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Soit ${\left({\lambda }_n\right)}_{n\ge 1}$ une suite de Blaschke du disque unité $𝔻$ et $\Theta$ une fonction intérieure. On suppose que la suite de noyaux reproduisants ${\left(k_{\Theta }(z,{\lambda }_n):=\frac{1-\overline{\Theta \left({\lambda }_n\right)}\Theta \left(z\right)}{1-\overline{{\lambda }_n}z}\right)}_{n\ge 1}$ est complète dans l’espace modèle $K_{\Theta }^p:=H^p\cap \Theta \overline{H_0^p}$, $1\<p\<+\infty$. On étudie, dans un premier temps, la stabilité de cette propriété de complétude, à la fois sous l’effet de perturbations des fréquences ${\left({\lambda }_n\right)}_{n\ge 1}$ mais également sous l’effet de perturbations de la fonction $\Theta$. On retrouve ainsi un certain nombre de résultats classiques sur les systèmes d’exponentielles. Puis, si on suppose de plus que la suite ...

In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges–Rovnyak spaces $\mathscr{H}\left(b\right)$, where $b$ is in the unit ball of $H^{\infty }\left({ℂ}_{+}\right)$. In particular, we generalize a result of Ahern–Clark obtained for functions of the model spaces $K_b$, where $b$ is an inner function. Using hypergeometric series, we obtain a nontrivial formula of combinatorics for sums of binomial coefficients. Then we apply this formula to show the norm convergence of reproducing kernel $k_{\omega ,n}^b$ of evaluation...

We discuss boundedness and compactness properties of the embedding $M_{\Lambda }^1\subset L^1\left(\mu \right)$, where $M_{\Lambda }^1$ is the closed linear span of the monomials $x^{{\lambda }_n}$ in $L^1\left([0,1]\right)$ and $\mu$ is a finite positive Borel measure on the interval $[0,1]$. In particular, we introduce a class of “sublinear” measures and provide a rather complete solution of the embedding problem for the class of quasilacunary sequences $\Lambda$. Finally, we show how one can recapture some of Al Alam’s results on boundedness and the essential norm of weighted composition operators from $M_{\Lambda }^1$...

Suppose $H^2$ is the Hardy space of the unit disc in the complex plane, while $\Theta$ is an inner function. We give conditions for a sequence of normalized reproducing kernels in the model space $K_{\Theta }=H^2\ominus \Theta H^2$ to be asymptotically close to an orthonormal sequence. The completeness problem is also investigated.