A Multidimensional Euler-Maclaurin Formulaand an Application

Abstract. We compare, in the multidimensional case, the Fourier integral of a function λ of bounded variation and the corresponding trigonometric series with the coefficients λ(k). Posing additional smoothness conditions on the function, we infer that the difference between the two mentioned values is controlled not only by the total variation as in dimension one and by certain multivariate versions of variations in our previous work with Trigub, but by a sum of more sophisticated values, some of which are related to certain types of variations. This is a generalization of the earlier one-dimensional result by Trigub that at zero turns out to be the celebrated Euler-Maclaurin formula. The results are applied to obtain bounds for Lebesgue constants.