Abstract
Let \({f \in C[-\omega, \omega]}\), \({0 < \omega < \pi}\), be nonlinear and nondecreasing. We wish to estimate the degree of approximation of f by trigonometric polynomials that are nondecreasing in \({[-\omega, \omega]}\). We obtain estimates involving the second modulus of smoothness of f and show that one in general cannot have estimates with the third modulus of smoothness.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Borwein, P., Erdelyi, T.: Polynomials and Polynomial Inequalities. Springer, Berlin, Heidelberg, New York, pp. x+480 (1995)
Bos L., Vianello M.: Subperiodic trigonometric interpolation and quadrature. Appl. Math. Comput. 218, 10630–10638 (2012)
Da Fies G., Vianello M.: Trigonometric Gaussian quadrature on subintervals of the period. Electr. Trans. Numer. Anal. 39, 102–112 (2012)
Dzyubenko G.A., Pleshakov M.G.: Comonotone approximation of periodic functions. Math. Zametki 83, 199–209 (2008)
Erdelyi T.: Markov type estimations for derivatives of polynomials of special type. Acta Math. Hungar. 51, 421–436 (1988)
Erdelyi T., Szabados J.: Bernstein type inequalities for a class of polynomials. Acta Math. Hungar. 53, 237–251 (1989)
Iliev G.L.: Exact estimates for partially mononte approximation. Anal. Math. 4, 181–197 (1978)
Kroó A.: On optimal polynomial messhes. J. Approx. Theory 163, 1107–1124 (2011)
Lorentz G.G., Zeller K.L.: Degree of approximation by monotone polynomials. J. Approx. Theory 1, 501–504 (1968)
Lorentz, G.G.: Approximation of functions, Holt, Rinehart and Winston, NewYork, Tornto, London, pp. ix+188 (1966)
Nagy B., Totik V.: Bernstein’s inequality for algebraic polynomials on cicular arcs. Constr. Approx. 37, 223–232 (2013)
Newman D.J.: Efficient co-monotone approximation. J. Approx. Theory 25, 189–192 (1979)
Pleshakov, M.G.: Comonotone approximation of periodic functions of Sobolev classes, Candidate’s Dissertation in Mathematics and Physics, Saratov State University, Russia (1997)
Pleshakov M.G., Shatalina A.V.: Piecewise coapproximation and the Whitney inequality. J. Approx. Theory 105, 189–210 (2000)
Rivlin, T.J.: An introduction to the approximation of functions. Dover Publ., New York, pp. vi+150 (1989)
Shvedov, A.S.: Orders of coapproximation of functions by algebraic polynomials. Mat. Zametki 29, 117–130 (1981) (in Russian)
Vianello M.: Norming meshes by Bernstein-like inequalities. Math. Inequ. Appl. 17, 929–936 (2014)
Videnskii V.S.: Markov and Bernstein type inequalities for derivatives of trigonometric polynomials on an interval shorter than the period. Dokl. Akad. Nauk. SSSR 130, 13–16 (1960)
Whitney H.: On functions with bounded nth differences. J. Math. Pure. Appl. 6, 67–95 (1957)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Leviatan, D., Sidon, J. Monotone Trigonometric Approximation. Mediterr. J. Math. 12, 877–887 (2015). https://doi.org/10.1007/s00009-014-0460-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-014-0460-8