Based on Bernstein＇s Theorem, Kalandia＇s Lemma describes the error estimate and the smoothness of the remainder under the second part of Hoelder norm when a HSlder function is approximated by its best polynomial app...Based on Bernstein＇s Theorem, Kalandia＇s Lemma describes the error estimate and the smoothness of the remainder under the second part of Hoelder norm when a HSlder function is approximated by its best polynomial approximation. In this paper, Kalandia＇s Lemma is generalized to the cases that the best polynomial is replaced by one of its four kinds of Chebyshev polynomial expansions, the error estimates of the remainder are given out under Hoeder norm or the weighted HSlder norms.展开更多
We obtain an upper bound for the average error of the quasi-Griinwald interpolation based on the zeros of Chebyshev polynomial of the second kind in the Wiener space.
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 corr...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.展开更多
For the weighted approximation in Lp-norm, we determine the asymptotic order for the p- average errors of Lagrange interpolation sequence based on the Chebyshev nodes on the Wiener space. We also determine its value i...For the weighted approximation in Lp-norm, we determine the asymptotic order for the p- average errors of Lagrange interpolation sequence based on the Chebyshev nodes on the Wiener space. We also determine its value in some special case.展开更多
基金Supported by National Natural Science Foundation of China(No. 10471107) SF of Wuhan University(No. 20127004).
文摘Based on Bernstein＇s Theorem, Kalandia＇s Lemma describes the error estimate and the smoothness of the remainder under the second part of Hoelder norm when a HSlder function is approximated by its best polynomial approximation. In this paper, Kalandia＇s Lemma is generalized to the cases that the best polynomial is replaced by one of its four kinds of Chebyshev polynomial expansions, the error estimates of the remainder are given out under Hoeder norm or the weighted HSlder norms.
基金Foundation item： Supported bv the National Natural Science Foundation of China（10471010）
文摘We obtain an upper bound for the average error of the quasi-Griinwald interpolation based on the zeros of Chebyshev polynomial of the second kind in the Wiener space.
基金supported by National Natural Science Foundation of China (Grant No.10471010)
文摘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.
基金Supported by National Natural Science Foundation of China(Grant No.10471010)
文摘For the weighted approximation in Lp-norm, we determine the asymptotic order for the p- average errors of Lagrange interpolation sequence based on the Chebyshev nodes on the Wiener space. We also determine its value in some special case.
