Myriam Ounaies
We use $L^2$ estimates for the $\bar\partial$ equation to find geometric conditions on discrete interpolating varieties for weighted spaces $A_p(\C)$ of entire functions such that $| f(z)|\le Ae^{Bp(z)}$ for some $A,B>0$. In particular, we give a characterization when $p(z)=e^{| z|}$ and more generally when $\ln p(e^r)$ is convex and $\ln p(r)$ is concave.
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We obtain sampling and interpolation theorems in radial weighted spaces of analytic functions for weights of arbitrary (more rapid than polynomial) growth. We give an application to invariant subspaces of arbitrary index in large weighted Bergman spaces.
We extend our work on nonseparated interpolating sequences, originally developed for Bergman spaces with weights of the form $(1 - |z|^2)^α$, to more general weights.
An approach to complex interpolation of compact subsets of $\Bbb C^n$, including Brunn-Minkowski type inequalities for the capacities of the interpolating sets, was developed recently by means of plurisubharmonic geodesics between relative extremal functions of the given sets. Here we show that a much better control can be achieved by means of the geodesics between weighted relative extremal functions. In particular, we establish convexity properties of the capacities that are stronger than those given by the Brunn-Minkowski inequalities.
Let $A_p(\C)$ be the space of entire functions such that $| f(z)|\le Ae^{Bp(z)}$ for some $A,B>0$ and let $V$ be a discrete sequence of complex numbers which is not a uniqueness set for $A_p(\C)$. We use $L^2$ estimates for the $\bar\partial$ equation to charaterize the trace of $A_p(\C)$ on $V$.
It is well known from a result by Shapiro-Shields that in the Hardy spaces, a sequence of reproducing kernels is uniformly minimal if and only if it is an unconditional basis in its span. This property which can be reformulated in terms of interpolation and so-called weak interpolation is not true in Paley-Wiener spaces in general. Here we show that the Carleson condition on a sequence $Λ$ together with minimality in Paley-Wiener spaces $PW_τ^{p}$ of the associated sequence of reproducing kernels implies the interpolation property of $Λ$ in $PW_{τ+ε}^{p}$, for every $ε>0$. With the same techni…
In this paper, we expand functions of specific $q$-exponential growth in terms of its even (odd) Askey- Wilson $q$-derivatives at $0$ and $η=(q^{1/4}+q^{-1/4})/2$. This expansion is a $q$-version of the celebrated Lidstone expansion theorem, where we expand the function in $q$-analogs of Lidstone polynomials, i.e., q-Bernoulli and $q$-Euler polynomials as in the classical case. We also raise and solve a $q$-extension of the problem of representing an entire function of the form $f(z)=g(z+1)-g(z)$, where $g(z)$ is also an entire function of the same order as $f(z)$.
The purpose of this paper is to prove an interpolation formula involving derivatives for entire functions of exponential type. We extend the interpolation formula derived by J. Vaaler in [37, Theorem 9] to general $L^p$ de Branges spaces. We extensively use techniques from de Branges' theory of Hilbert spaces of entire functions as developed in [6], but a crucial passage involves the Hilbert-type inequalities as derived in [15]. We give applications to homogeneous spaces of entire functions that involve Bessel functions and we prove a uniqueness result for extremal one-sided band-limited appro…
We study those smooth complex hypersurfaces W in C^n having the property that all holomorphic functions of finite weighted L^p norm on W extend to entire functions with finite weighted L^p norm. Such hypersurfaces are called interpolation hypersurfaces. We also examine the dual problem of finding all sampling hypersurfaces, i.e., smooth hypersurfaces W in C^n such that any entire function with finite weighted L^p norm is stably determined by its restriction to W. We provide sufficient geometric conditions on the hypersurface to be an interpolation and sampling hypersurface. The geometric condi…
We present sufficient conditions on a smooth uniformly flat hypersurface W in the unit ball to be an interpolation hypersurface or a sampling hypersurface for generalized Bergman spaces associated to the unit ball with its Bergman metric. The conditions are phrased in terms of a geometric Beurling-type densities, similar to the densities defined for hypersurfaces in C^n by Ortega-Cerda, Schuster and the second author. For the case of sampling, our proofs are different than those in the C^n case; we use an approach closer in spirit to the work of Berndtsson and Ortega-Cerda in the 1-dimensional…
We study multiple sampling and interpolation problems with unbounded multiplicities in the weighted Bergman space, both in the hilbertian case p = 2 and the uniform case p = +$\infty$.
We prove that square integrable holomorphic functions (with respect to a plurisubharmonic weight) can be extended in a square integrable manner from certain singular hypersurfaces (which include uniformly flat, normal crossing divisors) to entire functions in affine space. This provides evidence for a conjecture regarding the positivity of the curvature of the weight under consideration.
We give a description, in analytic and geometric terms, of the interpolation sequences for the algebra of entire functions of exponential type which are bounded on the real line.
We study the uniqueness sets, the weak interpolation sets, and convergence of the Lagrange interpolation series in radial weighted Fock spaces
In the space of all entire functions it is solved the problem of interpolation taking into account multiplicities by sums of the series of exponentials with the exponents from a given set. It is found a criterion of solubility of the problem in the case when all infinite portions of interpolation nodes are situated on a finite system of rays. It is also disclosed that the problem is equivalent to particular problems of simple interpolation, as well as of point-wise crude approximation, by sums of the series of exponentials.
We establish necessary and sufficient conditions for boundedness of composition operators on the most general class of Hilbert spaces of entire Dirichlet series with real frequencies. Depending on whether or not the space contains any nonzero constant function, different criteria for boundedness are developed. Thus, we complete the characterization of bounded composition operators on all known Hilbert spaces of entire Dirichlet series of one variable.
The Patterson-Sullivan construction is proved almost surely to recover a Bergman function from its values on a random discrete subset sampled with the determinantal point process induced by the Bergman kernel on the unit ball $\mathbb{D}_d$ in $\mathbb{C}^d$. For super-critical weighted Bergman spaces, the interpolation is uniform when the functions range over the unit ball of the weighted Bergman space. As main results, we obtain a necessary and sufficient condition for interpolation of a fixed pluriharmonic function in the complex hyperbolic space of arbitrary dimension (cf. Theorem 1.4 and …
Answering a question of Lindholm, we prove strict density inequalities for sampling and interpolation in Fock spaces of entire functions in several complex variables defined by a plurisubharmonic weight. In particular, these spaces do not admit a set that is simultaneously sampling and interpolating. To prove optimality of the density conditions, we construct sampling sets with a density arbitrarily close to the critical density. The techniques combine methods from several complex variables (estimates for $\bar \partial$) and the theory of localized frames in general reproducing kernel Hilbert…
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.
We consider a Nevanlinna-Pick interpolation problem on finite sequences of the unit disc D constrained by Hardy and radial-weighted Bergman norms. We find sharp asymptotics on the corresponding interpolation constants. As another application of our techniques we prove embedding theorems for rational functions. We find that the embedding of H $\infty$ into Hardy or radial-weighted Bergman spaces in D is invertible on the subset of rational functions of a given degree n whose poles are separated from the unit circle and obtain asymptotically sharp estimates of the corresponding embedding constan…
A pluripotential theoretic framework for polynomial interpolation of vector-valued functions and differential forms
本文提出修正的加权多势理论框架，引入向量值权重，将标量值函数插值的渐近结果部分推广到有限维复内积空间中的向量值函数，并应用于微分形式的多项式系数插值。
Let $Ω$ be a bounded, weakly pseudoconvex domain in C^n, n > 1, with real-analytic boundary. A real-analytic submanifold $M \subset bdΩ$ is called an analytic interpolation manifold if every real-analytic function on M extends to a function belonging to $\Cal{O}(\barΩ)$. We provide sufficient conditions for M to be an analytic interpolation manifold. We give examples showing that neither of these conditions can be relaxed, as well as examples of analytic interpolation manifolds lying entirely within the set of weakly pseudoconvex points of $bdΩ$.
For a real non-signdefinite function $B(z)$, $z\in \C$, we investigate the dimension of the space of entire analytical functions square integrable with weight $e^{\pm 2F}$, where the function $F(z)=F(x_1,x_2)$ satisfies the Poisson equation $\D F=B$. The answer is known for the function $B$ with constant sign. We discuss some classes of non-signdefinite positively homogeneous functions $B$, where both infinite and zero dimension may occur. In the former case we present a method of constructing entire functions with prescribed behavior at infinity in different directions. The topic is closely r…
We study the Bergman space interpolation problem of open Riemann surfaces obtained from a compact Riemann surface by removing a finite number of points. We equip such a surface with what we call an asymptotically flat conformal metric, i.e., a complete metric with zero curvature outside a compact subset. We then establish necessary and sufficient conditions for interpolation in weighted Bergman spaces over asymptotically flat Riemann surfaces.
Let $A$ be a sequence of points of $\mathbb{B}^n$ the unit ball in $\mathbb{C}^n.$ In terms of interpolating vectorial function (or Amar's function)[1], we give a necessary condition on $A$ to be interpolating for the weighted Bergman space $B^p_α(\mathbb{B}^n). In the particular case of Hardy space $H^p (\mathbb{B}^2)$, this condition is sufficient no optimal. In the main theorem proof, we resolve Gleason's problem (vectorial form) in $B^p_α(\mathbb{B}^n)$
Let $(M_j)_{j=1}^\infty\in\mathbb{N}$ and $(r_j)_{j=1}^\infty\in\mathbb{R}^+$ be increasing sequences satisfying some mild rate of growth conditions. We prove that there is an entire function $f: \mathbb{C} \rightarrow\mathbb{C}$ whose behavior in the large annuli $\{ z\in\mathbb{C} : r_{j}\cdot\exp(π/M_{j})\leq|z|\leq r_{j+1}\}$ is given by a perturbed rescaling of $z\mapsto z^{M_j}$, such that the only singular values of $f$ are rescalings of $\pm r_j^{M_j}$. We describe several applications to the dynamics of entire functions.
We investigate necessary and sufficient conditions under which entire functions in de Branges spaces can be recovered from function values and values of derivatives. Our main focus is on spaces with a structure function whose logarithmic derivative is bounded in the upper half-plane.
Carleson measures and interpolating and sampling sequences for weighted Bergman spaces on the unit disk are described for weights that are radial and grow faster than the standard weights $(1-|z|)^{-α}$, $0<α<1$. These results make the Hardy space $H^2$ appear naturally as a "degenerate" endpoint case for the class of Bergman spaces under study.
We find sufficient conditions for a discrete sequence to be interpolating or sampling for certain generalized Bergman spaces on open Riemann surfaces. As in previous work of Bendtsson, Ortega-Cerda, Seip, Wallsten and others, our conditions for interpolation and sampling are as follows: If a certain upper density of the sequence has value less that 1, then the sequence is interpolating, while if a certain lower density has value greater than 1, then the sequence is sampling. However, unlike previous work, we construct an infinite number of densities, naturally parameterized by $L^1_{loc} ([0,\…
According to Lidstone interpolation theory, an entire function of exponential type $<π$ is determined by it derivatives of even order at $0$ and $1$. This theory can be generalized to several variables. Here we survey the theory for a single variable. Complete proofs are given. This first paper of a trilogy is devoted to Univariate Lidstone interpolation; Bivariate and Multivariate Lidstone interpolation will be the topic of two forthcoming papers.
This paper concerns a commutant lifting theorem and a Nevanlinna-Pick type interpolation result in the setting of multipliers from vector-valued Drury-Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in $\mathbb{C}^n$. The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.
The de Branges spaces of entire functions generalise the classical Paley-Wiener space of square summable bandlimited functions. Specifically, the square norm is computed on the real line with respect to weights given by the values of certain entire functions. For the Paley-Wiener space, this can be chosen to be an exponential function where the phase increases linearly. As our main result, we establish a natural geometric characterisation, in terms of densities, for real sampling and interpolating sequences in the case when the derivative of the phase function merely gives a doubling measure o…
For $α>-1$ and $0<p<\infty$, we study weighted Bergman spaces $\mathcal B^p_α$ of harmonic functions on the real hyperbolic ball and obtain an atomic decomposition of these spaces in terms of reproducing kernels. We show that an $r$-separated sequence $\{a_m\}$ with sufficiently large $r$ is an interpolating sequence for $\mathcal B^p_α$. Using these we determine precisely when a Bergman space $\mathcal B^p_α$ is included in another Bergman space $\mathcal B^q_β$.
On strong growth conditions for weighted spaces of entire functions
该研究刻画快速递减增长整函数加权类的包含关系与权的强增长比较，通过相伴权序列将抽象权函数归约为序列情形，对比了伸缩型与指数型权系统定义的加权整函数空间。
In the spirit of very recent articles by J. Bonet, W. Lusky and J. Taskinen we are studying the so-called solid hulls and cores of spaces of weighted entire functions when the weights are given in terms of associated weight functions coming from weight sequences. These sequences are required to satisfy certain (standard) growth and regularity properties which are frequently arising and used in the theory of ultradifferentiable and ultraholomorphic function classes (where also the associated weight function plays a prominent role). Thanks to this additional information we are able to see which …
We use induction and interpolation techniques to prove that a composition operator induced by a map $ϕ$ is bounded on the weighted Bergman space $\A^2_α(\mathbb{H})$ of the right half-plane if and only if $ϕ$ fixes $\infty$ non-tangentially, and has a finite angular derivative $λ$ there. We further prove that in this case the norm, essential norm, and spectral radius of the operator are all equal, and given by $λ^{(2+α)/2}$.
We obtain ceratin estimates for the reproducing kernels of large weighted Bergman spaces. Applications of these estimates to boundedness of the Bergman projection on $L^p(\D,ω^{p/2})$, complex interpolation and duality of weighted Bergman spaces are given.
On inclusion relations between weighted spaces of entire functions
本文研究整函数加权空间的包含关系，通过关联权重函数与关联权重序列在权重函数和权重序列情形间建立刻画，并应用于逐点乘法封闭性。
We solve an interpolation problem in $A^p_α$ involving specifying a set of (possibly not distinct) $n$ points, where the $k^{\textrm{th}}$ derivative at the $k^{\textrm{th}}$ point is up to a constant as large as possible for functions of unit norm. The solution obtained has norm bounded by a constant independent of the points chosen. As a direct application, we obtain a characterization of the order-boundedness of a sum of products of weighted composition and differentiation operators acting between weighted Bergman spaces. We also characterize the compactness of such operators that map a wei…
Holomorphic Interpolation of Multivariate Completely Monotone Functions
通过将完全单调函数表示为正测度的Laplace或Stieltjes-Fantappiè变换，利用非交换Radon变换框架结合矩阵束实现与Weyl运算微积，实现有限点插值，得到方向完全单调的整函数或有理函数。
Let $0<p<\infty$, $β>-1$, and $Ω$ be a strongly pseudoconvex bounded domain with a smooth boundary in $\mathbb{C}^n$. We will study the interpolation problem for weighted Bergman spaces $A^p_β(Ω)$. In the case, $1\leq p<\infty$, and $β> \max \{n(2p-1)-1, n(2q-1)-1\}$, where $q$ is the conjugate exponent of $p$ (let $q=1$, for $p=1$), we show that a sequence in $\mathbb{B}_n$, the unit ball in $\mathbb{C}^n$, is interpolating for $A^p_β(\mathbb{B}_n)$ if and only if it is separated.
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