Let $P(\lambda ,D)={\sum }_{\left|\alpha \right|\le m}{a}_{\alpha }\left(\lambda \right)D^{\alpha }$ be a differential operator with constant coefficients $a_{\alpha }$ depending analytically on a parameter $\lambda$. Assume that the family $\{$ P($\lambda$,D)$\}$ is of constant strength. We investigate the equation $P(\lambda ,D){𝔣}_{\lambda }\equiv g_{\lambda }$ where ${𝔤}_{\lambda }$ is a given analytic function of $\lambda$ with values in some space of distributions and the solution ${𝔣}_{\lambda }$ is required to depend analytically on $\lambda$, too. As a special case we obtain a regular fundamental solution of P($\lambda$,D) which depends analytically on $\lambda$. This result answers a question of L. Hörmander. ...

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...

Soit $D$ un domaine plan ; y a-t-il des suites $z_n$ telles que toute suite bornée $w$ puisse être interpolée en $z_n$ par une fonction $f\left(z\right)$ régulière et bornée dans $D$ ? Dans l’affirmative est-il vrai que toute suite $z_n$ qui tend assez rapidement vers la frontière de $D$ possède cette propriété ? On répond affirmativement à ces deux questions dans le cas où $D$ est le cercle-unité.

Let $Q={\left(Q_k\right)}_{k\ge 0},Q_0=1,Q_{k+1}=q_k{Q}_k,q_k\ge 2$, be a Cantor scale, ${𝐙}_Q$ the compact projective limit group of the groups $𝐙/Q_k𝐙$, identified to ${\prod }_{0\le j\le k-1}𝐙/q_j𝐙$, and let $\mu$ be its normalized Haar measure. To an element $x=\{a_0,a_1,a_2,\cdots \},0\le a_k\le q_{k+1}-1$, of ${𝐙}_Q$ we associate the sequence of integral valued random variables $x_k={\sum }_{0\le j\le k}{a}_j{Q}_j$. The main result of this article is that, given a complex $𝐐$-multiplicative function $g$ of modulus $1$, we have $$\underset{x_k\to x}{lim}(\frac{1}{x_k}\sum _{n\le x_k-1}g\left(n\right)-\prod _{0\le j\le k}\frac{1}{q_j}\sum _{0\le a\le q_j}g\left(a{Q}_j\right))=0\mu \text{-a.e}.$$

Pseudodifferential operators $P(x,D)\sim {\sum }_{j=0}^{+\infty }{P}_{m-j}(x,D)$ are studied, from the viewpoint of local solvability and under the assumption that, micro-locally, the principal symbol factorizes as $P_m=Q{L}^2$ with $Q$ elliptic, homogeneous of degree $m-2$, and $L$ homogeneous of degree one, satisfying the following condition : there is a point $(x_0,{\xi }^0)$ in the characteristic variety $L=0$ and a complex number $z$ such that $d_{\xi }\mathrm{Re}\left(zL\right)\ne 0$ at $(x_0,{\xi }^0)$ and such that the restriction of $\mathrm{Im}\left(zL\right)$ to the bicharacteristic strip of $\mathrm{Re}\left(zL\right)$ vanishes of order $k\<+\infty$ at $(x_0,{\xi }^0)$, changing sign there from...

Let ⨍ be an analytic function on a compact subset K of the complex plane ℂ, and let $r_n\left(z\right)$ denote the rational function of degree n with poles at the points ${b_{ni}}_{i=1}^n$ and interpolating ⨍ at the points ${a_{ni}}_{i=0}^n$. We investigate how these points should be chosen to guarantee the convergence of $r_n$ to ⨍ as n → ∞ for all functions ⨍ analytic on K. When K has no “holes” (see [8] and [3]), it is possible to choose the poles ${b_{ni}}_{i,n}$ without limit points on K. In this paper we study the case of general compact sets K, when...

Given a large prime number $p$ and a rational function $r\left(X\right)$ defined over ${𝔽}_p=\mathbb{Z}/p\mathbb{Z}$, we investigate the size of the set $\left\{x\in {𝔽}_p:\tilde{r}\left(x\right)\>\tilde{r}(x+1)\right.$, where $\tilde{r}\left(x\right)$ and $\tilde{r}(x+1)$ denote the least positive representatives of $r\left(x\right)$ and $r(x+1)$ in $\mathbb{Z}$ modulo $p\mathbb{Z}$.