DOCSLIB.ORG

Dual of 2D Fractional Fourier Transform Associated to It\^ O--Hermite Polynomials

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Dual of 2D Fractional Fourier Transform Associated to It\^ O--Hermite Polynomials DUAL OF 2D FRACTIONAL FOURIER TRANSFORM ASSOCIATED TO ITÔ–HERMITE POLYNOMIALS ABDELHADI BENAHMADI AND ALLAL GHANMI Dedicated to the memory of Professor Elhachmia Ait Benhaddou Abstract. A class of integral transforms, on the planar Gaussian Hilbert space with range in the weighted Bergman space on the bi-disk, is defined as the dual transforms of the 2d fractional Fourier transform associated with the Mehler function for Itô– Hermite polynomials. Some spectral properties of these transforms are investigated. Namely, we study their boundedness and identify their null spaces as well as their ranges. Such identification depends on the zeros set of Itô–Hermite polynomials. Moreover, the explicit expressions of their singular values are given and compactness and membership in p-Schatten class are studied. The relationship to specific fractional Hankel transforms is also established 1 Introduction The role played by the classical Mehler formula [14], +∞ tnH (x)H (y) 1 −t2(x2 + y2) + 2txy ! X n n = √ exp , (1.1) n 2 2 n=0 2 n! 1 − t 1 − t n x2 dn −x2 for the real Hermite polynomials Hn(x) := (−1) e dxn (e ), is well known in the literature [14, 18, 3, 12, 15, 17, 13, 8, 16]. Its complex analogues for Itô–Hermite ν polynomials Hm,n have been obtained in [19, 4, 10, 6] and have been employed in [6] ν to establish integral reproducing property for Hm,n by a like Fourier transform, and to provide a closed expression of the heat kernel for the Cauchy initial value problem for 2 2,ν −ν|z|2 a special magnetic Laplacian acting on the Hilbert space Lν(C) := L (C; e dλ). In [20], Zayed has used the one in (2.1) below to construct a non trivial 2d fractional Fourier transform Z ν ν −ν|ζ|2 Fu,vψ(ξ) = ψ(ζ)Ku,v(ζ; ξ)e dxdy; ζ = x + iy, (1.2) C ν whose eigenfunctions are the Itô–Hermite polynomials. Here Ku,v(ζ; ξ) stands for the kernel function ν ν n o arXiv:2005.08490v1 [math.CV] 18 May 2020 Kν (ζ; ξ) = exp −uv(|ζ|2 + |ξ|2) + uζξ + vζξ . (1.3) u,v π(1 − uv) 1 − uv In the present paper, we explore further applications in the context of integral 2 2 2 transforms and weighted Bergman spaces Bα,β(D ) on the bi-disk D = D × D; D = {z ∈ C, zz < 1}, defined as the Hilbert space of all analytic functions on D2 that are square integrable with respect to the measure 2 2 dµα,β(z, w) = ωα,β(|z| , |w| )dλ(z, w), (1.4) 2010 Mathematics Subject Classification. Primary 44A20; 30G35; 30H20 Secondary 47B38; 30D55. Key words and phrases. Weighted Bergman space on bi-disk; Itô–Hermite polynomials; Mehler formula; Singular values; p-Schatten class; Segal–Bargmann transform. 1 2 ABDELHADI BENAHMADI AND ALLAL GHANMI where the weight function is given by α β ωα,β(s, t) := (1 − s) (1 − t) ; α, β > −1, (1.5) and dλ denotes the standard Lebesgue measure. To this end we follow the scheme already applied in [7] to introduce and study the dual transforms of fractional Hankel transforms with ranges in weighted Bergman space on the disk. Mainly, we consider the family of integral transforms Z ν ν −ν|z|2 Rwf(u, v) = f(z)Ku,v(z; w)e dλ(z) (1.6) C on D2, labeled by ν > 0 and w ∈ C and seen as the dual transform of the 2d fractional ν ν Fourier transform in (1.2), Rwf(u, v) = Fu,vf(w). The aim in this paper concern identification of the null space and the range of the ν transforms Rw. We also study their boundedness and provide complete description of their compactness and membership in p-Schatten class. Our main results can be stated as follows ν 2 Theorem 1.1. The integral transform Rw is well defined and bounded from Lν(C) 2 2 into the weighted Bergman space Bα,β(D ) if and only if α > 0 and β > 0. The ν characterization of its null space Ker(Rw) depends on the zeros set of Itô–Hermite ν ν polynomials Hm,n. Namely, if Nw(H) = {(m, n); m, n = 0, 1, 2, ··· ; Hm,n(w, w) = 0}, ν then Ker(Rw) is a vector space spanned as ν ν Ker(Rw) = Span{Hm,n;(m, n) ∈ Nw(H)}. ν 2 2 2 Theorem 1.2. For α, β > 0, the operator Rw : Lν(C) −→ Bα,β(D ) is compact and its singular values are given by νπΓ(α + 1)Γ(β + 1) !1/2 sν,α,β(w) = |Hν (w, w)|. m,n νm+nΓ(α + m + 2)Γ(β + n + 2) m,n Moreover, it belongs to the p-Schatten class for every p > max(2/(α + 1); 2/(β + 1)). The proof of Theorem 1.1 is contained in Propositions 3.1, 3.2, 3.3 and 3.5 presented in Section 3, while the one of Theorem 1.2 is given in Section 4. The next section is devoted to some preliminaries concerning weighted Bergman space on the bi-disk and Itô–Hermite polynomials. We conclude the paper by discussing the close connection of ν Rw to the fractional Hankel transforms √ α/2 ! 2 2 2ν u Z ∞ 2ν uv −ν(x +uvy ) ν,α 1−uv Hu,v (f)(y) = xf(x)Iα xy e dr, (1.7) 1 − uv v 0 1 − uv where Iα denotes the modified Bessel function [1, p.222]. 2 Theorem 1.3. Let f ∈ Lν(C) and gk the associated Fourier coefficients. Then, ν,α ν Hu,v (gk), for varying integer k, are the fractional Fourier coefficients of Fu,vf, the fractional Fourier transform of f. 2 Preliminaries 2 2 For fixed reals α, β > −1, the weighted Bergman space Bα,β(D ) is a closed subspace 2 2 2 of the Hilbert space Lα,β(D ) := L (D × D; dµα,β) endowed with the scaler product Z hf, giα,β = f(z, w)g(z, w)dµα,β(z, w). D2 DUAL OF 2D FRFT ASSOCIATED TO ITÔ–HERMITE POLYNOMIALS 3 2 2 We denote by k·kα,β the associated norm. An orthonormal basis of Bα,β(D ) is given by α,β α,β −1/2 ϕm,n := γm,n em,n m n α,β where em,n(z, w) := z w and γm,n is its square norm given by π2Γ(α + 1)Γ(β + 1)m!n! γα,β := = ke k2 . m,n Γ(α + m + 2)Γ(β + n + 2) m,n α,β 2 2 Subsequently, the sequential characterization of Bα,β(D ) is given by ∞ ∞ 2 2 X X α,β 2 Bα,β(D ) = am;nem,n; γm,n|am;n| < ∞ . m,n=0 m,n=0 2 2 2 2 Accordingly, B0,0(D ) is identified to be the Hardy space on the bi-disk, while B−1,−1(D ) 2 2 2 is the classical Bergman space on D . The reproducing kernel of Bα,β(D ) is given by (α + 1)(β + 1) K ((u, v); (z, w)) = . α,β π2(1 − uz)α+2(1 − vw)β+2 2 2 Thus Bα,β(D ) is obtained as the Bergman projection (α + 1)(β + 1) Z ϕ(z, w) P (ϕ)(u, v) = dµα,β(z, w) π2 D2 (1 − uz)α+2(1 − vw)β+2 2 2 of Lα,β(D ). Another realization is by means of the unitary two-dimensional second Bargmann transform Z ! 1 α β sw + tz − (s + t) Bα,βϕ(z, w) = α+1 β+1 s t exp ϕ(s, t)dsdt (1 − z) (1 − w) R+2 (1 − z)(1 − w) 2 +2 α β −x−y acting on the Hilbert space L (R ; x y e dxdy). The kernel function of Bα,β ap- pears as the tensor product of two copies of the kernel function of the standard one- dimensional second Bargmann transform [2, p. 203]. However, it can be seen as the bilinear generating function involving the product of generalized Laguerre polynomials (α) (β) Lm (s)Ln (t). 2 2 In the sequel, we will provide interesting realization of specific subspaces of Bα,β(D ) by invoking the complex Mehler function 2 0 2 0 ! 0 1 −uv(ν|z| + ν |w| ) + ν uzw + νvzw Kν;ν (z, w) = exp , (2.1) u,v 1 − uv 1 − uv 0 ν with ν, ν > 0 and u, v ∈ D, associated to Itô–Hermite polynomials Hm,n defined on the complex plane C by [11, 9, 5] ∂m+n Hν (z, z) = (−1)m+neνzz e−νzz . (2.2) m,n ∂zm∂zn ν 2 To this end, let recall that the Hm,n form an orthogonal basis of Lν(C) and that the Mehler function in (2.1) can be expanded in terms of normalized Itô–Hermite polyno- mials 1/2 ν 0 ψν := Hν (2.3) m,n πνm+nm!n! m,n as [6] ∞ ν;ν0 X m n ν ν Ku,v (z, w) := u v ψm,n(z)ψm,n(w). (2.4) m,n=0 4 ABDELHADI BENAHMADI AND ALLAL GHANMI For ν = ν0 = 1, this is exactly the one announced by Wünsche [19] and proved later by Ismail in [10, Theorem 3.3] as a specific case of his Kibble–Slepian formula [10, Theorem 1.1]. ν 3 Basic properties of Rw We begin by observing that the kernel function in (1.3) reads in terms of the one in (2.1) as ν Kν (z, w) = Kν;ν(z, w).
Load more

Recommended publications
  • DUALITY and APPROXIMATION of BERGMAN SPACES Introduction If Ω ⊂ C N Is a Domain and P > 0, Let a P(Ω) Denote the Bergma
    DUALITY AND APPROXIMATION OF BERGMAN SPACES D. CHAKRABARTI , L. D. EDHOLM & J. D. MCNEAL Abstract. Expected duality and approximation properties are shown to fail on Bergman n spaces of domains in C , via examples. When the domain admits an operator satisfying certain mapping properties, positive duality and approximation results are proved. Such operators are constructed on generalized Hartogs triangles. On a general bounded Rein- hardt domain, norm convergence of Laurent series of Bergman functions is shown. This extends a classical result on Hardy spaces of the unit disc. Introduction n p If Ω ⊂ C is a domain and p > 0, let A (Ω) denote the Bergman space of holomorphic functions f on Ω such that Z p p kfkLp(Ω) = jfj dV < 1; Ω where dV denotes Lebesgue measure. Three basic questions about function theory on Ap(Ω) motivate our work: (Q1): What is the dual space of Ap(Ω)? (Q2): Can an element in Ap(Ω) be norm approximated by holomorphic functions with better global behavior? (Q3): For g 2 Lp(Ω), how does one construct G 2 Ap(Ω) that is nearest to g? The questions are stated broadly at this point; precise formulations will accompany results in the sections below. At first glance (Q1-3) appear independent { one objective of the paper is to show the questions are highly interconnected. On planar domains some connections were shown in [14] and [8]. Our paper grew from the observation that irregularity of the Bergman projection described in [12] has several surprising consequences concerning (Q1-3). In particular: there 2 are bounded pseudoconvex domains D ⊂ C such that (a) the dual space of Ap(D) cannot be identified, even quasi-isometrically, with Aq(D) 1 1 where p + q = 1, (b) there are functions in Ap(D), p < 2, that cannot be Lp-approximated by functions in A2(D), and (c) the L2-nearest holomorphic function to a general g 2 Lp(D) is not in Ap(D).
    [Show full text]
  • Operator Theory Induced by Powers of the De Branges-Rovnyak Kernel and Its Application
    Operator theory induced by powers of the de Branges-Rovnyak kernel and its application∗ Shuhei KUWAHARA Sapporo Seishu High School, Sapporo 064-0916, Japan E-mail address: [email protected] and Michio SETO National Defense Academy, Yokosuka 239-8686, Japan E-mail address: [email protected] Abstract In this note, we give a new property of de Branges-Rovnyak kernels. As the main theorem, it is shown that the exponential of de Branges-Rovnyak kernel is strictly positive definite if the inner part of the corresponding Schur class function is nontrivial. 2010 Mathematical Subject Classification: Primary 30H45; Secondary 15B48 keywords: de Branges-Rovnyak kernel, positive definite kernel arXiv:2007.11217v3 [math.FA] 26 Aug 2021 1 Introduction Let D be the open unit disk in the complex plane C, and let H∞ be the Banach algebra consisting of all bounded analytic functions on D. Then, we set = ϕ H∞ : ϕ(λ) 1 (λ D) , S { ∈ | | ≤ ∈ } and which is called the Schur class. For any function ϕ in H∞, it is well known that ϕ belongs to if and only if S 1 ϕ(λ)ϕ(z) − (1) 1 λz − ∗This paper has been accepted by Canadian Mathematical Bulletin, in which the new title is “Expo- nentials of de Branges-Rovnyak kernels”. 1 is positive semi-definite. This equivalence relation based on the properties of the Szeg¨o kernel is crucial in the operator theory on the Hardy space over D, in particular, theories of Pick interpolation, de Branges-Rovnyak spaces and sub-Hardy Hilbert spaces (see Agler- McCarthy [2], Ball-Bolotnikov [4], Fricain-Mashreghi [6] and Sarason [15]).
    [Show full text]
  • Curriculum Vitae ˘Zeljko ˘Cu˘Ckovic
    Curriculum Vitae Zeljko˘ Cu˘ckovi´c˘ Department of Mathematics University of Toledo Toledo, OH 43606 (419)530-2132 E-mail: [email protected] Education Ph.D. in Mathematics, 9/87 - 3/91 Department of Mathematics Michigan State University, East Lansing M.S. in Mathematics, 9/80 - 3/85 Department of Mathematics University of Zagreb, Croatia B.S. in Mathematics, 9/74 - 4/79 Department of Mathematics University of Zagreb, Croatia Employment 8/05 - present: Professor, Department of Mathematics, University of Toledo. Also an Associate Chair from 2005-2010. 09/16 - 11/16: Visiting Professor, University of Iowa 01/08 - 05/08: Visiting Professor, Vanderbilt University 8/01 - 7/05: Associate Professor, Department of Mathematics, University of Toledo 8/00 - 7/01: Visiting Associate Professor, Department of Mathematics, University of Wisconsin-Madison 9/99 - 7/00: Associate Professor, Department of Mathematics, University of Toledo 9/94 - 8/99: Assistant Professor, Department of Mathematics, University of Toledo 9/92 - 8/94: Assistant Professor, Department of Mathematics, University of Wisconsin Centers, Waukesha 3/91 - 6/92: Instructor, Department of Mathematics, Michigan State University 9/87 - 2/91: Teaching Assistant, Department of Mathematics, Michigan State University 9/80 - 7/87: Teaching Assistant, Faculty of Technology, University of Zagreb Research Interests • Functional Analysis and Operator Theory • Complex Analysis Invited Lectures • \On the essential norm and regularity of the Berezin transform of Toeplitz op- erators on pseudoconvex domains in Cn", Workshop on Function spaces and operator theory, Hanoi, Vietnam, June 23-28, 2019. • \Compactness of Toeplitz operators on domains in Cn", Colloquium at the University of Central Florida, Orlando, March 1, 2019.
    [Show full text]
  • Bounded Extremal Problems in Bergman and Bergman-Vekua Spaces Briceyda Delgado, Juliette Leblond
    Bounded Extremal Problems in Bergman and Bergman-Vekua spaces Briceyda Delgado, Juliette Leblond To cite this version: Briceyda Delgado, Juliette Leblond. Bounded Extremal Problems in Bergman and Bergman-Vekua spaces. Complex Variables and Elliptic Equations, Taylor & Francis, In press. hal-02928903 HAL Id: hal-02928903 https://hal.archives-ouvertes.fr/hal-02928903 Submitted on 3 Sep 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Bounded Extremal Problems in Bergman and Bergman-Vekua spaces Briceyda B. Delgado* Juliette Leblond Abstract p We analyze Bergman spaces Af (D) of generalized analytic functions of solutions to the Vekua equation @w = (@f=f)w in the unit disc of the complex plane, for Lipschitz- smooth non-vanishing real valued functions f and 1 < p < 1. We consider a family of bounded extremal problems (best constrained approximation) in the Bergman space p p A (D) and in its generalized version Af (D), that consists in approximating a function p p in subsets of D by the restriction of a function belonging to A (D) or Af (D) subject to a norm constraint. Preliminary constructive results are provided for p = 2.
    [Show full text]
  • Coefficient Estimates on Weighted Bergman Spaces
    Coefficient estimates on weighted Bergman spaces John E. McCarthy ∗ Washington University, St. Louis, Missouri 63130, U.S.A. 0 Introduction Let A denote normalized area measure for the unit disk D in C. The Bergman 2 2 space La is the sub-space of the Hilbert space L (A) consisting of functions n 2 that are also analytic in D. The monomials z are orthogonal in La, and p 1 P1 n 2 have norm n+1 ; so a holomorphic function f(z) = n=0 anz is in La if and P1 2 1 PN n only if n=0 janj n+1 < 1, and if this is so, the partial sums n=0 anz are polynomials converging to f. The weighted Bergman spaces normally studied are obtained by replacing the measure dA(z) by the radial measure dAα(z) = (1 − jzj2)αdA(z); in these spaces the monomials are again orthogonal, and a function is approximable in norm by the partial sums of its power series at zero. In this paper we are interested in studying non-radial weights of the 2 1 form jm(z)j dAα(z), where m is the modulus of a function in H , the space of bounded analytic functions on D. There are two different ways of generalizing the Bergman space to these 2 2 weights. We shall use La(jmj Aα) to denote the space of analytic functions 2 2 2 2 on D that also lie in L (jmj Aα), and P (jmj Aα) to denote the closure of the ∗The author was partially supported by the National Science Foundation grant DMS 9296099.
    [Show full text]
  • Density of the Polynomials in Bergman Spaces
    Pacific Journal of Mathematics DENSITY OF THE POLYNOMIALS IN BERGMAN SPACES PAUL S. BOURDON Vol. 130, No. 2 October 1987 PACIFIC JOURNAL OF MATHEMATICS Vol. 130, No. 2,1987 DENSITY OF THE POLYNOMIALS IN BERGMAN SPACES PAUL S. BOURDON Let G be a bounded simply connected domain in the complex plane. Using a result of Hedberg, we show that the polynomials are dense in Bergman space L^(G) if G is the image of the unit disk U under a weak-star generator of H°°. We also show that density of the polynomi- 2 2 als in L a(G) implies density of the polynomials in H {G). As a consequence, we obtain new examples of cyclic analytic Toeplitz opera- tors on H2(U) and composition operators with dense range on H2(U). As an additional consequence, we show that if the polynomials are dense 2 in L a(G) and φ maps U univalently onto G, then φ is univalent almost everywhere on the unit circle C. 1. Introduction. Let Ω be an open, nonempty subset of the com- plex plane, and let dA be two-dimensional Lebesgue measure. The Berg- man space of Ω, L^(Ω), is the Hubert space of those functions / which are analytic on Ω and which satisfy I/I dA < oo. Let H°° denote the algebra of functions which are bounded and analytic on the open unit disk U. For any domain G in the plane, define the Caratheodory hull of G, G*, to be the complement of the closure of the unbounded component of the complement of the closure of G.
    [Show full text]
  • Arxiv:2010.06951V2 [Math.FA] 14 Dec 2020
    SOME APPLICATIONS OF INTERPOLATING SEQUENCES FOR BANACH SPACES OF ANALYTIC FUNCTIONS HAMZEH KESHAVARZI Abstract. M. J. Beltr´an-Meneua et al. [2] and E. Jord´aand A. Rodr´ıguez-Arenas [17] characterized the (uniformly) mean ergodic composition operators on H∞(D) and ∞ Hν (D), respectively. In this paper, by using the interpolating sequences, we give other necessary and sufficient conditions for the (uniformly) mean ergodicity of composition operators on these spaces. MSC(2010): primary: 47B33, secondary: 47B38; 47A35. Keywords: interpolating sequences, mean ergodic operators, composition operators, weighted Bergman space of infinite order. 1. Introduction Let H(D) be the space of all analytic functions on the open unit disk D. Let ϕ be an analytic self-map of D. The map ϕ induces a composition operator Cϕ on H(D) which is defined by Cϕf = f ◦ ϕ. We refer to [13, 21] for various aspects of the theory of composition operators on holomorphic function spaces. Also, H∞(D) is the Banach space of bounded functions in H(D) with supremum norm. Consider the continuous function ν : D → (0, ∞). For 0 <p< ∞, the weighted p D D Bergman space Aν ( ) is the space of analytic functions f on where p p kfkν,p = |f(z)| ν(z) < ∞. Zz∈D For p = ∞, the weighted Bergman space of infinite order is defined as: p D ∞ D D Aν( )= Hν ( )= {f ∈ H( ) : sup |f(z)|ν(z) < ∞}. z∈D Throughout this paper, ν has the following properties: arXiv:2010.06951v2 [math.FA] 14 Dec 2020 (i) it is radial, that is, ν(z)= ν(|z|)), (ii) it is decreasing, (iii) lim|z|→1 ν(z) = 0 and ν(1−2−n−1) (iv) it satisfies the Lusky condition; infn ν(1−2−n) > 0.
    [Show full text]
  • Bergman Space Methods and Integral Means Spectra of Univalent Functions
    Bergman space methods and integral means spectra of univalent functions ALAN SOLA Licentiate thesis Stockholm, Sweden 2007 TRITA-MAT-07-MA-04 KTH Matematik ISSN 1401-2278 SE-100 44 Stockholm ISBN 987-91-7178-630-2 SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie licentiatex- amen i matematik måndagen den 14 maj 2007 klockan 13.15 i seminarierum 3721, Institutionen för Matematik. © Alan Sola, April 2007 Tryck: Universitetsservice US-AB iii Abstract We study universal integral means spectra of certain classes of univalent func- tions defined on subsets of the complex plane. After reformulating the definition of the integral means spectrum of a univalent function in terms of membership in weighted Bergman spaces, we describe the Hilbert space techniques that can be used to estimate universal means spectra from above. Finally, we show that the method of norm expansion used in that context can be applied in a more general setting to reproducing kernel spaces in order to explicitly compute kernel functions. Contents Contents iv 1 Univalent functions . 1 2 Bergman spaces . 8 3 Estimating the universal means spectrum . 14 4 Discussion . 19 Bibliography 25 List of Papers Paper I: Spectral notions for conformal maps: a survey (with H. Hedenmalm) Submitted for publication Paper II: An estimate of the universal means spectrum of con- formal mappings Published in Comput. Methods Funct. Theory 6, No.2, 423-436 Paper III: Norm expansion along a zero variety (with H. Hedenmalm and S. Shimorin) Submitted for publication iv Introduction The present thesis for the degree of licentiate of technology consists of an introductory section and three research papers.
    [Show full text]
  • Random and Optimal Configurations in Complex Function Theory
    Random and optimal configurations in complex function theory ARON WENNMAN Doctoral Thesis Stockholm, Sweden 2018 KTH Institutionen för Matematik TRITA-SCI-FOU 2018:20 100 44 Stockholm ISBN 978-91-7729-783-3 SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i matematik Fredag den 8 juni 2018 kl 13.00 i sal E2, Kungl Tekniska högskolan, Lindstedtsvägen 3, Stockholm. c Aron Wennman, 2018 • Tryck: Universitetsservice US AB iii Abstract This thesis consists of six articles spanning over several areas of math- ematical analysis. The dominant theme is the study of random point pro- cesses and optimal point configurations, which may be though of as systems of charged particles with mutual repulsion. We are predominantly occupied with questions of universality, a phenomenon that appears in the study of random complex systems where seemingly unrelated microscopic laws pro- duce systems with striking similarities in various scaling limits. In particular, we obtain a complete asymptotic expansion of planar orthogonal polynomi- als with respect to exponentially varying weights, which yields universality for the microscopic boundary behavior in the random normal matrix (RNM) model (Paper A) as well as in the case of more general interfaces for Bergman kernels (Paper B). Still in the setting of RNM ensembles, we investigate prop- erties of scaling limits near singular points of the boundary of the spectrum, including cusps points (Paper C). We also obtain a central limit theorem for fluctuations of linear statistics in the polyanalytic Ginibre ensemble, using a new representation of the polyanalytic correlation kernel in terms of algebraic differential operators acting on the classical Ginibre kernel (Paper D).
    [Show full text]
  • Projections for Harmonic Bergman Spaces and Applications
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ARTICLE IN PRESS Journal of Functional Analysis 216 (2004) 388–421 Projections for harmonic Bergman spaces and applications Boo Rim Choe,a,1 Hyungwoon Koo,a,Ã,1 and Heungsu Yib,2 a Department of Mathematics, Korea University, Seoul 136-701, Republic of Korea b Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea Received 3 November 2003; accepted 24 February 2004 Available online 1 July 2004 Communicated by D. Sarason Abstract On the settingof generalbounded smooth domains in Rn; we construct L1-bounded nonorthogonal projections and obtain related reproducing formulas for the harmonic Bergman spaces. In addition, we show that those projections satisfy Sobolev Lp-estimates of any order even for p ¼ 1: Amongapplications are Gleason’s problems for the harmonic Bergman–Sobolev and (little) Bloch functions on star-shaped domains with strong reference points. r 2004 Elsevier Inc. All rights reserved. MSC: primary 31B05; secondary 31B10 Keywords: Projection; Harmonic Bergman function; Gleason’s problem 1. Introduction For a fixed integer nX2; let O be a bounded smooth domain in Rn: Let hðOÞ denote the class of all functions harmonic on O: For 1ppoN; the Lp-harmonic ÃCorrespondingauthor. E-mail addresses: [email protected] (B.R. Choe), [email protected] (H. Koo), hsyi@ kwangwoon.ac.kr (H. Yi). 1 Supported by KOSEF(R01-2003-000-10243-0). 2 Supported by the research grant of Kwangwoon University (2004). 0022-1236/$ - see front matter r 2004 Elsevier Inc.
    [Show full text]
  • Rational Approximation in the Bergman Spaces
    Rational Approximation in the Bergman Spaces Wei QU∗, Pei DANG November 9, 2018 Abstract It is known that adaptive Fourier decomposition (AFD) offers efficient rational approxima- tions to functions in the classical Hardy H2 spaces with significant applications. This study aims at rational approximation in Bergman, and more widely, in weighted Bergman spaces, the functions of which have more singularity than those in the Hardy spaces. Due to lack of an effective inner function theory, direct adaptation of the Hardy-space AFD is not performable. We, however, show that a pre-orthogonal method, being equivalent to AFD in the classical cases, is available for all weighted Bergman spaces. The theory in the Bergman spaces has equal force as AFD in the Hardy spaces. The methodology of approximation is via constructing the rational orthogonal systems of the Bergman type spaces, called Bergman space rational orthog- onal (BRO) system, that have the same role as the Takennaka-Malmquist (TM) system in the Hardy spaces. Subsequently, we prove a certain type direct sum decomposition of the Bergman spaces that reveals the orthogonal complement relation between the span of the BRO system and the zero-based invariant spaces. We provide a sequence of examples with different and ex- plicit singularities at the boundary along with a study on the inclusion relations of the weighted Bergman spaces. We finally present illustrative examples for effectiveness of the approximation. 1 Introduction The aim of this work is to propose an adaptive approximation method in the weighted Bergman 2 2 Aα spaces analogous with the one established in the Hardy H spaces of the unit disc and the upper-half complex plane (classical contexts) called adaptive Fourier decomposition (AFD) [17].
    [Show full text]
  • The Reachable Space of the Heat Equation for a Finite Rod As a Reproducing Kernel Hilbert Space
    THE REACHABLE SPACE OF THE HEAT EQUATION FOR A FINITE ROD AS A REPRODUCING KERNEL HILBERT SPACE MARCOS LOPEZ-GARC´ ´IA Abstract. We use some results from the theory of Reproducing Kernel Hilbert Spaces to show that the reachable space of the heat equation for a finite rod with either one or two Dirichlet boundary controls is a RKHS of analytic func- tions on a square, and we compute its reproducing kernel. We also show that the null reachable space of the heat equation for the half line with Dirichlet boundary data is a RKHS of analytic functions on a sector, whose reproducing kernel is (essentially) the sum of pullbacks of the Bergman and Hardy kernels on the half plane C+. We also consider the case with Neumann boundary data. 1. Introduction Let T > 0 fixed. Consider the following control system ∂ w ∂ w =0, 0 <x< 1, 0 < t < T, t − xx (1) w(0,t)= uℓ(t), w(1,t)= ur(t), 0 < t < T, w(x, 0)=0, 0 <x< 1, which models the temperature propagation in a rod with length 1. In control theory is an important issue to describe the so-called null reachable space, at time T > 0, defined as follows 2 := w( ,T ): w is solution of system (1) with controls u ,u LC(0,T ) . RT { · ℓ r ∈ } Using the null controllability of the system (1) in any positive time (see [4, Theorem 3.3]), one can show that the set of states w( ,T ) reached by solutions of system (1) from any initial datum w(x, 0) L2(0, 1) coincides· with .
    [Show full text]