scholarly journals An example of a non-exposed extreme function in the unit ball of H1

1994 ◽  
Vol 37 (1) ◽  
pp. 47-51 ◽  
Author(s):  
Jyunji Inoue
Keyword(s):  
Hardy Space ◽  
Unit Ball ◽  
Unit Disc ◽  
Inner Function ◽  
Classical Hardy Space

We construct a non-exposed extreme function f of the unit ball of H1, the classical Hardy space on the unit disc of the plane, which has the property: f(z)/(1−q(z))2 ∉ H1 for any nonconstant inner function q(z). This function constitutes a counterexample to a conjecture in D. Sarason [7].

scholarly journals Linear Differential Equations with Solutions in the Dirichlet Type Subspace of the Hardy Space

2007 ◽  
Vol 187 ◽  
pp. 91-113 ◽  
Author(s):  
J. Heittokangas ◽  
R. Korhonen ◽  
J. Rättyä
Keyword(s):  
Differential Equation ◽  
Hardy Space ◽  
Linear Differential Equation ◽  
Unit Disc ◽  
Sufficient Conditions ◽  
Dirichlet Type ◽  
All Solutions ◽  
Linear Differential ◽  
Classical Hardy Space ◽  
Uniformly Bounded

AbstractSufficient conditions for the analytic coefficients of the linear differential equationare found such that all solutions belong to a given -space, or to the Dirichlet type subspace Dp of the classical Hardy space Hp, where 0 < p ≤ 2. For 0 < q < ∞, the space consists of those functions f, analytic in the unit disc D, for which |f(z)|(1 – |z|2)q is uniformly bounded in D, and f ∈ Dp if the integral ∫D |f′(z)|p(1 – |z|2)p–1dσz converges.

scholarly journals The Numerical Range of C* ψ Cφ and Cφ C* ψ

2021 ◽  
Vol 8 (1) ◽  
pp. 13-23
Author(s):  
John Clifford ◽  
Michael Dabkowski ◽  
Alan Wiggins
Keyword(s):  
Quadratic Form ◽  
Hardy Space ◽  
Unit Disc ◽  
Numerical Range ◽  
Dimensional Subspace ◽  
Inner Function ◽  
Open Unit ◽  
Open Unit Disc ◽  
3 Dimensional ◽  
Unit Norm

Abstract In this paper we investigate the numerical range of C* bφ m Caφ n and Caφ n C* bφ m on the Hardy space where φ is an inner function fixing the origin and a and b are points in the open unit disc. In the case when |a| = |b| = 1 we characterize the numerical range of these operators by constructing lacunary polynomials of unit norm whose image under the quadratic form incrementally foliate the numerical range. In the case when a and b are small we show numerical range of both operators is equal to the numerical range of the operator restricted to a 3-dimensional subspace.

scholarly journals Power series of several variables with condition of logarithmical convexity

2021 ◽  
Vol 8 (1) ◽  
pp. 49-62
Author(s):  
Alexandr V. Zheleznyak ◽  
Keyword(s):  
Hilbert Space ◽  
Hardy Space ◽  
Power Series ◽  
Unit Disc ◽  
Reproducing Kernel ◽  
Space Of Analytic Functions ◽  
Several Variables ◽  
Positive Coefficients ◽  
Classical Hardy Space ◽  
Reproducing Kernel Function

We obtain a new version of Hardy theorem about power series of several variables reciprocal to the power series with positive coefficients. We prove that if the sequence {as} = as1,s2,...,sn, ||s|| ≥ K satisfies condition of logarithmically convexity and the first coefficient a0 is sufficiently large then reciprocal power series has only negative coefficients {bs} = bs1,s2,...,sn, except b0,0,...,0 for any K. The classical Hardy theorem corresponds to the case K = 0, n = 1. Such results are useful in Nevanlinna - Pick theory. For example, if function k(x, y) can be represented as power series Σn≥0 an(x-y)n, an > 0, and reciprocal function 1/k(x,y) can be represented as power series Σn≥0 bn(x-y)n such that bn < 0, n > 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna-Pick property. The reproducing kernel 1/1-x-y of the classical Hardy space H2(D) is a prime example for our theorems.

An argument of a function in H1/2

2012 ◽  
Vol 55 (2) ◽  
pp. 507-511
Author(s):  
Takahiko Nakazi ◽  
Takanori Yamamoto
Keyword(s):  
Hardy Space ◽  
Simple Proof ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc

AbstractLet H1/2 be the Hardy space on the open unit disc. For two non-zero functions f and g in H1/2, we study the relation between f and g when f/g ≥ 0 a.e. on ∂D. Then we generalize a theorem of Neuwirth and Newman and Helson and Sarason with a simple proof.

scholarly journals A note on a space Hp, a of holomorphic functions

1987 ◽  
Vol 35 (3) ◽  
pp. 471-479
Author(s):  
H. O. Kim ◽  
S. M. Kim ◽  
E. G. Kwon
Keyword(s):  
Hardy Space ◽  
Complex Plane ◽  
Unit Disc ◽  
Weak Type ◽  
Holomorphic Functions ◽  
Embedding Theorem ◽  
Type Operator ◽  
Taylor Coefficients ◽  
Bounded Holomorphic

For 0 < p < ∞ and 0 ≤a; ≤ 1, we define a space Hp, a of holomorphic functions on the unit disc of the complex plane, for which Hp, 0 = H∞, the space of all bounded holomorphic functions, and Hp, 1 = Hp, the usual Hardy space. We introduce a weak type operator whose boundedness extends the well-known Hardy-Littlewood embedding theorem to Hp, a, give some results on the Taylor coefficients of the functions of Hp, a and show by an example that the inner factor cannot be divisible in Hp, a.

scholarly journals Hardy space interpolation in the unit ball

1987 ◽  
Vol 90 (3) ◽  
pp. 325-351 ◽  
Author(s):  
Pascal J. Thomas
Keyword(s):  
Hardy Space ◽  
Unit Ball

Inner function images of radii

1979 ◽  
Vol 85 (2) ◽  
pp. 357-360 ◽  
Author(s):  
Walter Rudin
Keyword(s):  
Unit Disc ◽  
Inner Function ◽  
Classical Theorem ◽  
Boundary Behaviour

The main result of the present paper is the construction of a very oscillatory inner function F in the unit disc. The existence of such an F leads to a certain extension of Fatou's classical theorem concerning the boundary behaviour of H∞-functions.

Schwarz Problems for Poly-Hardy Space on the Unit Ball

2016 ◽  
Vol 71 (3-4) ◽  
pp. 801-823 ◽  
Author(s):  
Uwe Kähler ◽  
Min Ku ◽  
Tao Qian
Keyword(s):  
Hardy Space ◽  
Unit Ball

scholarly journals Resolution of the Identity of the Classical Hardy Space by Means of Barut-Girardello Coherent States

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Zouhaïr Mouayn
Keyword(s):  
Fourier Transform ◽  
Hardy Space ◽  
Real Line ◽  
Coherent States ◽  
Positive Real ◽  
Integrable Functions ◽  
Resolution Of The Identity ◽  
Square Integrable Functions ◽  
Complex Valued ◽  
Classical Hardy Space

We construct a one-parameter family of coherent states of Barut-Girdrardello type performing a resolution of the identity of the classical Hardy space of complex-valued square integrable functions on the real line, whose Fourier transform is supported by the positive real semiaxis.

The reproducing kernel of $\mathcal H^2$ and radial eigenfunctions of the hyperbolic Laplacian

2019 ◽  
Vol 124 (1) ◽  
pp. 81-101
Author(s):  
Manfred Stoll
Keyword(s):  
Hardy Space ◽  
Unit Ball ◽  
Hypergeometric Function ◽  
Compact Subset ◽  
Spherical Harmonics ◽  
Harmonic Functions ◽  
Reproducing Kernel ◽  
Diagonal Function ◽  
Hyperbolic Harmonic Functions

In the paper we characterize the reproducing kernel $\mathcal {K}_{n,h}$ for the Hardy space $\mathcal {H}^2$ of hyperbolic harmonic functions on the unit ball $\mathbb {B}$ in $\mathbb {R}^n$. Specifically we prove that \[ \mathcal {K}_{n,h}(x,y) = \sum _{\alpha =0}^\infty S_{n,\alpha }(\lvert x\rvert )S_{n,\alpha }(\lvert y\rvert ) Z_\alpha (x,y), \] where the series converges absolutely and uniformly on $K\times \mathbb {B}$ for every compact subset $K$ of $\mathbb {B}$. In the above, $S_{n,\alpha }$ is a hypergeometric function and $Z_\alpha $ is the reproducing kernel of the space of spherical harmonics of degree α. In the paper we prove that \[ 0\le \mathcal K_{n,h}(x,y) \le \frac {C_n}{(1-2\langle x,y\rangle + \lvert x \rvert^2 \lvert y \rvert^2)^{n-1}}, \] where $C_n$ is a constant depending only on $n$. It is known that the diagonal function $\mathcal K_{n,h}(x,x)$ is a radial eigenfunction of the hyperbolic Laplacian $\varDelta_h $ on $\mathbb{B} $ with eigenvalue $\lambda _2 = 8(n-1)^2$. The result for $n=4$ provides motivation that leads to an explicit characterization of all radial eigenfunctions of $\varDelta_h $ on $\mathbb{B} $. Specifically, if $g$ is a radial eigenfunction of $\varDelta_h $ with eigenvalue $\lambda _\alpha = 4(n-1)^2\alpha (\alpha -1)$, then \[ g(r) = g(0) \frac {p_{n,\alpha }(r^2)}{(1-r^2)^{(\alpha -1)(n-1)}}, \] where $p_{n,\alpha }$ is again a hypergeometric function. If α is an integer, then $p_{n,\alpha }(r^2)$ is a polynomial of degree $2(\alpha -1)(n-1)$.

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