Assume we want to estimate $$I=\int f(x):{\rm d}x=\hat f(0).$$ This can be done by an estimator of the form $$\tilde I=\sum_iw_if(x_i).$$ Introducing the "sampling function" $$S(x)=\sum_iw_i\delta(x-x_i),$$ this estimator can be expressed as $$\tilde I=\int Sf=\widehat{Sf}(0)=\left(\hat S\ast\hat f\right)(0)=\langle\hat S,\hat f\rangle_{L^2}.$$ The error of the estimation is $\tilde I-I$.
Now, I've read that the following and don't really understand what's actually meant. If the sampling pattern sums to $1$ (I guess it's meant that $\sum_iw_i=1$), then the error is the dot product between the spectrum of $f$ and the spectrum of $S$ where the DC is replaced by zero. If the sampling pattern doesn't sum to $1$, then the error has an additional term, which is the true integral multiplied by the difference between the DC of the sampling pattern and one.
Could anybody express in mathematical formulas what's actually meant by this? The statements can be found on p. 4 of this paper.