Ill-posed integral equation problem using Fourier Transfom

Question

By using the Fourier Transform, show that the following equation $$\int_{-\infty}^{+\infty} K(x-y) g(y) dy = f(x), \qquad -\infty < x < \infty$$ is ill-posed.

For overcoming ill-posedness of above problem what we can do?

Answer 1

By the convolution theorem, the solution to the above equation is the inverse FT of

$$G(k) = \frac{F(k)}{\mathbf{K}(k)} $$

where $\mathbf{K}(k)$ is the FT of $K(x)$. This solution is ill-posed because $G(k)$ is infinite whenever $\mathbf{K}(k)$ is zero.

To get around this, you may use additional information in the problem of which the original equation is a model. For example, if $K$ is the impulse response of a linear system exhibiting some statistical noise, you may use an expectation-maximization technique to find the most likely value of $G(k)$. See this as a primer of deconvolution.