Solving a problem of L. Schwartz, those constant coefficient partial differential operators $P\left(D\right)$ are characterized that admit a continuous linear right inverse on $ℰ\left(\Omega \right)$ or ${𝒟}^{\text{'}}\left(\Omega \right)$, $\Omega$ an open set in ${\mathbf{R}}^n$. For bounded $\Omega$ with $C^1$-boundary these properties are equivalent to $P\left(D\right)$ being very hyperbolic. For $\Omega ={\mathbf{R}}^n$ they are equivalent to a Phragmen-Lindelöf condition holding on the zero variety of the polynomial $P$.

To a plurisubharmonic function $u$ on ${\mathbf{C}}^n$ with logarithmic growth at infinity, we may associate the Robin function $${\displaystyle {\rho }_u\left(z\right)=\underset{\lambda \to \infty }{lim\; sup}u\left(\lambda z\right)-log\left(\lambda z\right)}$$ defined on ${\mathbf{P}}^{n-1}$, the hyperplane at infinity. We study the classes $L_{+}$, and (respectively) $L_p$ of plurisubharmonic functions which have the form $u=log(1+|z\left|\right)+O\left(1\right)$ and (respectively) for which the function ${\rho }_u$ is not identically $-\infty$. We obtain an integral formula which connects the Monge-Ampère measure on the space ${\mathbf{C}}^n$ with the Robin function on ${\mathbf{P}}^{n-1}$. As an application we obtain a criterion...

Given an irreducible algebraic curves $A$ in ${ℂ}^N$, let $m_d$ be the dimension of the complex vector space of all holomorphic polynomials of degree at most $d$ restricted to $A$. Let $K$ be a nonpolar compact subset of $A$, and for each $d=1,2,...,$ choose $m_d$ points ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$ in $K$. Finally, let ${\Lambda }_d$ be the $d$-th Lebesgue constant of the array $\left\{A_{dj}\right\}$; i.e., ${\Lambda }_d$ is the operator norm of the Lagrange interpolation operator $L_d$ acting on $C\left(K\right)$, where $L_d\left(f\right)$ is the Lagrange interpolating polynomial for $f$ of degree $d$ at the points ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$. Using techniques...

For $a\in {\mathbf{R}}^n\setminus \left\{0\right\}$ and $\Omega$ an open bounded subset of ${\mathbf{R}}^n$ definie $L^p(\Omega ,a)$ as the closed subset of $L^p\left(\Omega \right)$ consisting of all functions that are constant almost everywhere on almost all lines parallel to $a$. For a given set of directions $a^{\nu }\in {\mathbf{R}}^n\setminus \left\{0\right\}$, $\nu =1,...,m$, we study for which $\Omega$ it is true that the vector space $$(*)L^p(\Omega ,a^1)+\cdots +L^p(\Omega ,a^m)\text{is}\text{a}\text{closed}\text{subspace}\text{of}{L}^p\left(\Omega \right).$$ This problem arizes naturally in the study of image reconstruction from projections (tomography). An essentially equivalent problem is to decide whether a certain matrix-valued differential operator...

In this paper we prove a lower bound for the linear dependence of three positive rational numbers under certain weak linear independence conditions on the coefficients of the linear forms. Let $\Lambda =b_{2}log{\alpha }_2-b_{1}log{\alpha }_1-b_{3}log{\alpha }_3\ne 0$ with $b_1,b_2,b_3$ positive integers and ${\alpha }_1,{\alpha }_2,{\alpha }_3$ positive multiplicatively independent rational numbers greater than $1$. Let ${\alpha }_{j1}={\alpha }_{j1}/{\alpha }_{j2}$ with ${\alpha }_{j1},{\alpha }_{j2}$ coprime positive integers $(j=1,2,3)$. Let ${\alpha }_j\ge \text{max}\{{\alpha }_{j1},e\}$ and assume that gcd$(b_1,b_2,b_3)=1.$ Let $$b^{\text{'}}=\left(\frac{b_2}{log{\alpha }_1}+\frac{b_1}{log{\alpha }_2}\right)\left(\frac{b_2}{log{\alpha }_3}+\frac{b_3}{log{\alpha }_2}\right)$$ and assume that $B\ge \text{max}\{10,log{b}^{\text{'}}\}.$ We prove that either $\{b_1,b_2,b_3\}$ is $\left(c_4,B\right)$-linearly dependent over $\mathbb{Z}$ (with respect to $\left(a_1,a_2,a_3\right)$)...

For $x\in ℂ$, $\left|x\right|\<1$, $s\in \mathbb{N}$, let ${\mathrm{Li}}_s\left(x\right)$ be the $s$-th polylogarithm of $x$. We prove that for any non-zero algebraic number $\alpha$ such that $\left|\alpha \right|\<1$, the $\mathbb{Q}\left(\alpha \right)$-vector space spanned by $1,{\mathrm{Li}}_1\left(\alpha \right),{\mathrm{Li}}_2\left(\alpha \right),\cdots$ has infinite dimension. This result extends a previous one by Rivoal for rational $\alpha$. The main tool is a method introduced by Fischler and Rivoal, which shows the coefficients of the polylogarithms in the relevant series to be the unique solution of a suitable Padé approximation problem.