The average errors for Hermite-Fejr interpolation on the Wiener space |
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摘 要: | For 1≤ p ∞, firstly we prove that for an arbitrary set of distinct nodes in -1, 1], it is impossible that the errors of the Hermite-Fejr interpolation approximation in L p -norm are weakly equivalent to the corresponding errors of the best polynomial approximation for all continuous functions on -1, 1]. Secondly, on the ground of probability theory, we discuss the p-average errors of Hermite-Fejr interpolation sequence based on the extended Chebyshev nodes of the second kind on the Wiener space. By our results we know that for 1≤ p ∞ and 2≤ q ∞, the p-average errors of Hermite-Fejr interpolation approximation sequence based on the extended Chebyshev nodes of the second kind are weakly equivalent to the p-average errors of the corresponding best polynomial approximation sequence for L q -norm approximation. In comparison with these results, we discuss the p-average errors of Hermite-Fejr interpolation approximation sequence based on the Chebyshev nodes of the second kind and the p-average errors of the well-known Bernstein polynomial approximation sequence on the Wiener space.
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The average errors for Hermite-Fejér interpolation on the Wiener space |
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Authors: | GuiQiao Xu YingFang Du |
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Institution: | 1. School of Mathematical Science, Tianjin Normal University, Tianjin, 300387, China 2. College of Chemistry and Life Science, Tianjin Normal University, Tianjin, 300387, China
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Abstract: | For 1 ? p < ∞, firstly we prove that for an arbitrary set of distinct nodes in ?1, 1], it is impossible that the errors of the Hermite-Fejér interpolation approximation in L p -norm are weakly equivalent to the corresponding errors of the best polynomial approximation for all continuous functions on ?1, 1]. Secondly, on the ground of probability theory, we discuss the p-average errors of Hermite-Fejér interpolation sequence based on the extended Chebyshev nodes of the second kind on the Wiener space. By our results we know that for 1 ? p < ∞ and 2 ? q < ∞, the p-average errors of Hermite-Fejér interpolation approximation sequence based on the extended Chebyshev nodes of the second kind are weakly equivalent to the p-average errors of the corresponding best polynomial approximation sequence for L q -norm approximation. In comparison with these results, we discuss the p-average errors of Hermite-Fejér interpolation approximation sequence based on the Chebyshev nodes of the second kind and the p-average errors of the well-known Bernstein polynomial approximation sequence on the Wiener space. |
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