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 Mat. Zametki, 2015, Volume 97, Issue 6, Pages 815–831 (Mi mz10661)

Approximation of the Multidimensional Jacobi Transform in $L_2$ by Partial Integrals

R. A. Veprintsev

Tula State University

Abstract: In the space $L_2(\mathbb{R}^d_+)$ with hyperbolic weight, the exact Jackson inequality with the optimal argument in the modulus of continuity is proved. The optimal argument is the least value of the argument in the modulus of continuity for which the exact constant in the Jackson inequality takes the minimum value. The approximation is carried out by partial integrals of the multidimensional Jacobi transform. In the study of the optimal argument, the geometry of the domain of the partial integral and the geometry of the neighborhood of zero in the definition of the modulus of continuity are taken into account. The optimal argument is obtained for the case in which the first skew field is an $l_p$-ball for $1\leq p\leq 2$ and the second, a parallelepiped.

Keywords: Jacobi transform, Jackson inequality, modulus of continuity, $l_p$-ball, parallelepiped, Jacobi function, Lebesgue-measurable function, Bernstein's inequality.

 Funding Agency Grant Number Russian Foundation for Basic Research 13-01-00045 Ministry of Education and Science of the Russian Federation 1.1333.2014Ê This work was supported by the Russian Foundation for Basic Research (grant no. 13-01-00045) and by the Ministry of Education and Science of the Russian Federation (grant no. 1.1333.2014K).

DOI: https://doi.org/10.4213/mzm10661

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English version:
Mathematical Notes, 2015, 97:6, 831–845

Bibliographic databases:

Document Type: Article
UDC: 517.5

Citation: R. A. Veprintsev, “Approximation of the Multidimensional Jacobi Transform in $L_2$ by Partial Integrals”, Mat. Zametki, 97:6 (2015), 815–831; Math. Notes, 97:6 (2015), 831–845

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz10661
• https://doi.org/10.4213/mzm10661
• http://mi.mathnet.ru/eng/mz/v97/i6/p815

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This publication is cited in the following articles:
1. D. V. Gorbachev, V. I. Ivanov, “Approximation in $L_2$ by Partial Integrals of the Fourier Transform over the Eigenfunctions of the Sturm–Liouville Operator”, Math. Notes, 100:4 (2016), 540–549
2. D. V. Gorbachev, V. I. Ivanov, R. A. Veprintsev, “Approximation in $L_2$ by partial integrals of the multidimensional Fourier transform in the eigenfunctions of the Sturm–Liouville operator”, Proc. Steklov Inst. Math. (Suppl.), 300, suppl. 1 (2018), 97–113
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