Let $f:\mathbb{R}^n \to \mathbb{R}$ be a non-negative Lebesgue integrable function with integral on $\mathbb{R}^n$ equals to 1. Let $\tilde{f}(x)=f(-x)$. Suppose $f$ satisfies the following equation: $$ f \ast \left( \log \frac{1}{f \ast \tilde{f}} \right) = c, $$ where $c \in \mathbb{R}$ is some constant. How do I find such a function $f$ which solves the above condition?
I have tried using Laplace transforms but the logarithmic function is making it hard to obtain any progress. Can any one help me solve this?
One can also take $n=1$ for simplicity.
Taking Fourier transforms turns the outer convolution into a product and the constant into a Dirac delta. Since the Fourier transform of f is continuous this shows that that of $\log f*\tilde{f}$ must be a Dirac delta. Hence the logarithm must be a constant and so must the inner convolution. Take Fourier transforms of this inner convolution and you see the product of the Fourier transforms must be a Dirac delta which is not possible for the product of two continuous functions