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The sinc function occuring in the classical Shannon sampling theorem is replaced by algebraic polynomials. It turns out that it is not necessary to take the algebraic polynomial of best approximation to sinc a finite sum of the Taylor series expansion already suffices. The basic problem is to couple the number of terms taken of the truncated sampling expansion with that of the Taylor expansion in such a fashion that convergence of the modified sampling expansion is guaranteed. The resulting error is additive to the truncation error and is negligible in comparison with the latter. The results are illustrated by two representative examples.