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This article is cited in 1 scientific paper (total in 1 paper)
Spline polynomials with a prescribed sequence of extrema
M. B. Korobkova A. I. Gertsen Leningrad Pedagogical Institute
Abstract:
In the present note a theorem about strong suitability of the space of algebraic polynomials of degree $\leqslant n$ in $C_{[a,b]}$ (Theorem A in [1]) is generalized to the space of spline polynomials $\mathcal{S}^{n,k}_{[a,b]}$ ($n\geqslant2$, $k\geqslant0$) in $C_{[a,b]}$. Namely, it is shown that the following theorem is valid: for arbitrary numbers $\eta_0,\eta_1,\dots,\eta_{n+k}$, satisfying the conditions $(\eta_i-\eta_{i-1})(\eta_{i+1}-\eta_i)<0$ ($i=1,\dots,n+k-1$), there is a unique polynomial $s_{n,k}(t)\in \mathcal{S}^{n,k}_{[a,b]}$ and points $a=\xi_0<\xi_1<\dots<\xi_{n+k-1}<\xi_{n+k}=b$ ($\xi_1<z_1<\xi_n,\dots\xi_k<z_k<\xi_{n+k-1}$), such that $s_{n,k}(\xi_i)=\eta_i$ ($i=0,\dots,n+k$), $s'_{n,k}(\xi_i)=0$ ($i=1,\dots,n+k-1$).
Received: 09.11.1970
Citation:
M. B. Korobkova, “Spline polynomials with a prescribed sequence of extrema”, Mat. Zametki, 11:3 (1972), 251–258; Math. Notes, 11:3 (1972), 158–162
Linking options:
https://www.mathnet.ru/eng/mzm9786 https://www.mathnet.ru/eng/mzm/v11/i3/p251
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