Turkish Journal of Mathematics




Recently Fang and Li established a sampling formula that involves only samples from the function and its first partial derivatives for functions from Bernstein space, $B^{p}_{\sigma}(\mathbb{R}^{2})$. In this paper, we derive a general bivariate sampling series for the entire function of two variables that satisfy certain growth conditions. This general bivariate sampling formula involves samples from the function and its mixed and nonmixed partial derivatives. Some known sampling series will be special cases of our formula, like the sampling series of Parzen, Peterson and Middleton, and Gosselin. The truncated series of this formula are used to approximate functions from the Bernstein space so we establish a bound for the truncation error of this series based on localized sampling without decay assumption. Numerically, we compare our approximation results with the results of Fang and Li's sampling formula. Our formula gives us highly accurate approximations in comparison with the results of Fang and Li's formula.


Sampling series, contour integral, truncation error

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