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In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by $$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well define …

I was wondering if it possible to find a measure $\eta$ on $\mathbb{R}$ such that $$ L(x):=\log\frac{1}{|x|}=\int e^{itx}\;d\eta(t) $$ for every $x\in [0,1/2]$. On a structural ground, this questio …

For me, infinite matrix means an operator on $\ell^2(\mathbb N)$ (or sometimes $\ell^2(\mathbb Z)$, but usually referred to bi-infinite matrix). Concerning the eigenvalues, you thus may just look at t …

Let $(X_N)_N$ be a sequence of trace class operators acting on, say, $L^2(\mathbb{R})$. What are the minimal assumptions in order to have the convergence of their Fredholm determinant $$ \lim_N\det( …

Consider a probability measure $\mu$ on, let's say, $\mathbb R$. Is there a necessary and sufficient condition so that $\mu$ has compact support $Supp(\mu)$ ? I agree this question is too vague, an …

To deal with this singular integral is rather subtle. If you think about it, it may be possible that your integral is actually not well-defined under the only assumptions you suggest. One way to deal …

Imagine you investigate a set of objects $\mathcal{E}$, and you just realize this that $\mathcal{E}$ possesses an affine structure with respect to some real vector space $\mathcal{V}$ having a scalar …

Stieljes transform is about taking convolution of your signal $s(t)$ with the $\frac1 t$, so as to obtain $$ C_s(z)=\int \frac{s(t)}{z-t}dt.$$ It is of course well defined at least when $z$ is a compl …

I came up with this operation after playing with determinantal point processes: Given two probability densities $f,g$ defined on some measurable space with reference measure $\mu$, set $$ f\star g(x …