A Hilbert transform representation of the error in Lagrange interpolation

Kubayi, D. G.; Lubinsky, D. S.

doi:10.1016/j.jat.2004.05.002

Cited by 4 publications

(2 citation statements)

References 5 publications

(7 reference statements)

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“…We remark that more general approaches to the error in Lagrange interpolation were discussed in [13,14].…”

Section: Proof Of Theoremmentioning

confidence: 99%

Polynomial interpolation, an L-function, and pointwise approximation of continuous functions

Ganzburg

2008

Journal of Approximation Theory

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We show that if {s k } ∞ k=1 is the sequence of all zeros of the L-function L(s, ) := ∞ k=0 (−1) k (2k + 1) −s satisfying Re s k ∈ (0, 1), k = 1, 2, . . . , then any function from span {|x| s k } ∞ k=1 satisfies the pointwise rapid convergence property, i.e. there exists a sequence of polynomials Q n (f, x) of degree at most n such that f − Q n C[−1,1] C(f )E n (f ), n=1, 2, . . . , and for every x ∈ [−1, 1], lim n→∞ (|f (x)−Q n (f, x)|)/E n (f )= 0, where E n (f ) is the error of best polynomial approximation of f in C[−1, 1]. The proof is based on Lagrange polynomial interpolation to |x| s , Re s > 0, at the Chebyshev nodes. We also establish a new representation for |L(x, )|.

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“…We remark that more general approaches to the error in Lagrange interpolation were discussed in [13,14].…”

Section: Proof Of Theoremmentioning

confidence: 99%

Polynomial interpolation, an L-function, and pointwise approximation of continuous functions

Ganzburg

2008

Journal of Approximation Theory

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“…3.194.4]. Similar ideas were used in [16], [19]. Recall that α is the least integer exceeding α. and…”

Section: Interpolation Identitiesmentioning

confidence: 99%

On the Bernstein Constants of Polynomial Approximation

Lubinsky

2006

Constr Approx

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Let α > 0 not be an integer. In papers published in 1913 and 1938, S. N. Bernstein established the limit.denotes the error in best uniform approximation of |x| α on [−1, 1] by polynomials of degree ≤ n. Bernstein proved that Λ * ∞,α is itself the error in best uniform approximation of |x| α by entire functions of exponential type at most 1, on the whole real line. We prove that the best approximating entire function is unique, and satisfies an alternation property. We show that the scaled polynomials of best approximation converge to this unique entire function. We derive a representation for Λ

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Approximation of Continuous Functions

Gal

Sabadini

2019

Frontiers in Mathematics

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A Hilbert transform representation of the error in Lagrange interpolation

Cited by 4 publications

References 5 publications

Polynomial interpolation, an L-function, and pointwise approximation of continuous functions

Polynomial interpolation, an L-function, and pointwise approximation of continuous functions

On the Bernstein Constants of Polynomial Approximation

Approximation of Continuous Functions

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