Let $\Gamma$ be a rectifiable Jordan curve in the finite complex plane $ℂ$ which is regular in the sense of Ahlfors and David. Denote by $L_C^2\left(\Gamma \right)$ the space of all complex-valued functions on $\Gamma$ which are square integrable w.r. to the arc-length on $\Gamma$. Let $L^2\left(\Gamma \right)$ stand for the space of all real-valued functions in $L_C^2\left(\Gamma \right)$ and put $$L_0^2\left(\Gamma \right)=\{h\in L^2\left(\Gamma \right){\int }_{\Gamma }h\left(\zeta \right)\left|\mathrm{d}\zeta \right|=0\}.$$ Since the Cauchy singular operator is bounded on $L_C^2\left(\Gamma \right)$, the Neumann-Poincaré operator $C_1^{\Gamma }$ sending each $h\in L^2\left(\Gamma \right)$ into $$C_1^{\Gamma }h\left({\zeta }_0\right):=\Re {\left(\pi \mathrm{i}\right)}^{-1}\mathrm{P}.V.{\int }_{\Gamma }\frac{h\left(\zeta \right)}{\zeta -{\zeta }_0}\mathrm{d}\zeta ,{\zeta }_0\in \Gamma ,$$ is bounded on $L^2\left(\Gamma \right)$. We show that the inclusion $$C_1^{\Gamma }\left(L_0^2\left(\Gamma \right)\right)\subset L_0^2\left(\Gamma \right)$$ characterizes the circle in the class of all...

We give several characterizations of the symmetrized n-disc Gₙ which generalize to the case n ≥ 3 the characterizations of the symmetrized bidisc that were used in order to solve the two-point spectral Nevanlinna-Pick problem in ℳ ₂(ℂ). Using these characterizations of the symmetrized n-disc, which give necessary and sufficient conditions for an element to belong to Gₙ, we obtain necessary conditions of interpolation for the general spectral Nevanlinna-Pick problem. They also allow us to give a...

We consider the space of holomorphic functions at the origin which extend analytically on the universal covering of $ℂ\setminus \omega \mathbb{Z}$, $\omega \in {ℂ}^{☆}$. We show that this space is stable by convolution product, thus is a resurgent algebra.

We present some results on the location of zeros of regular polynomials of a quaternionic variable. We derive new bounds of Eneström-Kakeya type for the zeros of these polynomials by virtue of a maximum modulus theorem and the structure of the zero sets of a regular product established in the newly developed theory of regular functions and polynomials of a quaternionic variable. Our results extend some classical results from complex to the quaternionic setting as well.

In this paper we completely characterize those weighted Hardy spaces that are Poletsky-Stessin Hardy spaces $H_u^p$. We also provide a reduction of $H^{\infty }$ problems to $H_u^p$ problems and demonstrate how such a reduction can be used to make shortcuts in the proofs of the interpolation theorem and corona problem.