The deconvolution problem is to find a function $u(\mathbf x)$ from the equation: $$ K * u(\mathbf x) = \int_{M} K (\mathbf x - \xi) u(\xi) d \xi = f(\mathbf x) $$ It is said in my lectures that in the case of finite $f$ and $K$ the problem is ill-posed. From the convolution theorem follows, that: $$ u(\mathbf x) = \int_{M} e^{i \omega \mathbf x} \frac{\tilde{f}(\omega)}{\tilde{K}(\omega)} $$ Then it is stated, that in the latter integration the integration is over the whole space, and the frequencies $\omega$ can be very high due to the noise, and the integral may fail to converge.
In the first equation the action on the $u$ is well defined, however, do I correctly understand, that there can be none or more than one functional satisfying this equation, and this is reflected in the fact, that the lattter integral may be not defined?