Recently, several papers have considered a nonlinear analogue of Fourier series in signal analysis, referred to as either nonlinear phase unwinding or adaptive Fourier decomposition. In these processes, a signal is represented as the real component of a complex function $F: \partial\mathbb{D}\to \mathbb{C}$, and by performing an iterative method to obtain a sequence of Blaschke decompositions, the signal can be approximated using only a few terms. To better understand the convergence of these methods, the study of Blaschke decompositions on weighted Hardy spaces was studied by Coifman and Steinerberger, under the assumption that the complex valued function $F$ has an analytic extension to $\mathbb{D}_{1+ε}$ for some $ε>0$. This provided bounds on weighted Hardy norms involving a function and its Blaschke decomposition. That work also noted that in many examples, the nonlinear unwinding series of a function converges at an exponential rate to the original signal, which when coupled with an efficient algorithm to perform a Blaschke decomposition, has lead to a new and efficient way to approximate signals.
In this work, we continue the study of Blaschke decompositions on weighted Hardy Spaces for functions in the larger space $\mathcal{H}^2(\mathbb{D})$ under the assumption that the function has finitely many roots in $\mathbb{D}$. By studying the growth rate of the weights, we improve the bounds provided by Coifman and Steinerberger. This provides us with new insights into Blaschke decompositions on classical function spaces including the Hardy-Sobolev spaces and weighted Bergman spaces. Further, we state a sufficient condition on the weights for our improved bounds to hold for any function in the Hardy space, $\mathcal{H}^2(\mathbb{D})$. These results may help to better explain why the exponential convergence of the unwinding series is seen in many numerical examples.
A sequence which is a finite union of interpolating sequences for $H^\infty$ have turned out to be especially important in the study of Bergman spaces. The Blaschke products $B(z)$ with such zero sequences have been shown to be exactly those such that the multiplication $f \mapsto fB$ defines an operator with closed range on the Bergman space. Similarly, they are exactly those Blaschke products that boundedly divide functions in the Bergman space which vanish on their zero sequence. There are several characterizations of these sequences, and here we add two more to those already known. We also…
We continue the work of \cite{TLNT}. Let $E$ be a non-Blaschke subset of the unit disc $\mathbb{D}$ of the complex plane $\mathbb{C}$. Fixed $1\leq p\leq \infty$, let $H^p(\mathbb{D})$ be the Hardy space of holomorphic functions in the disk whose boundary value function is in $L^p(\partial \mathbb{D})$. Fixed $0<R<1$. For $ε>0$ define C_p(\varepsilon, R) = \sup \{\sup_{|z| \leq R}|g(z)|: g\in H^p, \|g\|_p\leq 1, |g(ζ)| \leq \varepsilon \forall ζ\in E\}. In this paper we find upper and lower bounds for $C_p(ε, R)$ when $ε$ is small for any non-Blaschke set $E$. The bounds are nearly-optimal for…
We give a generalization of the notion of finite Blaschke products from the perspective of generalized inner functions in various reproducing kernel Hilbert spaces. Further, we study precisely how these functions relate to the so-called Shapiro--Shields functions and shift-invariant subspaces generated by polynomials. Applying our results, we show that the only entire inner functions on weighted Hardy spaces over the unit disk are multiples of monomials, extending recent work of Cobos and Seco.
Inner functions play a central role in function theory and operator theory on the Hardy space over the unit disk. Motivated by recent works of C. Bénéteau et al. and of D. Seco, we discuss inner functions on more general weighted Hardy spaces and investigate a method to construct analogues of finite Blaschke products.
An inspiration at the origin of wavelet analysis (when Grossmann, Morlet, Meyer and collaborators were interacting and exploring versions of multiscale representations) was provided by the analysis of holomorphic signals, for which the images of the phase of Cauchy wavelets were remarkable in their ability to reveal intricate singularities or dynamic structures, such as instantaneous frequency jumps in musical recordings. Our goal is to follow their seminal work and introduce recent developments in nonlinear analysis. In particular we sketch methods extending conventional Fourier analysis, exp…
The relationship between the distribution of zeros of an infinite Blaschke product $B$ and the inclusion in weighted Bergman spaces $A_α^p$ of the derivative of $B$ or the derivative of functions in its model space $H^2 \ominus BH^2$ is investigated.
The maximum selection principle allows to give expansions, in an adaptive way, of functions in the Hardy space $\mathbf H_2$ of the disk in terms of Blaschke products. The expansion is specific to the given function. Blaschke factors and products have counterparts in the unit ball of $\mathbb C^N$, and this fact allows us to extend in the present paper the maximum selection principle to the case of functions in the Drury-Arveson space of functions analytic in the unit ball of $\mathbb C^N$. This will give rise to an algorithm which is a variation in this higher dimensional case of the greedy a…
We provide a characterization of the commutant of analytic Toeplitz operators $T_B$ induced by finite Blachke products $B$ acting on weighted Bergman spaces which, as a particular instance, yields the case $B(z)=z^n$ on the Bergman space solved recently by by Abkar, Cao and Zhu. Moreover, it extends previous results by Cowen and Wahl in this context and applies to other Banach spaces of analytic functions such as Hardy spaces $H^p$ for $1<p<\infty$. Finally, we apply this approach to study the reducing subspaces of $T_{B}$ in weighted Bergman spaces.
The theory of slice regular functions of a quaternionic variable, introduced in 2006 by Gentili and Struppa, extends the notion of holomorphic function to the quaternionic setting. This fast growing theory is already rich of many results and has interesting applications. In this setting, the present paper is devoted to introduce and study the quaternionic counterparts of Hardy spaces of holomorphic functions of one complex variable. The basic properties of the theory of quaternionic Hardy spaces are investigated, and in particular a Poisson-type representation formula, the notions of outer fun…
In this paper we give sharp norm estimates for the Bergman operator acting from weighted mixed-norm spaces to weighted Hardy spaces in the ball, endowed with natural norms.
Consider a scaled Nevanlinna-Pick interpolation problem and let $Π$ be the Blaschke product whose zeros are the nodes of the problem. It is proved that if $Π$ belongs to a certain class of inner functions, then the extremal solutions of the problem or most of them, are in the same class. Three different classical classes are considered: inner functions whose derivative is in a certain Hardy space, exponential Blaschke products and also the well known class of $α$-Blaschke products, for $0<α<1$.
The main goal of this paper is to give an unified proof of the corona problem on weighted Hardy spaces and on Morrey spaces. We use a technique that allows to reduce the problem to the Hardy spaces $H^2(θ)$
We obtain characterizations of positive Borel measures $μ$ on $\B^n$ so that some weighted Hardy-Sobolev are imbedded in $L^p(dμ)$, where $w$ is an $A_p$ weight in the unit sphere of $\C^n$.
In this paper we completely characterize those weighted Hardy spaces that are Poletsky--Stessin Hardy spaces $H^p_u$. We also provide a reduction of $H^\infty$ problems to $H^p_u$ problems and demonstrate how such a reduction can be used to make shortcuts in the proofs of the interpolation theorem and corona problem.
Let $0<p<\infty$ and $0\leq q<\infty$. For each $f$ in the weighted Hardy space $H_{p, q}$,\ we show that $d\|f_r\|_{p,q}^p/dr$ grows at most like $o(1/1- r)$ as $r\rightarrow 1$.
In this paper, we investigate some properties of planar harmonic mappings. First, we generalize the main results in \cite{CPW3} and \cite{HT}, and then discuss the relationship between area integral means and harmonic Hardy spaces or harmonic weighted Bergman spaces. At the end, coefficient estimates of mappings in weighted Bergman spaces are obtained.
In this paper we study the boundary values of harmonic and holo- morphic functions in the weighted Hardy spaces on the unit disk $\mathbb{D}$. These spaces were introduced by Poletsky and Stessin in [6] for plurisubharmonic functions on hyperconvex domains $D \subset \mathbb{C}^n$ as generalizations of classical Hardy spaces. We show that in the case when $D$ is the unit disk $\mathbb{D}$ the theory of boundary values for functions in these spaces is analogous to the classical one.
Free interpolation in Hardy spaces is caracterized by the well-known Carleson condition. The result extends to Hardy-Orlicz spaces contained in the scale of classical Hardy spaces $H^p$, $p>0$. For the Smirnov and the Nevanlinna classes, interpolating sequences have been characterized in a recent paper in terms of the existence of harmonic majorants (quasi-bounded in the case of the Smirnov class). Since the Smirnov class can be regarded as the union over all Hardy-Orlicz spaces associated with a so-called strongly convex function, it is natural to ask how the condition changes from the Carles…
In this paper, we consider the weighted Hardy space $\mathcal{H}^p(ω)$ induced by an $A_1$ weight $ω.$ We characterize the positive Borel measure $μ$ such that the identical operator maps $\mathcal{H}^p(ω)$ into $L^q(dμ)$ boundedly when $0<p, q<\infty.$ As an application, we obtain necessary and sufficient conditions for the boundedness of generalized area operators $A_{μ,ν}$ from $\mathcal{H}^p(ω)$ to $L^q(ω).$
The Dirichlet--Hardy space $\Ht$ consists of those Dirichlet series $\sum_n a_n n^{-s}$ for which $\sum_n |a_n|^2<\infty$. It is shown that the Blaschke condition in the half-plane $\operatorname{Re} s>1/2$ is a necessary and sufficient condition for the existence of a nontrivial function $f$ in $\Ht$ vanishing on a given bounded sequence. The proof implies in fact a stronger result: every function in the Hardy space $H^2$ of the half-plane $\operatorname{Re} s>1/2$ can be interpolated by a function in $\Ht$ on such a Blaschke sequence. Analogous results are proved for the Hilbert space $\Da$ …
We prove that a conformal mapping defined on the unit disk belongs to a weighted Bergman space if and only if certain integrals involving the harmonic measure converge. With the aid of this theorem, we give a geometric characterization of conformal mappings in Hardy or weighted Bergman spaces by studying Euclidean areas. Applying these results, we prove several consequences for such mappings that extend known results for Hardy spaces to weighted Bergman spaces. Moreover, we introduce a number which is the analogue of the Hardy number for weighted Bergman spaces. We derive various expressions f…
In this article, we consider two classes of weighted Hardy spaces on products of planar domains and their corresponding kernel functions, and we prove product versions of Saitoh's conjecture related to the two classes of weighted Hardy spaces.
In this paper we mainly discuss three things. First, there is no canonical norm on the space $H^p_u(\mathbb{D})$. Second, we improve the weak-$*$ convergence of the measures $μ_{u,r}$. Third, the dilations $f_t$ of the function $f\in H^p_u(\mathbb{D})$ converge to $f$ in $H^p_u$-norm and hence the polynomials are dense in $H^p_u(\mathbb{D})$.
In this paper, we introduce a discrete analogue of weighted Hardy spaces on rooted trees and study weighted composition operators between them in detail. In particular, we characterize bounded and compact weighted composition operators between discrete Hardy spaces. We also consider isometric weighted composition operators between these spaces.
In this paper, we completely characterize the order boundedness of weighted composition operators between different weighted Dirichlet spaces and different derivative Hardy spaces.
In this paper, we give some estimates for the essential norm and a new characterization for the boundedness and compactness of weighted composition operators from weighted Bergman spaces and Hardy spaces to the Bloch space.
In this paper, we investigate the complex symmetric structure of generalized weighted composition operators $D_{m,ψ,φ}$ on the weighted Hardy space $H^2(β)$. We obtain explicit conditions for $ D_{m,ψ,φ}$ to be complex symmetric with the conjugation $J_w$. Under the assumption that $ D_{m,ψ,φ}$ is $J_w$-symmetric, some sufficient and necessary conditions for $D_{m,ψ,φ}$ to be Hermitian and normal are given.
In this paper, the boundedness and compactness of the difference of composition-differentiation operators $D_φ-D_ψ$ acting from Hardy spaces $H^p$ to weighted Bergman spaces $A^q_α$ are completely characterize for all $0<p,q<\infty$.
In this paper, we investigate the spectra of invertible weighted composition operators with automorphism symbols, on Hardy space $H^2(\mathbb{B}_N)$ and weighted Bergman spaces $A_α^2(\mathbb{B}_N)$, where $\mathbb{B}_N$ is the unit ball of the $N$-dimensional complex space. By taking $N=1$, $\mathbb{B}_N=\mathbb{D}$ the unit disc, we also complete the discussion about the spectrum of a weighted composition operator when it is invertible on $H^2(\mathbb{D})$ or $A_α^2(\mathbb{D})$.
We prove that the weighted composition operator $W_{ϕ,φ}$ fixes an isomorphic copy of $\ell^p$ if the operator $W_{ϕ,φ}$ is not compact on the derivative Hardy space $S^p$. In particular, this implies that the strict singularity of the operator $W_{ϕ,φ}$ coincides with the compactness of it on $S^p$. Moreover, when $p\neq2$, we characterize the conditions for those weighted composition operators $W_{ϕ,φ}$ on $S^p$ which fix an isomorphic copy of $\ell^2$ .
We study reproducing kernel Hilbert and Pontryagin spaces of slice hyperholomorphic functions which are analogs of the Hilbert spaces of analytic functions introduced by de Branges and Rovnyak. In the first part of the paper we focus on the case of Hilbert spaces, and introduce in particular a version of the Hardy space. Then we define Blaschke factors and Blaschke products and we consider an interpolation problem. In the second part of the paper we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in partic…
Let $\mathcal{D}$ be the class of radial weights on the unit disk which satisfy both forward and reverse doubling conditions. Let $g$ be an analytic function on the unit disk $\mathbb{D}$. We characterize bounded and compact Volterra type integration operators \[ J_{g}(f)(z)=\int_{0}^{z}f(λ)g'(λ)dλ\] between weighted Bergman spaces $L_{a}^{p}(ω)$ induced by $\mathcal{D}$ weights and Hardy spaces $H^{q}$ for $0<p,q<\infty$.
Suppose $n\geq 3$ and let $B$ be the open unit ball in $\mathbb{R}^n$. Let $φ: B\to B$ be a $C^2$ map whose Jacobian does not change sign, and let $ψ$ be a $C^2$ function on $B$. We characterize bounded weighted composition operators $W_{φ,ψ}$ acting on harmonic Hardy spaces $h^p(B)$. In addition, we compute the operator norm of $W_{φ,ψ}$ on $h^p(B)$ when $φ$ is a Möbius transformation of $B$.
Let $B$ be a meromorphic Blaschke product in the upper half-plane with zeros $z_n$ and let $K_B=H^2\ominus BH^2$ be the associated model subspace of the Hardy class. In other words, $K_B$ is the space of square summable meromorphic functions with the poles at the points $\bar z_n$. A nonnegative function $w$ on the real line is said to be an admissible majorant for $K_B$ if there is a non-zero function $f\in K_B$ such that $|f|\le w$ a.e. on $\mathbb{R}$. We study the relations between the distribution of the zeros of a Blaschke product $B$ and the class of admissible majorants for the space $…
In this paper, we investigate the boundedness, compactness, essential norm and the Schatten class of weighted composition operators $uC_φ$ on Bergman type spaces $A_ω^p $ with double weight $ω$. Let $X=\{u\in H(D): uC_φ:A_ω^p\to A_ω^p \mbox{ is bounded}\}.$ For some regular weights $ω$, we obtain that $X=H^\infty$ if and only if $φ$ is a finite Blaschke product.
A classical result due to Blaschke states that for every analytic self-map $f$ of the open unit disk of the complex plane there exists a Blaschke product $B$ such that the zero sets of $f$ and $B$ agree. In this paper we show that there is an analogue statement for critical sets, i.e. for every analytic self-map $f$ of the open unit disk there is even an indestructible Blaschke product $B$ such that the critical sets of $f$ and $B$ coincide. We further relate the problem of describing the critical sets of bounded analytic functions to the problem of characterizing the zero sets of some weighte…
