The $N$-dimensional Gaussian integral
$$\int \mathrm{d}^N x \, \mathrm{e}^{-\frac{1}{2}\boldsymbol{x}^\mathrm{T}A\boldsymbol{x}+\boldsymbol{b}^\text{T}\boldsymbol{x}}=\left(\frac{(2\pi)^N}{\det A}\right)^\frac{1}{2}e^{\frac{1}{2}\boldsymbol{b}^\text{T} A^{-1}\boldsymbol{b}}$$
can be extended to functions by considering them as infinite dimensional vectors giving the path integral (in dodgy notation)
$$\int\mathcal{D}f\,\mathrm{e}^{-\frac{1}{2}\int fAf + \int bf}=N\mathrm{e}^{\frac{1}{2}\int b A^{-1} b}$$
where $A^{-1}$ is the Green's function of $A$. However, is there an easy way to see the effect of imposing boundary conditions on the functions? For example, if they are periodic such that $f(0)=f(\beta)$, then it is also the case that $A^{-1}(0)=A^{-1}(\beta)$, but can this be seen from the discrete version without having to go through the full functional derivation, integrating by parts etc.?
