The Romberg method of numerical integration is extended to multidimensional integration. The elements of the m th column of the Romberg array are shown to approximate up to 0 ( h 2 m +2 ) provided the integrand has 2 m +2 bounded partial derivatives in each variable.

Let the transfer function for the numerical filter be H(w) = ~__, h,,e(~'~,~5 where h,~ is the nth weight and f,~ is the sampling frequency.The weights are giwm by h,~ = 1/2rrf,~f'~/;Ti, H(w)e(-*"./f~ ) dw.If we assume that H(w) has a continuous first derivatiw; a~d the second derivative exists, an error bound is given by e(N, f,) =< 2fffrrNf~tc I H'(w) I &.~, where w~ is the angular cut-off frequency, wt the termination frequency, and 2N q-1 is the number of weights used.This is a bound for the error in recovering the transfer fmmtion by using a finite number of weights.