Related papers: Remarks on one-component inner functions

To an ideal in $\mathbb{C}[x,y]$ one can associate a topological zeta function. This is an extension of the topological zeta function associated to one polynomial. But in this case we use a principalization of the ideal instead of an…

We define Hardy spaces $H^p(\Omega_\pm)$ on half-strip domain~$\Omega_+$ and $\Omega_-= \mathbb{C}\setminus\overline{\Omega_+}$, where $0<p<\infty$, and prove that functions in $H^p(\Omega_\pm)$ has non-tangential boundary limit a.e. on…

Suppose that $f$ belongs to a suitably defined complete metric space $ {{\cal C}}^{{\alpha}}$ of H\"older $ {\alpha}$-functions defined on $[0,1]$. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper…

Let $p$ be an analytic function defined on the open unit disc $\mathbb{D}$ with $p(0)=1$ and $0< \alpha \leq 1$. The conditions on complex valued functions $C$, $D$ and $E$ are obtained for $p$ to be subordinate to $((1+z)/(1-z))^{\alpha}$…

If E is a separable symmetric sequence space with trivial Boyd indices and $\cC^E$ is the corresponding ideal of compact operators, then there exists a $C^1$-function $f_E$, a self-adjoint element $W\in \cC^E$ and a densely defined closed…

Let $H^2(\mathbb{D}^n)$ denote the Hardy space over the polydisc $\mathbb{D}^n$, $n \geq 2$. A closed subspace $\mathcal{Q} \subseteq H^2(\mathbb{D}^n)$ is called Beurling quotient module if there exists an inner function $\theta \in…

We consider the partial theta function $\theta (q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $z\in \mathbb{C}$ is a variable and $q\in \mathbb{C}$, $0<|q|<1$, is a parameter. Set $\alpha _0~:=~\sqrt{3}/2\pi ~=~0.2756644477\ldots$. We…

The Jacobi theta-functions admit a definition through the autonomous differential equations (dynamical system); not only through the famous Fourier theta-series. We study this system in the framework of Hamiltonian dynamics and find…

Concrete sharp constants in a pointwise estimate of the gradient of a harmonic function in the unit disk are obtained under the assumption that function belong to Hardy space $h^p$, $p\ge 1$. This generalizes some recent result of Maz'ya &…

If $Z(t) = \chi^{-1/2}(1/2+it)\zeta(1/2+it)$ denotes Hardy's function, where $\zeta(s) = \chi(s)\zeta(1-s)$ is the functional equation of the Riemann zeta-function, then it is proved that $$ \int_0^T Z(t)\d t = O_\e(T^{1/4+\e}). $$

Let $\theta$ be an inner function satisfying the connected level set condition of B. Cohn, and let $K^{1}_{\theta}$ be the shift-coinvariant subspace of the Hardy space $H^1$ generated by $\theta$. We describe the dual space to…

Given a symmetric linear transformation on a Hilbert space, a natural problem to consider is the characterization of its set of symmetric extensions. This problem is equivalent to the study of the partial isometric extensions of a fixed…

Bloch and Okounkov introduced an $n$-point correlation function on the fermionic Fock space and found a closed formula in terms of theta functions. This function affords several distinguished interpretations and in particular can be…

We introduce a generalization of the Bourgain-Rosenthal-Schechtman $R_{\omega}^p$ space: Let $Y$ be a Haar system Hardy space, i.e., a separable rearrangement-invariant function space on the unit interval or an associated Hardy space…

Let $n$ be a positive integer. Let $\mathbf U$ be the unit disk, $p\ge 1$ and let $h^p(\mathbf U)$ be the Hardy space of harmonic functions. Kresin and Maz'ya in a recent paper found the representation for the function $H_{n,p}(z)$ in the…

We introduce a variable exponent version of the Hardy space of analytic functions on the unit disk, we show some properties of the space, and give an example of a variable exponent $p(\cdot)$ that satisfies the $\log$-H\"older condition…

Theta functions play a major role in many current researches and are powerful tools for studying integrable systems. The purpose of this paper is to provide a short and quick exposition of some aspects of meromorphic theta functions for…

This paper aims to obtain decompositions of higher dimensional $L^p(\mathbb{R}^n)$ functions into sums of non-tangential boundary limits of the corresponding Hardy space functions on tubes for the index range $0<p<1$. In the one-dimensional…

The purpose of this note is to show that the set functions defined in \cite{trong-tuyen} can be suitably extended to all subsets $E$ of the unit disk $\mathbb{D}$. In particular we obtain uniform nearly-optimal estimates for the following…

Using an integral formula on a homogeneous Siegel domain, we show a necessary and sufficient condition for composition operators on the weighted Bergman space of a minimal bounded homogeneous domain to be compact. To describe the…