Henkin measures, Riesz products and singular sets
type: Item
instance of: scholarly article
title: Henkin measures, Riesz products and singular sets
zbMATH Open document ID: 0913.31003
author: Evgueni Doubtsov
publication date: 1998-09-14T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75299
review text: The principal objects of this paper are measures defined on the complex sphere \(S=\{\zeta\in{\mathbb{C}}^n:|\zeta| =1\}, n\geq 2.\) For \((p, q)\in{\mathbb{Z}}^2_+\), \(H(p, q)\) is the vector space of all harmonic homogeneous polynomials in \(S\) of total degree \(p+q,\) of degree \(p\) in \(z_1,\dots,z_n,\) and of degree \(q\) in \(\overline z_,\dots,\overline z_n.\) The author develops the spectral function theory on \(S\) in terms of \(H(p, q).\) If \(M(S)\) is the set of all regular complex Borel measures on \(S,\) then the spectrum of \(\mu\in M(S)\) is \[ \text{spec}(\mu)=\Biggl\{(p, q)\in{\mathbb{Z}}^2_+: \mu_{pq}(z)=\int _SK_{pq}(z,\zeta)d\mu (\zeta)\neq 0,\;z\in S\Biggr\}, \] where \(K_{pq}(z, \zeta)\) is the reproducing kernel for \(H(p, q)\subset L^2(S).\) Given a set (a spectrum) \(\Lambda\subset{\mathbb{Z}}^2_+,\) the general problem is to investigate the properties of the space \(M_{\Lambda}(S)=\{\mu\in M(S):\text{spec}(\mu)\subset\Lambda\}.\) For \(d\in{\mathbb{Z}}^2_+,\) the author defines \(\Lambda (d)=\{(p, q)\in{\mathbb{Z}}^2_+: (p-d)(q-d)=0\), \(p\geq d\), \(q\geq d\}.\) In complex analysis, \(\text{spec}(\mu)\subset\Lambda =\Lambda (0)\) iff the Poisson integral \(P[\mu ]\) is a pluriharmonic function. By analogy, if \(\text{spec}(\mu)\subset\Lambda (d)\) or \(\text{spec}(\mu)\subset\Lambda (d)\cup\{(0, 0)\},\) he calls \(\mu \) \(d\)-pluriharmonic. He proved in [Zap. Nauchn. Semin. POMI 217, 54-58 (Russian) (1994); English transl. in J. Math. Sci., New York 85, No. 2, 1790-1793 (1997; Zbl 0907.31007)] that \(\Lambda (0)\) is singular. In the paper under review, he establishes (Theorem A) that also ``the set \(\Lambda (d)\) is singular \(\forall d\in{\mathbb{Z}}^2_+\)''. The classical Riesz product on the unit circle \({\mathcal T}\) is \[ \mu:=\prod_{k=1}^{\infty} \Biggl({{\overline a_k\overline z^{jk}}\over{2}}+1+{{a_kz^{jk}}\over{2}} \Biggr), \quad z\in{\mathcal T},\quad | a_k|\leq 1,\quad {{j_{k+1}}\over{j_k}}\geq 3. \] The author introduc…
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 32A38
Mathematics Subject Classification ID: 42A55
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 1199201
zbMATH Keywords: Henkin measures
zbMATH Keywords: pluriharmonic Riesz products
zbMATH Keywords: complex spherical harmonics
zbMATH Keywords: singular sets
reviewed by: Petru Caraman
MaRDI profile type: Publication
OpenAlex ID: W2319438331
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DOI: 10.5802/AIF.1635
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1199201