Browsing by Subject "distributions"

Now showing 1 - 4 of 4

Access status: Open Access ,
A distribution associated with the Kontorovich-Lebedev transform
(2006) Yakubovich, Semyon B.
We show that in a sense of distributions $\lim_{\varepsilon\to 0+} {1\over \pi^2} \tau\sinh\pi\tau \int_{\varepsilon}^{\infty} K_{i\tau}(y)K_{ix}(y){dy\over y} =\delta(\tau-x),$ where $\delta$ is the Dirac distribution, $\tau#, $x\in\mathbb{R}$ and $K_{\nu}(x)$ is the modified Bessel function. The convergence is in $\mathcal{E}^{\prime}(\mathbb{R})$ for any even $\varphi(x)\in\mathcal{E}(\mathbb{R})$ being a restriction to $\mathbb{R}$ of a function $\varphi(z)$ analytic in a horizontal open strip $G_a=\{z\in\mathbb{C}\colon\,|\text{Im}\,z|\lt a, \ a\gt 0\}$ and continuous in the strip closure. Moreover, it satisfies the condition $\varphi(z)=O\bigl(|z|^{-\text{Im}\,z-\alpha}e^{-\pi|\text{Re}\,z|/2}\bigr)$, $|\text{Re}\,z|\to\infty$ uniformly in $\overline{G_a}$. The result is applied to prove the representation theorem for the inverse Kontorovich-Lebedev transformation on distributions.
Access status: Open Access ,
A two cones support theorem
(2016) Estrada, Ricardo
We show that if the Radon transform of a distribution $f$ vanishes outside of an acute cone $C_{0}$, the support of the distribution is contained in the union of $C_{0}$ and another acute cone $C_{1}$, the cones are in a suitable position, and $f$ vanishes distributionally in the direction of the axis of $C_{1}$, then actually supp $\operatorname*{supp}f\subset C_{0}$. We show by examples that this result is sharp.
Access status: Open Access ,
Limit-point criteria for the matrix Sturm-Liouville operator and its powers
(2017) Brojtigam, Irina Nikolaevna
We consider matrix Sturm-Liouville operators generated by the formal expression $l[y]=-(P(y^{\prime}-Ry))^{\prime}-R^*P(y^{\prime}-Ry)+Qy,$ in the space $L^2_n(I)$, $I:=[0, \infty)$. Let the matrix functions $P:=P(x)$, $Q:=Q(x)$ and $R:=R(x)$ of order $n \in \mathbb{N}$ be defined on $I$, $P$ is a nondegenerate matrix, $P$ and $Q$ are Hermitian matrices for $x \in I$ and the entries of the matrix functions $P^{-1}$, $Q$ and $R$ are measurable on $I$ and integrable on each of its closed finite subintervals. The main purpose of this paper is to find conditions on the matrices $P$, $Q$ and $R$ that ensure the realization of the limit-point case for the minimal closed symmetric operator generated by $l^k[y]$ $k \in \mathbb{N}$. In particular, we obtain limit-point conditions for Sturm-Liouville operators with matrix-valued distributional coefficients.
Access status: Restricted ,
Ogólna postać funkcjonałów na przestrzeniach liniowo-topologicznych
(Data obrony: 2013-09-20) Pazgan, Marcin
Wydział Matematyki Stosowanej