The question is how to calculate $$ I = \int_\Omega \exp\left(-\frac{1}{2} x^\mathrm{T} A x + J^\mathrm{T} x\right) d^n x $$ the space $\Omega$ is defined by the intersection of $m$ hyperplanes as $$ B x = b $$ where $B$ is a $m \times n$ martrix and $b$ is a $m \times 1$ vector.
If $m = 1$, I can use $$ \int_{\mathbb{R}^n} f(\mathbf{x}) \delta(g(\mathbf{x})) \mathrm{d} \mathbf{x}=\int_{g^{-1}(0)} \frac{f(\mathbf{x})}{|\nabla g|} \mathrm{d} \sigma(\mathbf{x}) $$ and replace Dirac delta by its Fourier transform thus the intergral can be calculated. However, if $m > 1$ the same method meet the multiplication of two Dirac deltas which is not defined in the standard theory of generalized functions.
So do we have another method to deal with such integral?
Many thanks !!
Multiple Deltas are fine, since you have multiple coordinates, use a Delta for each constraint $\delta (B_{i}^T x - b_i)$ and use the Fourier representation as before.