Skip to main content

Marcinkiewicz–Zygmund type results in multivariate domains


We investigate Marcinkiewicz–Zygmund type inequalities for multivariate polynomials on various compact domains in \({\mathbb{R}^d}\). These inequalities provide a basic tool for the discretization of the L p norm and are widely used in the study of the convergence properties of Fourier series, interpolation processes and orthogonal expansions. Recently Marcinkiewicz–Zygmund type inequalities were verified for univariate polynomials for the general class of doubling weights, and for multivariate polynomials on the ball and sphere with doubling weights. The main goal of the present paper is to extend these considerations to more general multidimensional domains, which in particular include polytopes, cones, spherical sectors, toruses, etc. Our approach will rely on application of various polynomial inequalities, such as Bernstein–Markov, Schur and Videnskii type estimates, and also using symmetry and rotation in order to generate results on new domains.

This is a preview of subscription content, access via your institution.


  1. Arestov V.V.: On integral inequalities for trigonometric polynomials and their derivatives. Izv. Akad. Nauk SSSR, Ser. Mat. 45, 3–22 (1981)

    MathSciNet  Google Scholar 

  2. Bos L., De Marchi S., Sommariva A., Vianello M.: admissible meshes and discrete extremal sets. Numer. Math. Theory Methods Appl. 4, 1–12 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Calvi J.P., Levenberg N.: Uiform approximation by discrete least squares polynomials. J. Approx. Theory 152, 82–100 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  4. De Marchi S., Marchioro M., Sommariva A.: Polynomial approximation and cubature at approximate Fekete and Leja points of the cylinder. Appl. Math. Comput. 218, 10617–10629 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Jetter K., Stöckler J., Ward J.D.: Error estimates for scattered data interpolation. Math. Comp. 68, 733–747 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  6. T. Erdélyi, Markov–Bernstein type inequality for trigonometric polynomials with respect to doubling weights on \({[-\omega, \omega]}\), Constr. Approx., 19 (2003), 329–338.

  7. Feng Dai, Multivariate polynomial inequalities with respect to doubling weights and \({A_{\infty}}\) weights, J. Funct. Anal., 235 (2006), 137–170.

  8. Kroó A.: On optimal polynomial meshes. J. Approx. Theory. 163, 1107–1124 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  9. D. Lubinsky, Marcinkiewicz–Zygmund inequalities: in: Methods and Results, in Recent Progress in Inequalities (ed. G. V. Milovanovic et al.), Kluwer Academic Publishers (Dordrecht, 1998), pp. 213–240.

  10. Lubinsky D.: L p Markov–Bernstein inequalities on arcs of the unit circle. J. Approx. Theory 108, 1–17 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  11. Marcinkiewicz J., Zygmund A.: Mean values of trigonometric polynomials. Fund. Math. 28, 131–166 (1937)

    MATH  Google Scholar 

  12. Mastroianni G., Totik V.: Weighted polynomial inequalities with doubling and \({A_{\infty}}\) weights. Constr. Approx. 16, 37–71 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  13. H. N. Mhaskar, F. J. Narcowich and J. D. Ward, Spherical Marcinkiewicz–Zygmund inequalities and positive quadrature, Math. Comp., 70 (2001), 1113–1130; Corrigendum: Math. Comp., 71(2001), 453–454.

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to A. Kroó.

Additional information

This work has been partially supported by the BIRD163015 and DOR funds of the University of Padova. The second author was supported by the Visiting Professors program year 2017 of the Department of Mathematics “Tullio Levi-Civita” of the University of Padova and by the OTKA Grant K111742. This research has been accomplished within the RITA “Research ITalian network on Approximation”.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

De Marchi, S., Kroó, A. Marcinkiewicz–Zygmund type results in multivariate domains. Acta Math. Hungar. 154, 69–89 (2018).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Key words and phrases

  • multivariate polynomial
  • Marcinkiewicz–Zygmund type inequality
  • L p optimal mesh

Mathematics Subject Classification

  • 41A17
  • 41A63