This should really be well-known, but I was not able to find a definite answer to this question: Is the Fourier transform of a bounded function always a borel measure (i.e. an ord …

Let $g\in \mathcal{E}'(\mathbb{R} )$ be a distribution on $\mathbb{R}$ having compact support. Define $f\in \mathcal{D}'(\mathbb{R})$ by $$ f = \varphi \ast g , $$ where $\ast $ d …

A fractional derivative of distributions is usually introduced using definition of fractional integral as a convolution of two distributions. However there is another approach base …

Is there some regularizing version of Schwartz kernel theorem for topological spaces, i.e., in the form of Every continuous linear map $A\colon C\prime(X_2) \to C(X_1)$ is give …

I note here $\mathcal{D}'$ the space of all distributions and $\mathcal{S}'$ the space of tempered distributions, I am considering the following question: Let $u \in \mathcal{D}'$ …

Let $E_{\alpha}(x^{\alpha})$ be a Mittag-Leffler function, $\alpha \in (0,1)$. It is an eigenfunction for nonlocal fractional derivative, defined as a convolution with $$ \Phi_{ …

(Note that I am interested in the Gelfand-Pettis integral specifically, as opposed to, for example, the Bochner integral.) I have tried Googling things like "integral topological …

This question is related to my previous question about the regularity of the Maxwell equations. Assume we are working on a space where there are only electric point charges, $(q_i …

A few questions, hopefully to spark some discussion. How can one define a product of measures? We could use Colombeau products by embedding the measures into the distributions? …

Consider the fourier transform of $e^{it|\xi|^{2\alpha}}$ ($\alpha>0$)in $\mathbb{R}^n$,let $K_{\alpha}=\mathcal{F}(e^{it|\xi|^{\alpha}})$,so $K$ is a tempered distribution.Now I w …

What is the notion of a principal value of an integral when the singularity appears at one endpoint? Namely, $ PV \int_a^b f(t) dt = ? $, where the integral is convergent in the …

Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give …

If we equip the Schwartz space $\mathcal{S}$ with its usual Fréchet space topology, then the space of continuous linear functionals $\mathcal{S}^\ast$ is known as the space of Schw …

Let $\mathcal S (\mathbb R)$ denote the space of Schwartz functions on $\mathbb R$ and $\mathcal S^* (\mathbb R)$ denote the dual space of Schwartz (a.k.a tempered) distributions. …

I imagine most people who frequent MO have been indoctrinated into the point of view that the Riemann integral can be safely discarded once one has taken the time to develop the Le …