For functions in $C^k(\mathbb {R})$ which commute with a translation, we prove a theorem on approximation by entire functions which commute with the same translation, with a requirement that the values of the entire function and its derivatives on a specified countable set belong to specified dense sets. Using this theorem, we show that if A and B are countable dense subsets of the unit circle $T\subseteq \mathbb {C}$ with $1\notin A$, $1\notin B$, then there is an analytic function $h\colon \mathbb {C}\setminus \{0\}\to \mathbb {C}$ that restricts to an order isomorphism of the arc $T\setminus \{1\}$ onto itself and satisfies $h(A)=B$ and $h'(z)\not =0$ when $z\in T$. This answers a question of P. M. Gauthier.

Complex linear differential equations with entire coefficients are studied in the situation where one of the coefficients is an exponential polynomial and dominates the growth of all the other coefficients. If such an equation has an exponential polynomial solution $f$, then the order of $f$ and of the dominant coefficient are equal, and the two functions possess a certain duality property. The results presented in this paper improve earlier results by some of the present authors, and the paper adjoins with two open problems.

We consider the problem of computing the partition function $\sum _x e^{f(x)}$, where $f: \{-1, 1\}^n \longrightarrow {\mathbb R}$ is a quadratic or cubic polynomial on the Boolean cube $\{-1, 1\}^n$. In the case of a quadratic polynomial f, we show that the partition function can be approximated within relative error $0 < \epsilon < 1$ in quasi-polynomial $n^{O(\ln n - \ln \epsilon )}$ time if the Lipschitz constant of the non-linear part of f with respect to the $\ell ^1$ metric on the Boolean cube does not exceed $1-\delta $, for any $\delta>0$, fixed in advance. For a cubic polynomial f, we get the same result under a somewhat stronger condition. We apply the method of polynomial interpolation, for which we prove that $\sum _x e^{\tilde {f}(x)} \ne 0$ for complex-valued polynomials $\tilde {f}$ in a neighborhood of a real-valued f satisfying the above mentioned conditions. The bounds are asymptotically optimal. Results on the zero-free region are interpreted as the absence of a phase transition in the Lee–Yang sense in the corresponding Ising model. The novel feature of the bounds is that they control the total interaction of each vertex but not every single interaction of sets of vertices.

Let f and g be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type and also satisfy $f\circ g =g \circ f$. We show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form f and $g = \phi \circ f$, where $\phi$ is a quasiconformal map, have the same Julia sets and that polynomial type quasiregular maps cannot commute with transcendental type ones unless their degree is less than or equal to their dilatation.

En s’appuyant sur la notion d’équivalence au sens de Bohr entre polynômes de Dirichlet et sur le fait que sur un corps quadratique la fonction zeta de Dedekind peut s’écrire comme produit de la fonction zeta de Riemann et d’une fonction L, nous montrons que, pour certaines valeurs du discriminant du corps quadratique, les sommes partielles de la fonction zeta de Dedekind ont leurs zéros dans des bandes verticales du plan complexe appelées bandes critiques et que les parties réelles de leurs zéros y sont denses.

In 1955, Lehto showed that, for every measurable function $\psi $ on the unit circle ${\mathbb T}$, there is a function f holomorphic in the unit disc ${{\mathbb D}}$, having $\psi $ as radial limit a.e. on ${\mathbb T}$. We consider an analogous boundary value problem, where the unit disc is replaced by a Stein domain on a complex manifold and radial approach to a boundary point p is replaced by (asymptotically) total approach to p.

For a nondecreasing function $K: [0, \infty)\rightarrow [0, \infty)$ and $0<s<\infty $, we introduce a Morrey type space of functions analytic in the unit disk $\mathbb {D}$, denoted by $\mathcal {D}^s_K$. Some characterizations of $\mathcal {D}^s_K$ are obtained in terms of K-Carleson measures. A relationship between two spaces $\mathcal {D}^{s_1}_K$ and $\mathcal {D}^{s_2}_K$ is given by fractional order derivatives. As an extension of some known results, for a positive Borel measure $\mu $ on $\mathbb {D}$, we find sufficient or necessary condition for the embedding map $I: \mathcal {D}^{s}_{K}\mapsto \mathcal {T}^s_{K}(\mu)$ to be bounded.

Let $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk ${\mathbb {D}}\subset {{\mathbb C}}$. We show that the dense stable rank of $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ is $1$ and, using this fact, prove some nonlinear Runge-type approximation theorems for $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra $H^\infty ({\mathbb {D}})$.

The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.

Classical results about peaking from complex interpolation theory are extended to polynomials on a closed disk, and on the complement of its interior. New results are obtained concerning interpolation by univalent polynomials on a Jordan domain whose boundary satisfies certain smoothness conditions.

Let $(z_k)$ be a sequence of distinct points in the unit disc $\mathbb {D}$ without limit points there. We are looking for a function $a(z)$ analytic in $\mathbb {D}$ and such that possesses a solution having zeros precisely at the points $z_k$, and the resulting function $a(z)$ has ‘minimal’ growth. We focus on the case of non-separated sequences $(z_k)$ in terms of the pseudohyperbolic distance when the coefficient $a(z)$ is of zero order, but $\sup _{z\in {\mathbb D}}(1-|z|)^p|a(z)| = + \infty$ for any $p > 0$. We established a new estimate for the maximum modulus of $a(z)$ in terms of the functions $n_z(t)=\sum \nolimits _{|z_k-z|\le t} 1$ and $N_z(r) = \int_0^r {{(n_z(t)-1)}^ + } /t{\rm d}t.$ The estimate is sharp in some sense. The main result relies on a new interpolation theorem.

We prove a number of results concerning the Hausdorff and packing dimension of sets of points which escape (at least in average) to infinity at a given rate under non-autonomous iteration of exponential maps. In particular, we generalize the results proved by Sixsmith in 2016 and answer his question on annular itineraries for exponential maps.

This paper presents an approach, based on interpolation theory of operators, to the study of interpolating sequences for interpolation Banach spaces between Hardy spaces. It is shown that the famous Carleson result for H∞ can be lifted to a large class of abstract Hardy spaces. A description is provided of the range of the Carleson operator defined on interpolation spaces between the classical Hardy spaces in terms of uniformly separated sequences. A key role in this description is played by some general interpolation results proved in the paper. As by-products, novel results are obtained which extend the Shapiro–Shields result on the characterisation of interpolation sequences for the classical Hardy spaces Hp. Applications to Hardy–Lorentz, Hardy–Marcinkiewicz and Hardy–Orlicz spaces are presented.

Let f be analytic in the unit disk $\mathbb {D}=\{z\in \mathbb {C}:|z|<1 \}$ and let ${\mathcal S}$ be the subclass of normalised univalent functions with $f(0)=0$ and $f'(0)=1$, given by $f(z)=z+\sum _{n=2}^{\infty }a_n z^n$. Let F be the inverse function of f, given by $F(\omega )=\omega +\sum _{n=2}^{\infty }A_n \omega ^n$ for $|\omega |\le r_0(f)$. Denote by $ \mathcal {S}_p^{* }(\alpha )$ the subset of $ \mathcal {S}$ consisting of the spirallike functions of order $\alpha $ in $\mathbb {D}$, that is, functions satisfying

$$\begin{align*}{\mathrm{Re}} \ \bigg\{e^{-i\gamma}\dfrac{zf'(z)}{f(z)}\bigg\}>\alpha\cos \gamma, \end{align*}$$

for $z\in \mathbb {D}$, $0\le \alpha <1$ and $\gamma \in (-\pi /2,\pi /2)$. We give sharp upper and lower bounds for both $ |a_3|-|a_2| $ and $ |A_3|-|A_2| $ when $f\in \mathcal {S}_p^{* }(\alpha )$, thus solving an open problem and presenting some new inequalities for coefficient differences.

In this paper, we will show that the unitary equivalence of two multiplication operators on the Bergman spaces on polygons depends on the geometry of the polygon.

Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are functions for which the postsingular set is a compact subset of the Fatou set. Equivalently, they are characterized as being expanding. Mihaljević-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalize these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics.

We establish the global regularity of multilinear Fourier integral operators that are associated to nonlinear wave equations on products of $L^p$ spaces by proving endpoint boundedness on suitable product spaces containing combinations of the local Hardy space, the local BMO and the $L^2$ spaces.

This note characterizes, in terms of interpolating Blaschke products, the symbols of Hankel operators essentially commuting with all quasicontinuous Toeplitz operators on the Hardy space of the unit circle. It also shows that such symbols do not contain the complex conjugate of any nonconstant singular inner function.

In this paper we investigate the range of validity of Ruelle’s property. First, we show that every finitely generated Fuchsian group has Ruelle’s property. We also prove the existence of an infinitely generated Fuchsian group satisfying Ruelle’s property. Concerning the negative results, we first generalize Astala and Zinsmeister’s results [Mostow rigidity and Fuchsian groups. C. R. Math. Acad. Sci. Paris 311 (1990), 301–306; Teichmüller spaces and BMOA. Math. Ann. 289 (1991), 613–625] by proving that all convergence-type Fuchsian groups of the first kind fail to have Ruelle’s property. Finally, we give some results about second-kind Fuchsian groups. [-3.2pc]

We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$.