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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
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