Convergence of the Derivatives of Hermite-Fejér Interpolation Polynomials of Higher Order Based at the Zeros of Freud Polynomials

Kanjin, Yûichi; Sakai, Ryozi

doi:10.1006/jath.1995.1024

Cited by 8 publications

(5 citation statements)

References 8 publications

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“…If we consider the higher order Hermite-Fejér interpolation polynomial L n (ν, f ; x) on a finite interval, then we can see a remarkable difference between the cases of an odd number ν and of an even number ν, for example, as between the Lagrange interpolation polynomial L n (1, f ; x) and the Hermite-Fejér interpolation polynomial L n (2, f ; x) ([17]- [22]). We can also see a similar phenomenon in the cases of the infinite intervals ([7]- [14]). In this paper, we consider the even case in L p -norm.…”

Section: Suppose Thatsupporting

confidence: 68%

Convergence and Divergence of Higher-Order Hermite or Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

Jung

Nakamura

Sakai

et al. 2012

ISRN Mathematical Analysis

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Let and let , where and is an even function. Then we can construct the orthonormal polynomials of degree for . In this paper for an even integer we investigate the convergence theorems with respect to the higher-order Hermite and Hermite-Fejér interpolation polynomials and related approximation process based at the zeros of . Moreover, for an odd integer , we give a certain divergence theorem with respect to the higher-order Hermite-Fejér interpolation polynomials based at the zeros of .

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Section: Suppose Thatsupporting

confidence: 68%

Convergence and Divergence of Higher-Order Hermite or Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

Jung

Nakamura

Sakai

et al. 2012

ISRN Mathematical Analysis

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“…Y n has the advantage of being a positive operator, which makes convergence more general, though possibly at a slower rate than interpolatory operators. Convergence of Hermite and Hermite-Féjer polynomials associated with exponential weights has been investigated in [20], [22], [54], [55], [58], [59], [60], [62], [63], [64], [89], [109], [110], [142], [180]. In some cases, processes involving higher order derivatives have been studied, and the derivatives of the interpolating polynomials have also been investigated.…”

Section: Hermite and Hermite-féjer Interpolationmentioning

confidence: 99%

Weighted polynomial approximation with Freud weights

Lubinsky

Тотик

1994

Constr. Approx

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Let W : R → (0, 1] be continuous. Bernstein's approximation problem, posed in 1924, deals with approximation by polynomials in the weighted uniform norm f → f W L∞(R) . The qualitative form of this problem was solved by Achieser, Mergelyan, and Pollard, in the 1950's. Quantitative forms of the problem were actively investigated starting from the 1960's. We survey old and recent aspects of this topic, including the Bernstein problem, weighted Jackson and Bernstein Theorems, Markov-Bernstein and Nikolskii inequalities, orthogonal expansions and Lagrange interpolation. We present the main ideas used in many of the proofs, and different techniques of proof, though not the full proofs. The class of weights we consider is typically even, and supported on the whole real line, so we exclude Laguerre type weights on [0, ∞). Nor do we discuss Saff's weighted approximation problem, nor the asymptotics of orthogonal polynomials.MSC: 41A10, 41A17, 41A25, 42C10

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“…The results are important for studies of convergence or divergence of the higher order Hermite-Feje´r interpolation polynomials. For the typical case W m ðxÞ ¼ expðÀjxj m Þ; m ¼ 1; y; we have obtained some convergence or divergence theorems in [KS1,KS2]. We can also obtain the same result for L n ðn; f ; xÞ with the weights (0.3).…”

Section: Hermite-feje´r Interpolation Polynomialsmentioning

confidence: 59%

“…Let nX1 be an odd integer, and let Q satisfy the condition Cðn þ 1Þ: Then there is a function f ACðRÞ such that for any fixed constant M40; ðn; f ; xÞj ¼ N:Theorem 5.4 (Cf. Kanjin and Sakai[KS2]). Let Q satisfy the condition CðnÞ; and let I be any compact interval.…”

mentioning

confidence: 97%

Orthonormal polynomials for generalized Freud-type weights and higher-order Hermite–Fejér interpolation polynomials

Kasuga

Sakai²

2004

Journal of Approximation Theory

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Let Q : R-R be even, nonnegative and continuous, Q 0 be continuous, Q 0 40 in ð0; NÞ; and let Q 00 be continuous in ð0; NÞ: Furthermore, Q satisfies further conditions. We consider a certain generalized Freud-type weight W 2 rQ ðxÞ ¼ jxj 2r expðÀ2QðxÞÞ: In previous paper (J. Approx. Theory 121 (2003) 13) we studied the properties of orthonormal polynomials fP n ðW 2 rQ ; xÞg N n¼0 with the generalized Freud-type weight W 2 rQ ðxÞ on R: In this paper we treat three themes. Firstly, we give an estimate of P n ðW 2 rQ ; xÞ in the L p -space, 0oppN: Secondly, we obtain the Markov inequalities, and third we study the higher-order Hermite-Feje´r interpolation polynomials based at the zeros fx kn g n k¼1 of P n ðW 2 rQ ; xÞ: In Section 5 we show that our results are applicable to the study of approximation for continuous functions by the higher-order Hermite-Feje´r interpolation polynomials. r 2004 Elsevier Inc. All rights reserved.

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Convergence of the Derivatives of Hermite-Fejér Interpolation Polynomials of Higher Order Based at the Zeros of Freud Polynomials

Cited by 8 publications

References 8 publications

Convergence and Divergence of Higher-Order Hermite or Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

Convergence and Divergence of Higher-Order Hermite or Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

Weighted polynomial approximation with Freud weights

Orthonormal polynomials for generalized Freud-type weights and higher-order Hermite–Fejér interpolation polynomials

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