Abstract
We research the simultaneous approximation problem of the higher-order Hermite interpolation based on the zeros of the second Chebyshev polynomials under weighted Lp-norm. The estimation is sharp.
Highlights
For 0 < p < +∞ and a non-negative measurable function u, the space Lup is defined to be the set of measurable f, such that ( ) ∫ = f p,u1 f (t ) p u (t ) dt 1 p, 0 < p < +∞−1 is finite
When 0 < p < 1, is not a norm; we keep this notation for convenience
For a given integer r ≥ 0, s ≥ 0 and m ≥ 1, the Hermite interpolation is defined to be the unique polynomial of degree N= mn + r + s −1, denoted by Hn,m,r,s (ω, f ), satisfying n,m,r (t) n,m,r
Summary
If ω is a Jacobi weight function, we write ω ∈ J. (2015) The Approximation of Hermite Interpolation on the Weighted Mean Norm. For a given integer r ≥ 0, s ≥ 0 and m ≥ 1 , the Hermite interpolation is defined to be the unique polynomial of degree N= mn + r + s −1 , denoted by Hn,m,r,s (ω, f ) , satisfying. We shall fix the integers m, r and s for the rest of the paper, and omit them from the notations. We have researched the simultaneous approximation problem of the lower-order Hermite interpolation based on the zeros of Chebyshev polynomials under weighted Lp-norm in references [3]-[5]. We will research the simultaneous approximation problem of the higher-order Hermite interpolation in this article.
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