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Volume 1, Issue 4
Local Explicit Many-Knot Spline Hermite Approximation Schemes

Dong-Xu Qi & Shu-Zi Zhou

J. Comp. Math., 1 (1983), pp. 317-321.

Published online: 1983-01

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  • Abstract

If $f^{(i))}(\alpha)(\alpha=a, i=0,1,...,k-2)$ are given, then we get a class of the Hermite approximation operator Qf=F satisfying $F^{(i)}(\alpha)=f^{(i)}(\alpha)$, where F is the many-knot spline function whose knots are at points $y_i:$=$y_0$<$y_1$<$\cdots$<$y_{k-1}=b$, and $F\in P_k$ on $[y_{i-1},y_i]$. The operator is of the form $Qf:=\sum\limits_{i=0}^{k-2}[f^{(i)}(a)\phi_i+f^{(i)}(b)\psi _i]$. We give an explicit representation of $\phi_i$ and $\psi_i$ in terms of B-splines $N_{i,k}$. We show that Q reproduces appropriate classes of polynomials.

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@Article{JCM-1-317, author = {Dong-Xu Qi and Shu-Zi Zhou}, title = {Local Explicit Many-Knot Spline Hermite Approximation Schemes}, journal = {Journal of Computational Mathematics}, year = {1983}, volume = {1}, number = {4}, pages = {317--321}, abstract = {

If $f^{(i))}(\alpha)(\alpha=a, i=0,1,...,k-2)$ are given, then we get a class of the Hermite approximation operator Qf=F satisfying $F^{(i)}(\alpha)=f^{(i)}(\alpha)$, where F is the many-knot spline function whose knots are at points $y_i:$=$y_0$<$y_1$<$\cdots$<$y_{k-1}=b$, and $F\in P_k$ on $[y_{i-1},y_i]$. The operator is of the form $Qf:=\sum\limits_{i=0}^{k-2}[f^{(i)}(a)\phi_i+f^{(i)}(b)\psi _i]$. We give an explicit representation of $\phi_i$ and $\psi_i$ in terms of B-splines $N_{i,k}$. We show that Q reproduces appropriate classes of polynomials.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9707.html} }
TY - JOUR T1 - Local Explicit Many-Knot Spline Hermite Approximation Schemes AU - Dong-Xu Qi & Shu-Zi Zhou JO - Journal of Computational Mathematics VL - 4 SP - 317 EP - 321 PY - 1983 DA - 1983/01 SN - 1 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9707.html KW - AB -

If $f^{(i))}(\alpha)(\alpha=a, i=0,1,...,k-2)$ are given, then we get a class of the Hermite approximation operator Qf=F satisfying $F^{(i)}(\alpha)=f^{(i)}(\alpha)$, where F is the many-knot spline function whose knots are at points $y_i:$=$y_0$<$y_1$<$\cdots$<$y_{k-1}=b$, and $F\in P_k$ on $[y_{i-1},y_i]$. The operator is of the form $Qf:=\sum\limits_{i=0}^{k-2}[f^{(i)}(a)\phi_i+f^{(i)}(b)\psi _i]$. We give an explicit representation of $\phi_i$ and $\psi_i$ in terms of B-splines $N_{i,k}$. We show that Q reproduces appropriate classes of polynomials.

Dong-Xu Qi and Shu-Zi Zhou. (1983). Local Explicit Many-Knot Spline Hermite Approximation Schemes. Journal of Computational Mathematics. 1 (4). 317-321. doi:
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