When we have a system of of $n$ linear equations represented by $$A \overrightarrow{x} = \overrightarrow{b} $$ with $\overrightarrow{x} = (x_{1}, x_{2}, \dots, x_{n})^{\intercal} $, we can solve for each component of this vector by means of Cramer's Rule: $$x_{i} = \frac{\det(A_{i})}{\det(A)}. \qquad \qquad (1) $$ Here, $A_{i}$ is formed by replacing the $i$'th column of the column vector of $A$ by the vector $\overrightarrow{b}$.
I'm curious whether a similar rule exists for “continuous” matrix equations. When we consider the Fredholm equation of the first kind:
$$g(t) = \int_{-\infty}^{\infty} K(t,s) f(s) ds ,$$ the solution is given by $$f(s) = \int_{-\infty}^{\infty} \frac{\mathcal{F}_{t} [g(t)](\omega) }{\mathcal{F}_{t} [K(t)](\omega) } e^{2\pi i \omega s} d \omega. \qquad \qquad (2)$$ Here, $\mathcal{F}_{t}$ is the Fourier transform.
My questions are:
- Can $(2)$ be considered to be a continuous analogue of $(1)$, when one sets $s = s^{*}$ (a particular value of $s$) ?
- If not, what is a continuous analogue of Cramer's rule for continuous matrix equations (involving integration against a kernel) ?