INTEGRAL REPRESENTATION OF THE n TH DERIVATIVE IN DE BRANGES ROVNYAK SPACES AND THE NORM CONVERGENCE

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INTEGRAL REPRESENTATION OF THE n-TH DERIVATIVE IN DE BRANGES-ROVNYAK SPACES AND THE NORM CONVERGENCE OF ITS REPRODUCING KERNEL EMMANUEL FRICAIN, JAVAD MASHREGHI Abstract. In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges–Rovnyak spaces H(b), where b is in the unit ball of H∞(C+). In particular, we generalize a result of Ahern–Clark obtained for functions of the model spaces Kb, where b is an inner function. Using hypergeometric series, we obtain a nontrivial formula of combinatorics for sums of binomial coefficients. Then we apply this formula to show the norm convergence of reproducing kernel kb?,n of the evaluation of n-th derivative of elements of H(b) at the point ? as it tends radially to a point of the real axis. 1. Introduction Let C+ denote the upper half plane in the complex plane and let H2(C+) denote the usual Hardy space consisting of analytic functions f on C+ which satisfy ?f?2 := sup y>0 (∫ R |f(x+ iy)|2 dx )1/2 < +∞. P. Fatou [12] proved that, for any function f in H2(C+) and for almost all x0 in R, f?(x0) := lim t?0+ f(x0 + it) exists.

  • interesting relations

  • hypergeometric functions

  • space contractions

  • blaschke product

  • hilbert space

  • radial limits

  • schwarz-pick matrix

  • branges-rovnyak spaces


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INTEGRAL REPRESENTATION OF THEn-TH DERIVATIVE IN DE
BRANGES-ROVNYAK SPACES AND THE NORM CONVERGENCE
OF ITS REPRODUCING KERNEL

EMMANUEL FRICAIN, JAVAD MASHREGHI

Abstract.In this paper, we give an integral representation for the boundary values
of derivatives of functions of the de Branges–Rovnyak spacesH(b), wherebis in the

unit ball ofH(C+). Inparticular, we generalize a result of Ahern–Clark obtained for
functions of the model spacesKb, wherebUsing hypergeometricis an inner function.
series, we obtain a nontrivial formula of combinatorics for sums of binomial coefficients.
b
Then we apply this formula to show the norm convergence of reproducing kernelkω,nof
the evaluation ofn-th derivative of elements ofH(b) at the pointωas it tends radially
to a point of the real axis.

1.Introduction

2
LetC+denote the upper half plane in the complex plane and letH(C+) denote the
usual Hardy space consisting of analytic functionsfonC+which satisfy

Z1/2
2
kfk2:= sup|f(x+iy)|dx <+∞.
y>0R
2
P. Fatou [12] proved that, for any functionfinH(C+) and for almost allx0inR,


f(x0lim) :=f(x0+it)
+
t→0
∗2∗
exists. Moreover,we havef∈L(R),Ff= 0 on (−∞,0), whereFis the Fourier–

Plancherel transformation, andkfk2=kfk2course the boundary points where. Of
the radial limit exists depend on the functionf. However wecannot say more about the

2000Mathematics Subject Classification.Primary: 46E22, Secondary: 47A15, 33C05, 05A19.
2
Key words and phrases.de Branges-Rovnyak spaces, model subspaces ofH, integral representation,
hypergeometric functions.
This work was supported by funds from NSERC (Canada) and the Jacques Cartier Center (France).
1

2

EMMANUEL FRICAIN, JAVAD MASHREGHI

2
boundary behavior of a typical element ofH(C+[16, 1, 2, 14],many authors, e.g.). Then
have studied this question by restricting the class of functions.A particularly interesting
2
class of subspaces ofH(C+) consists of de Branges–Rovnyak spaces.
∞2
Forϕ∈L(R), letTϕstand for the Toeplitz operator defined onH(C+) by

Tϕ(f) :=P+(ϕf),

2
(f∈H(C+)),

2 2∞
whereP+denotes the orthogonal projection ofL(R) ontoH(C+for). Then,ϕ∈L(R),
kϕk∞≤1, the de Branges–Rovnyak spaceH(ϕ), associated withϕ, consists of those
2 1/2
H(C+) functions which are in the range of the operator (Id−TϕTϕ) .It is a Hilbert
space when equipped with the inner product

1/2 1/2
h(Id−TϕTϕ)f,(Id−TϕTϕ)giϕ=hf, gi2,

2 1/2
wheref, g∈H(C+)⊖ker (Id−TϕTϕ) .
These spaces (and more precisely their general vector-valued version) appeared first in L.
de Branges and J. Rovnyak [7, 8] as universal model spaces for Hilbert space contractions.
As a special case, whenbis an inner function (that is|b|on= 1 a.e.R), the operator
(Id−TbT) is an orthogonal projection andH(b) becomes a closed (ordinary) subspace
b
2 22
ofH(C+) which coincides with the so-called model spacesKb=H(C+)⊖bH(C+).
Thanks to the pioneer works of Sarason, e.g.[18], we know that de Branges-Rovnyak
spaces have an important role to be played in numerous questions of complex analysis and
operator theory.We mention a recent paper of A. Hartmann, D. Sarason and K. Seip [15]
who give a nice characterization of surjectivity of Toeplitz operator and the proof involves
the de Branges-Rovnyak spaces.We also refer to works of J. Shapiro [19, 20] concerning
the notion of angular derivative for holomorphic self-maps of the unit disk.See also a
paper of J. Anderson and J. Rovnyak [3], where generalized Schwarz-Pick estimates are
given and a paper of M. Jury [17], where composition operators are studied by methods
based onH(b) spaces.
In the case wherebis an inner function, H. Helson [16] studied the problem of analytic
continuation across the boundary for functions inKb. Then,still whenbis an inner
function, P. Ahern and D. Clark [1] characterized those pointsx0ofRwhere every function