Abstract: In this note we establish integral formulas for polyanalytic functions in several variables. More precisely, given a positive integer q, we provide explicit expressions for the reproducing kernels of the weighted Bergman spaces of q-analytic functions on the unit ball in \(\mathbb {C}^n\) and that of q-analytic Fock space in \(\mathbb {C}^n\) . PubDate: 2021-10-07

Abstract: The exponential Teichmüller spaces \(E_p\) , \(0\le p \le \infty\) , interpolate between the classical Teichmüller space ( \(p=\infty\) ) and the space of harmonic diffeomorphisms \((p=0)\) . In this article, we prove the existence of non-variational critical points for the associated functional: mappings f of the disk whose distortion is p-exponentially integrable, \(0<p<\infty\) , yet for any diffeomorphism g(z) of \({\mathbb {D}}\) with \(g \partial {\mathbb {D}}=identity\) and \(g\ne identity\) we have \(f\circ g\) is not of p-exponentially integrable distortion. PubDate: 2021-08-06

Abstract: We show that a K-quasiregular \(\omega\) -curve from a Euclidean domain to a Euclidean space with respect to a covector \(\omega\) is locally \((1/K)(\Vert \omega \Vert / \omega _{\ell _1})\) -Hölder continuous. We also show that quasiregular curves enjoy higher integrability. PubDate: 2021-07-22

Abstract: We consider compact Leviflat homogeneous Cauchy–Riemann (CR) manifolds. In this setting, the Levi-foliation exists and we show that all its leaves are homogeneous and biholomorphic. We analyze separately the structure of orbits in complex projective spaces and parallelizable homogeneous CR-manifolds in our context and then combine the projective and parallelizable cases. In codimensions one and two, we also give a classification. PubDate: 2021-07-15

Abstract: In this paper we introduce a new distance by means of the so-called Szegő kernel and examine some basic properties and its relationship with the so-called Skwarczyński distance. We also examine the relationship between this distance, and the so-called Bergman distance and Szegő distance. PubDate: 2021-07-13

Abstract: The systematic study of CR manifolds originated in two pioneering 1932 papers of Élie Cartan. In the first, Cartan classifies all homogeneous CR 3-manifolds, the most well-known case of which is a one-parameter family of left-invariant CR structures on \(\mathrm {SU}_2= S^3\) , deforming the standard ‘spherical’ structure. In this paper, mostly expository, we illustrate and clarify Cartan’s results and methods by providing detailed classification results in modern language for four 3-dimensional Lie groups. In particular, we find that \({\mathrm {SL}_2({\mathbb {R}})}\) admits two one-parameter families of left-invariant CR structures, called the elliptic and hyperbolic families, characterized by the incidence of the contact distribution with the null cone of the Killing metric. Low dimensional complex representations of \({\mathrm {SL}_2({\mathbb {R}})}\) provide CR embedding or immersions of these structures. The same methods apply to all other 3-dimensional Lie groups and are illustrated by descriptions of the left-invariant CR structures for \(\mathrm {SU}_2\) , the Heisenberg group, and the Euclidean group. PubDate: 2021-06-25

Abstract: Let \(\Omega\) be a bounded open subset of \({\mathbb {R}}^n\) and \(\Delta\) the Euclidean Laplace operator \(\sum _{i=1}^n \partial ^2/\partial x_i^2\) . Let \(\beta (x)\) denote the number of eigenvalues less or equal to x with respect to the eigenvalue problem \(\Delta f = -x f\) on \(\Omega\) with \(f=0\) on the boundary of \(\Omega\) . A well-known result due to Hermann Weyl gives the asymptotic formula \(\beta (x)= (2 \pi )^{-n} B_n m_n(\Omega ) x^{n/2}+ o(x^{n/2})\) as \(x\rightarrow \infty\) , where \(B_n\) is the volume of the unit ball in \({\mathbb {R}}^n\) and \(m_n(\Omega )\) is the volume of \(\Omega\) . In this work, we consider the analogous problem for radial functions in the discrete setting of the homogeneous isotropic tree T of homogeneity \(q+1\) ( \(q\ge 2\) ). As the volume of T with respect to the hyperbolic metric is infinite, we don’t expect and indeed we show that there is no analogous result for the commonly used hyperbolic Laplacian on T. We consider instead the eigenvalue problem for radial functions on T with respect to the Euclidean Laplacian on T introduced in [6], where the boundary condition \(f=0\) means that f converges radially to 0 at \(\infty\) . We prove that \(\beta (x)\) is within 2 of \(\log _q \sqrt{x}\) . We also consider other boundary conditions and pose some open questions. PubDate: 2021-06-10

Abstract: In this article, we survey the recent literature surrounding the geometry of complex polynomials. Specific areas surveyed are (i) Generalizations of the Gauss–Lucas Theorem, (ii) Geometry of Polynomials Level Sets, and (iii) Shape Analysis and Conformal Equivalence. PubDate: 2021-05-17

Abstract: Using a recent Mergelyan type theorem for products of planar compact sets, we establish generic existence of universal Taylor series on products of planar simply connected domains \({\varOmega }_i\) , \(i=1,\ldots ,d\) . The universal approximation is realized by partial sums of the Taylor development of the universal function on products of planar compact sets \(K_i\) , \(i=1,\ldots ,d\) such that \({\mathbb {C}}-K_i\) is connected and for at least one \(i_0\) the set \(K_{i_0}\) is disjoint from \({\varOmega }_{i_0}\) . PubDate: 2021-05-15

Abstract: For a subharmonic function defined in the unit disc, we consider the relation between angular boundary value, angular cluster set, and “angular normality” at a point of the unit circle. PubDate: 2021-04-30

Abstract: In this paper, we first show that the canonical solution operator \(S_1\) to \({\bar{\partial }}\) restricted to (0,1)-forms with holomorphic function coefficients can be expressed by an integral operator using the Dirichlet kernel. Then we prove that operator \(S_k\,(k\ge 1)\) is a Hilbert–Schmidt operator on the Dirichlet space of \({\mathbb {D}}\) , but fails to be the Hilbert–Schmidt operator on the Dirichlet space of \({\mathbb {D}}^{2}\) . Finally we show that the concomitant operator \(P_{k}\) of \(S_{k}\,(k\ge 2)\) is similar to the direct sum of k copies of the concomitant operator \(P_{1}\) of \(S_{1}\) . PubDate: 2021-04-26

Abstract: In the last few years, the notion of optimal polynomial approximant has appeared in the mathematics literature in connection with the Hilbert spaces of analytic functions of one or more variables. In the 1970s, researchers in engineering and applied mathematics introduced least-squares inverses in the context of digital filters in signal processing. It turns out that in the Hardy space \(H^{2},\) these objects are identical. This paper is a survey of known results about optimal polynomial approximants. In particular, we will examine their connections with orthogonal polynomials and reproducing kernels in weighted spaces and digital filter design. We will also describe what is known about the zeros of optimal polynomial approximants, their rates of decay, and convergence results. Throughout the paper, we state many open questions that may be of interest. PubDate: 2021-04-17

Abstract: Let \(u_{k}\) be a solution of the Helmholtz equation with the wave number k, \(\varDelta u_{k}+k^{2} u_{k}=0\) , on (a small ball in) either \({\mathbb {R}}^{n}\) , \({\mathbb {S}}^{n}\) , or \({\mathbb {H}}^{n}\) . For a fixed point p, we define \(M_{u_{k}}(r)=\max _{d(x,p)\le r} u_{k}(x) .\) The following three ball inequality $$M_{u_{k}}(2r)\le C(k,r,\alpha )M_{u_{k}}(r)^{\alpha }M_{u_{k}}(4r)^{1-\alpha }$$ is well known, it holds for some \(\alpha \in (0,1)\) and \(C(k,r,\alpha )>0\) independent of \(u_{k}\) . We show that the constant \(C(k,r,\alpha )\) grows exponentially in k (when r is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds. PubDate: 2021-04-13

Abstract: In this paper, we show that one-sided extendability of functions in certain \(L^{\infty }\) spaces of a rectifiable Jordan arc is a rare phenomenon. We also discuss possible generalizations. PubDate: 2021-04-13

Abstract: It is shown that the Laurent series of a holomorphic function smooth up to the boundary on a Reinhardt domain in \({\mathbb {C}}^n\) converges unconditionally to the function in the Fréchet topology of the space of functions smooth up to the boundary. PubDate: 2021-04-12

Abstract: Let \(\Omega \subset {{\mathbb{C}}}\) be a bounded domain. In this note, we use complex variable methods to study the number of critical points of the function \(v=v_\Omega\) that solves the elliptic problem \(\Delta v = -2\) in \(\Omega ,\) with boundary values \(v=0\) on \(\partial \Omega .\) This problem has a classical flavor but is especially motivated by recent studies on localization of eigenfunctions. We provide an upper bound on the number of critical points of v when \(\Omega\) belongs to a special class of domains in the plane, namely, domains for which the boundary \(\partial \Omega\) is contained in \(\{z: z ^2 = f(z) + \overline{f(z)}\},\) where \(f^{\prime}(z)\) is a rational function. We furnish examples of domains where this bound is attained. We also prove a bound on the number of critical points in the case when \(\Omega\) is a quadrature domain. The note concludes with the statement of some open problems and conjectures. PubDate: 2021-04-09

Abstract: We give an elementary proof of an analogue of Fejér’s theorem in weighted Dirichlet spaces with superharmonic weights. This provides a simple way of seeing that polynomials are dense in such spaces. PubDate: 2021-04-05