Geodesic
On a refinement of Marcinkiewicz-Zygmund type inequalities
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 4, pp. 196-209 Cet article a éte moissonné depuis la source Math-Net.Ru

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The main goal of this paper is to verify a refined Marcinkiewicz–Zygmund type inequality with a quadratic error term $$ \frac{1}{2}\sum_{j=0}^{nm-1}(x_{j+1}-x_{j-1})w(x_j)|t_n(x_{j})|^q=(1+O(m^{-2}))\int\limits_{-\pi}^{\pi}w(x)|t_n(x)|^q\,dx, \quad 2\leq q\infty, $$ where $t_n$ is any trigonometric polynomial of degree at most $n, \ -\pi=x_0$, and $w$ is a Jacobi type weight. Moreover, the quadratic error term $O(m^{-2})$ is shown to be sharp, in general. In addition, similar results are given for $q=\infty$ and in the multivariate case.
Keywords: multivariate polynomials; Marcinkiewicz-Zygmund, Bernstein, and Schur type inequalities; discretization of $L^p$ norm; doubling and Jacobi type weights. 
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A. V. Kroó. On a refinement of Marcinkiewicz-Zygmund type inequalities. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 4, pp. 196-209. http://geodesic.mathdoc.fr/item/TIMM_2020_26_4_a12/

[1] Arestov V.V., “On integral inequalities for trigonometric polynomials and their derivatives”, Math. USSR-Izv., 18:1 (1982), 1–17 | DOI

[2] Calvi J.P., Levenberg N., “Uniform approximation by discrete least squares polynomials”, J. Approx. Theory, 152 (2008), 82–100 | DOI

[3] Dai F., Prymak A., Temlyakov V.N., Tikhonov V.N., “Integral norm discretization and related problems”, Russian Math. Surveys, 74:4 (2019), 579–630 | DOI

[4] De Marchi S., Kro$\acute{\mathrm{o}}$ A., “Marcinkiewicz - Zygmund type results in multivariate domains”, Acta Math. Hungar., 154 (2018), 69–89 | DOI

[5] DeVore R.A., Lorentz G.G., Constructive Approximation, Springer-Verlag, Berlin; Heidelberg; New York, 1993, 452 pp.

[6] Jetter K., St$\ddot{\mathrm{o}}$ckler J., Ward J.D., “Error Estimates for Scattered Data Interpolation”, Math. Comp., 68 (1999), 733–747 | DOI

[7] John F., “Extremum problems with inequalities as subsidiary conditions”, Courant Anniversary Volume, Interscience, N Y, 1948, 187–204

[8] Kro$\acute{\mathrm{o}}$ A., “On optimal polynomial meshes”, J. Approx. Theory, 163 (2011), 1107–1124 | DOI

[9] Kro$\acute{\mathrm{o}}$ A., “On the existence of optimal meshes in every convex domain on the plane”, J. Approx. Theory, 238 (2019), 26–37 | DOI

[10] Lubinsky D., “Marcinkiewicz - Zygmund Inequalities: Methods and Results”, Recent Progress in Inequalities, eds. G.V. Milovanovic et al., Kluwer Acad. Publ., Dordrecht, 1998, 213–240 | DOI

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

[12] Mastroianni G., Totik V., “Weighted polynomial inequalities with doubling and $A_\infty$ weights”, Constr. Approx., 16 (2000), 37–71 | DOI

[13] Piazzon F., Vianello M., “Markov inequalities, Dubiner distance, norming meshes and polynomial optimization on convex bodies”, Optimization Letters, 13 (2019), 1325–1343 | DOI