IJPAM: Volume 46, No. 3 (2008)

BIVARIATE OPERATORS WHICH INTERPOLATE SOME
PARTIAL DERIVATIVES OF A FUNCTION
ON THE VERTICES OF A SQUARE

Marius M. Birou
Department of Mathematics
Technical University of Cluj-Napoca
Daicoviciu Str. 15, Cluj-Napoca, 400020, ROMANIA
e-mail: marius.birou@math.utcluj.ro


Abstract.In this article we study the operator

\begin{displaymath}B_{r}=P_{1}^{\prime}Q_{r}^{\prime\prime}\oplus P_{2}^{\prime}... ...\prime}\oplus\cdots\oplus P_{r}^{\prime}Q_{1}^{\prime\prime}\,,\end{displaymath}

where $P_i$, $Q_j$ are univariate Birkhoff interpolation projectors with two nodes (see [#!ag!#]) and $P_{i}^{\prime}$, $Q_{j}^{\prime\prime}$ are the corresponding parametric extensions (see [#!desh!#]). The univariate projectors $P_i$, $Q_j$ are chosen so that the bivariate projectors $P_{i}^{\prime}$, $Q_{j}^{\prime\prime}$ form the chains in a lattice of projectors, i.e.

\begin{displaymath}P'_1\le \dots \le P'_r,\quad Q''_1\le\dots \le Q''_r.\end{displaymath}

We give the range space, the interpolation properties and the remainder term for this operator. The operator $B_r$ interpolates some partial derivatives of a function on the vertices of the square. Some comparations about approximation order and information used with tensor product operator are given.

Received: March 19, 2008

AMS Subject Classification: 41A05, 41A10, 41A63

Key Words and Phrases: Birkhoff interpolation, interpolation projectors, approximation order

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 46
Issue: 3