Construction of bilateral polynomials of the Lagrange type for multidimensional functions.(Ukrainian)Zbl 0961.41003

Let $$(x_{i},y_{j})$$ be points of approximation. For a two-dimensional function $$F(x,y)$$ the authors propose bilateral formulas of approximation $$\widehat L_{mn}= \sum_{i=0}^{m}\sum_{j=0}^{n}\widehat\omega_{i}(x) \widehat\psi_{j}(y)\phi(x_{i},y_{j})-C(x+x_{0})^{m+1}/(m+1)!$$, $$\widetilde L_{mn}= \sum_{i=1}^{m+1}\sum_{j=1}^{n+1}\widetilde\omega_{i}(x) \widetilde\psi_{j}(y)\phi(x_{i},y_{j})-C(x+x_{0})^{m+1}/(m+1)!$$, where $$\widehat\omega_{i}(x)= \prod_{\nu=0,\nu\neq i}^{m}(x-x_{\nu})/(x_{i}-x_{\nu})$$, $$\widehat\psi_{j}(y)= \prod_{\mu=0,\mu\neq j}^{n}(y-y_{\mu})/(y_{j}-y_{\mu})$$, $$\widetilde\omega_{i}(x)= \prod_{\nu=1,\nu\neq i}^{m+1}(x-x_{\nu})/(x_{i}-x_{\nu})$$, $$\widetilde\psi_{j}(y)= \prod_{\mu=1,\mu\neq j}^{n+1}(y-y_{\mu})/(y_{j}-y_{\mu})$$. For the functions $$\exp\{xy\}$$ and $$\exp\{x+y\}$$ some numerical analysis is presented.

MSC:

 41A10 Approximation by polynomials