|
On interpolation theory in the complex domain
D. L. Berman
Abstract:
It is shown that for the nodes $z_k^{(n)}=e^{i\theta_k^{(n)}}$, where $\theta_k^{(n)}=\frac{(2k+1)\pi}n$, $k=1,\dots,n$; $n=1,2,\dots$, the following statements hold: 1) The Hermite–Fejér interpolation process for an arbitrary polynomial converges in $|z|\leqslant1$ with rapidity $O\bigl(\frac1n\bigr)$. 2) The process $R_n(f,z)=\sum_{k=1}^nf\bigl(z_k^{(n)}\bigl)\bigl[l_k^{(n)}(z)\bigr]^2$, where $\bigl\{l_k^{(n)}(z)\bigr\}$ are Lagrange fundamental polynomials with nodes $\bigl\{z_k^{(n)}\bigr\}$, diverges at all points $z\ne0$ of $|z|\leqslant1$ for every function $f(z)=z^s$, $s=0,1,2,\dots$ .
Received: 03.05.1971
Citation:
D. L. Berman, “On interpolation theory in the complex domain”, Izv. Akad. Nauk SSSR Ser. Mat., 36:4 (1972), 789–794; Math. USSR-Izv., 6:4 (1972), 782–787
Linking options:
https://www.mathnet.ru/eng/im2334https://doi.org/10.1070/IM1972v006n04ABEH001900 https://www.mathnet.ru/eng/im/v36/i4/p789
|
Statistics & downloads: |
Abstract page: | 303 | Russian version PDF: | 79 | English version PDF: | 3 | References: | 29 | First page: | 1 |
|