K. Scherer: On the equivalence of the K-functional and moduli of continuity and some applications, in “Constructive Theory of Functions of Several Variables” (Proc. Conf. Oberwolfach 1976, W. Schempp, K. Zeller, eds., Springer, Berlin-Heidelberg-New York 1977, 119–140.
G.G. Lorentz: “Approximation of Functions”, Holt, Rinehart and Winston, New York, 1966.
L.L. Schumaker: “Spline Functions: Basic Theory”, Wiley & Sons, New York, 1981.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cottin, C. Directional K- Functionals Claudia Cottin. Results. Math. 24, 211–221 (1993). https://doi.org/10.1007/BF03322331
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03322331