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Hermite-Fejér type interpolation of higher order. (English) Zbl 0935.41003
Let $$X:= \{x_{nk}, y_{n\mu}\}$$ be a matrix of nodes satisfying $-1\leq x_{n1}<\cdots< x_{nn}\leq 1,\quad -1\leq y_{n1}<\cdots< y_{n\ell}\leq 1,$ where $$x_{nk}\neq y_{n\mu}$$, $$k= 1,\dots, n$$, $$\mu= 1,\dots,\ell$$, $$n\geq 1$$, $$\ell= \ell(n)\geq 0$$.
Let $$\pi_n$$ denote the set of polynomials of degree $$\leq n$$. If $$f\in C^{n-1}[-1, 1]$$, let $$S^*_{mn}(f)$$ denote Hermite type polynomials of higher order satisfying \begin{aligned} S^*_{mn}(f)\in \pi_{mn+ \ell-1},\;S^{*(\nu)}_{m,n}(f, x_k) & = f^{(\nu)}(x_k),\;k=1,\dots, n;\;\nu= 0,1,\dots,m- 1,\\ S^*_{mn}(f, y_\mu) & = f(y_\mu),\;\mu= 1,\dots,\ell.\end{aligned} Similarly let $$S_{mn}(f)$$ denote Hermite-Fejér type interpolation polynomials satisfying \begin{aligned} S_{mn}(f) & \in \pi_{mn+ \ell-1},\;S_{mn}(f, x_k)= f(x_k),\quad k= 1,\dots,n,\\ S^{(\nu)}_{mn}(f, x_k) & = 0,\quad k= 1,\dots, n,\quad \nu= 1,\dots, m-1,\\ S_{mn}(f, y_\mu) & = f(y_\mu),\quad \mu= 1,\dots,\ell.\end{aligned} For $$\ell= 0$$, $$S^*_{mn}(f, x)$$ is the usual Hermite interpolation polynomial and $$S_{mn}(f, x)$$ is the usual Hermite-Fejér interpolant. In case $$\ell= 2$$, $$y_{n1}= y_{n2}= 1$$, $$S_{mn}(f, x)$$ is the quasi-Hermite interpolant.
Then one can write \begin{aligned} S^*_{mn}(f, x) & = \sum^n_{i=1} \sum^{m-1}_{j= 0} f^{(j)}(x_i)B_{ij}(x)+ \sum^\ell_{\mu= 1} f(y_\mu)C_\mu(x),\\ S_{mn}(f, x) & = \sum^n_{i= 1}f(x_i) B_{i0}(x)+ \sum^\ell_{\mu= 1} f(y_\mu) C_\mu(x),\end{aligned} where $$\{B_{ij}(x)\}^{n,m-1}_{i=1,j= 0}$$ and $$\{C_\mu(x)\}$$ are the fundamental polynomials of this interpolation problem. Using suitable variations on the method of J. Szabados [Acta Math. Hung. 61, 357-368 (1993; Zbl 0801.41001)], the authors find in Section 2, the explicit forms for the fundamental polynomials $$\{B_{ij}(x)\}$$ and $$\{C_\mu(x)\}^\ell_{\mu= 1}$$ and obtains some estimates for them.
The main results are given in Section 5. Let $$R_{mn}(f)= f(x)- S_{mn}(f)$$ be the error of approximation of $$f$$ by $$S_{mn}(f)$$. Then the authors show
Proposition 8. For $$p\in \pi_{mn+\ell- 1}$$, $\|R^{(j- 1)}_{mn}(p)\|\geq (mn+ \ell-1)^{-1} \sup_i (1- x^2_{ni})^{1/2}|p^{(j)}(x_{ni})|,\quad j= 1,\dots, n-1.$ From this the authors prove the following saturation theorem for Hermite-Fejér interpolation:
Proposition 9. If $$X$$ has infinitely many limit points and a polynomial $$p(x)$$ satisfies for a fixed $$j$$ $$(1\leq j\leq m-1)$$, $\|R^{(j)}_{mn}(p)\|= o((mn+ \ell-1)^{-1})$ then $$p(x)$$ must be a polynomial of degree $$j-1$$.
Proposition 10. Suppose $$\|S_{mn}\|= o(1)$$ and $$\overline X:=\text{closure of }X= [-1,1]$$. If a function $$f(x)$$ is such that $$f^{(j)}\in C$$, $$1\leq j\leq m-1$$, satisfies $\|R_{mn}(f^{(j-1)})\|= o((mn+ \ell-1)^{-1}),$ then $$f(x)$$ is a polynomial of degree $$j-1$$.
##### MSC:
 41A05 Interpolation in approximation theory
##### Keywords:
convergence; saturation; Hermite-Fejér type interpolation
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##### References:
 [1] Shi, Y.G., Some notes on Hermite-Fejér type interpolation, Approx. Theory Appl., 1991, 7: 28–39. · Zbl 0757.41006 [2] Szabados, J., On the order of magnitude of fundamental polynomials of Hermite interpolation, Acta Math. Hungar., 1993, 61: 357–368. · Zbl 0801.41001 · doi:10.1007/BF01874691 [3] Vértesi, P., Hermite-Fejér interpolation of higher order. I, Acta Math. Hungar., 1989, 54: 135–152. · Zbl 0703.41004 · doi:10.1007/BF01950715 [4] Zygmund, A., Trigonometric Series, Vol. 1, Cambridge University Press, London, 1979. · JFM 58.0296.09 [5] Shi, Y.G., A theorem of Grünwald-type for Hermite-Fejér interpolation of higher order, Constr. Approx., 1994, 10: 439–450. · Zbl 0820.41003 · doi:10.1007/BF01303521
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