ON INTERPOLATION AND SAMPLING IN
HILBERT SPACES OF ANALYTIC FUNCTIONS
Bo Be nd sson and Joaquim O ega
Abs ac .
In his pape wegi e new p o o s o some heo ems due o Seip, Seip-Walls en and
Lyuba skii-Seip on sequences o in e p ola ion and sampling o spaces o analy ic unc ions
ha a e squa e in eg able wi h esp ec o ce ain weigh s. The esul s a e also gi en in a
somewha mo e gene al se ing.
1. In oduc ion
In a se ies o ecen pap e s Seip S], Seip-Walls en S-W] and Lyuaba skii-Seip L-S] ha e
s udied se s o in e p ola ion and sampling o a ious spaces o analy ic unc ions o one
a iable. Pa o hese esul s conce n Hilb e spaces o unc ions ha a e squa e in eg able
agains ce ain weigh s, and ano he , closely ela ed pa deals wi h simila spaces wi h
uni o m no ms. The me ho ds used in he ci ed pap e s a e based on classical- yp e bu
in ica e cons uc ions o one- a iable na u e ha o some ex en goback o Beu ling B].
In O] Ohsawa has sugges ed he use o
L
2
- echniques o

@
o p o e esul s o he
ab o e yp e. In pa icula , Ohsawa gi es a p o o o he suciency pa o he heo em o
Seip-Walls en conce ning in e p ola ion in he space o en i e unc ions in
C
sa is ying
Z
j
j
2
e
;j
z
j
2
<
1
:
As in he app oach ini ia ed byBombie i, Ho mande and Sko da (see H]), he main di-
cul y in such a p o o is he cons uc ion o a (plu i)-subha monic unc ion wi h p esc ib ed
singula i ies a he poin s whe e one wishes o in e p ola e. Fo his Ohsawa uses pa o
he cons uc ions o Seip-Walls en, ul ima ely going back o Beu ling, and he p oses as a
p oblem o gi e a mo e elemen a y p o o . One pu p ose o his no e is o show how ha
can be done. As i u ns ou , he me ho d we use also wo ks o mo e gene al weigh s and
he e o e also implies he suciency pa o he heo em by Lyuba skii-Seip. Fu he mo e,
we shall show how he p osi i e di ec ion o he sampling heo em can be ob ained in a
simila manne , and we shall also pe mi somewha mo e gene al g ow h condi ions han
Lyuba skii-Seip (see Theo em 2).
In O] Ohsawa also gi es a new p o o o he heo em o Seip ab ou in e p ola ion in
Be gman spaces o he disk. This pa o Ohsawa's wo k con ains wo essen ial ing edien s.
The  s one is, as in he case o en i e space, he cons uc ion o a subha monic unc ion
Fi s au ho suppo ed by he NFR
Second au ho was pa ially supp o ed by he DGICYT g an PB92-08084-C02-01. The esea ch nec-
essa y o conduc his wo k has b een pa ially suppo ed by he Comissiona p e Uni e si a s i Rece ca
de la Gene ali a de Ca alunya
Typ ese by
A
M
S
-T
E
X
1
2 BO BERNDTSSON AND JOAQUIM ORTEGA
wi h p esc ib ed singula i ies. In O] no de ails o his cons uc ion is gi en, so in sec ion 3
o his pap e we shall showhow he cons uc ion in en i e space can b e adap ed o he disk
case. Fo his we use a so called in a ian con olu ion in he disk, in o duced by Ul ich
U]. The o he ing edien in Ohsawa's p o o is a gene alized e sion o Ho mande 's
L
2
-
es ima es o

@
, inspi ed by a heo em o Donelly and Fee man D-F]. Ohsawa deduces his
L
2
-es ima e om a mo e gene al e sion in ol ing ec o bundles o e Kahle mani olds.
Since his e minology p obably is no so well known among sp ecialis s in one complex
a iable, we shall also ake his opp o uni y o gi e a di ec p o o o he disk (see also
Be1] o a ela ed a gumen ). In sec ion 3 we shall also p o e he p osi i e pa o he
sampling heo em, and show ha bo h he in e p ola ion and he sampling heo ems o
Seip S2] hold o mo e gene al weigh s.
The heo ems o Seip e al also conce n spaces ha a e dened by o he no ms han
L
2
, no ably uni o m no ms. In sec ion 4 we show ha he me ho ds o his pap e also
gi e esul s o his yp e, e en in ou somewha mo e gene al se ing. The gene al lines
o he p o o s a e he same as in he
L
2
case, wi h he die ence ha he
L
2
-es ima es
o Ho mande ha e o be eplaced by simila es ima es in uni o m no ms om Be2] and
Be3].
One commen is in o de . The esul s o Seip e al a e a ac i e pa ly b ecause hey
a e so p ecise and gi e necessa y and sucien condi ions o in e p ola ion and sampling.
This pap e gi es die en p o o s o he posi i e di ec ions o hese heo ems, bu we ha e
no new ideas ab ou p o o s o he con e se di ec ions. In pa icula , we do no knowi he
condi ions in Theo em 2 and 4 a e also necessa y o sampling and in e p ola ion, al hough
his seems likely, and p e haps can be p o ed along he lines o Beu ling B], and Seip S],
S2].
Acknoledgemen .
Pa o his wo k was done while he second au ho was isi ing
he Ma h. Depa men a Go eb o g. He wan s o exp ess his hea el g a i ude o he
ins i u ion o i s in i a ion and o i s memb e s o hei hospi ali y.
2. In e pola ion and sampling in
C
.
Le

be a subha monic unc ion in
C
, and le
F
2

be dened by
F
2

=
h
2
H
(
C
)
k
k
2

=:
Z
j
j
2
e
;

<
1g
:
By deni ion, a sequence ;=
z
j
g
C
is
sampling
o
F
2

i he e a e cons an s
A
and
B
such ha o any
h
2
F
2

A
k
h
k
2


X
j
h
(
z
j
)
j
2
e
;

(
z
j
)

B
k
h
k
2

:
The sequence ; is called
in e pola ing
i o any sequence
c
j
g
such ha
X
j
c
j
j
2
e
;

(
z
j
)
<
1
we can nd an
h
2
F
2

such ha
h
(
z
j
) =
c
j
. In o ducing he no a ion
l
2

o he space
o space o all sequences

=
c
j
g
such ha
k

k
2
=
P
j
c
j
j
2
e
;

(
z
j
)
<
1
we see ha ; is
ON INTERPOLATION AND SAMPLING 3
in e p ola ing i he na u al es ic ion map om
F
2

o
l
2

is su jec i e, and sampling i i
is b ounded and injec i e, wi h closed ange.
In S] is dened he no ion o upp e and lowe densi y o a sequence in he ollowing
way:
D
+
(;) = lim sup
!1
sup
z
2
C
n
(;
(
z
))
=
2
and
D
;
(;) = lim in
!1
in
z
2
C
n
(;
(
z
))
=
2
:
(I
E
is a se
n
(
E
) deno es he numbe o elemen s in
E
.) Finally a sequence is called
uni o mly sepa a ed i he inmum o he dis ances be ween dis inc p oin s is s ic ly
p osi i e. The main esul s o Seip S] and Seip-Walls en S-W] a e as ollows.
Theo em A.
Asequence
;
is in e pola ing o
F
2

wi h

=

j
z
j
2
i and only i i is
uni o mly sepa a ed and
D
+
(;)
< = :
Theo em B.
A sequence
;
is sampling o
F
2

wi h

=

j
z
j
2
i and only i i can
be w i en as a ni e union o uni o mly sepa a edsequences and mo eo e con ains a
uni o mly sepa a ed subsequence
;
0
sa is ying
D
;
(;
0
)
> = :
In Lyuba skii-Seip L-S], an analogous no ion o densi y dep ending on an angle is in-
o duced and shown o cha ac e ize in e p ola ing and sampling sequences o
F
2

when

is subha monic, o class
C
2
and p osi i ely homogenous o deg ee 2 (i.e.

(
z
)=
2

(
z
) o
>
0).
Unwinding he deni ions o
D
+
and
D
;
we see ha
D
+
< 
i and only i o some
>
0and all sucien ly la ge
and all
z
i holds ha
(1)
n
(;
(
z
))
=
2
<
;

and ha
D
;
>
i and only i o some
>
0 and all sucien ly la ge
and all
z
i holds
ha
(2)
n
(;
(
z
))
=
2
>
+
:
I is no ha d o see ha bo h hese condi ions a e equi alen o saying ha (1) and
(2) hold o
some
la ge
. Fo a gene al subha monic unc ion

we now dene a simila
no ion. He e,aswell as in he es o his pap e we le he Laplace op e a o  b e dened
as =
@
2
=@ z @

z
, a con en ion which die s om he s anda d one by a ac o 4.
Deni ion.
The sequence
;
is dense wi h espec o


i o some
<
1
and
 >
0
i
holds ha
n
(;
(
z
))
=
2
>


(
z
)+

o al l
z
.
;
is hin wi h espec o


i o some
<
1
and
>
0
i holds ha
n
(;
(
z
))
=
2
<


(
z
)
;

o al l
z
We a e now eady o o mula e he  s e sion o ou main esul .
4 BO BERNDTSSON AND JOAQUIM ORTEGA
Theo em 1.
Suppose

is subha monic in
C
and ha


is uni o mly bounded. Then a
uni o mly sepa a ed sequence
;
is
(a) in e pola ing o
F
2

i
;
is hin wi h espec o


,
and
(b) sampling o
F
2

i
;
is dense wi h espec o


.
In he ligh o wha wejus said i is clea ha when

=

j
z
j
2
his is jus a eph asing
o he suciency pa o he heo em o Seip-Walls en. Mo eo e , one can check ha when

is p osi i ely homogenous o deg ee 2 we ge he esul o Lyuba skii-Seip (no e ha such
a unc ion always has a uni o mly bounded laplacian i i is o class
C
2
).
Theo em 1 has as a consequence he ollowing, p e haps mo e na u al, heo em .
Theo em 2.
Suppose

is subha monic in
C
and ha


is uni o mly bounded. Then a
uni o mly sepa a ed sequence
;
is
(a) in e pola ing o
F
2

i o some
<
1
and
>
0
i holds ha
n
(;
(
z
))
=
2
<
1

2
Z
j

;
z
j
<


(

)
;

o al l
z
and
(b) sampling o
F
2

i o some
<
1
and
>
0
i holds ha
n
(;
(
z
))
=
2
>
1

2
Z
j

;
z
j
<


(

)+

o al l
z
.
Assume ha he condi ion in Theo em 2a (o b) holds o a ce ain alue o
.Le

=
1

2

(0

)
and

=




be he a e ages o

o e disks wi h adius
.

is again subha monic, and he condi ions
mean p ecisely ha ; is hin (o dense) wi h esp ec o 

. F om Theo em 1 we conclude
ha ; is in e p ola ing (sampling) o
F
2

.Bu i is easily seen ha , i he Laplacian o

is uni o mly b ounded, hen

;

=
O
(
2
). Since
is a xed numb e , his p o es Theo em
2, gi en Theo em 1.
To p epa e o he p o o o Theo em 1 le

=
P

z
j
be he measu e consis ing o
apoin mass a each poin in ou sequence, which we assume om now on is uni o mly
sepa a ed. Le
E
=1
=
log
j
z
j
2
b e he undamen al solu ion o he Laplace op e a o (wi h
ou con en ion 
E
=

0
). We now dene an auxilia y unc ion
=(

;



)

E:
This unc ion is ce ainly well dened i ;is ni e. No ice ha he alue o
a
z
hen
dep ends only on he p oin s in ; wi h
j
z
j
;
z
j
<
, since o he o he p oin s he wo e ms
in he deni ion o
cancel by he mean alue p op e y o ha monic unc ions. The e o e
ON INTERPOLATION AND SAMPLING 5
we can dene
o a bi a y sequences by a limi ing p o cedu e. An al e na i e way o
dene
is o  s le
u
=(

;

0
)

E:
This unc ion sa ises (and o cou se is cha ac e ized by) he p op e ies

u
=1
=
2
;

0
in
j
z
j
<
, and
u
=
@u
=@ n
=0
on
j
z
j
=
. Mo eo e
u
=0 in
j
z
j
>
. Explici ly,
u
is gi en by
u
=
1


j
z
j
2
2
;
1+log

2
j
z
j
2

i
j
z
j
<
and
u
=0 o he wise.
In pa icula
u
has compac supp o so we can dene
by
=
;
u

:
Since
E
is subha monic i ollows om he submean alue p op e y ha
u

0 so we
always ha e

0. O cou se i holds ha

=

;



:
Assume now ha ; is hin wi h esp ec o 

.This means p ecisely ha o some la ge
enough
and some p osi i e




<


;
:
Le

=

+
:
Then




+


:

sa ises he es ima es
(i)



in
C
and
(ii)
j

;
log
j
z
;
z
j
j
2
;

j
C
in
j
z
;
z
j
j
<
0
, i

0
is chosen so ha
j
z
j
;
z
k
j
>
2

0
o
j
6
=
k
.(ii) jus says ha
j

;
log
j
z
;
z
j
j
2
j
C
which is clea since
=
;
u


in iew o ou explici o mula o
u
.

6 BO BERNDTSSON AND JOAQUIM ORTEGA
The p o o o Theo em 1 now ollows s anda d lines. Le
c
j
g
be a sequence o alues
such ha
X
j
c
j
j
2
e
;

(
z
j
)

A<
1
:
The  s s ep in he cons uc ion o an in e p ola ing unc ion is o in e p ola e lo cally nea
each p oin in he sequence
z
j
g
. Fo each
j
we apply he Riesz decomp osi ion o mula in
adisk 
j
o adius

0
and cen e
z
j
. W i e

=
h
j
+
G


]
whe e
h
j
is ha monic and
G


]is a G een p o en ial. W i e
h
j
= 2
<
H
j
,whe e
H
j
is holomo phic. Le
G
j
=
H
j
;
H
j
(
z
j
). Then
G
j
is a holomo phic unc ion sa is ying
G
j
(
z
j
)= 0 and
j

;

(
z
j
)
;
2
<
G
j
j
C
in 
j
. This means ha
j
=
c
j
e
G
j
sol es
j
(
z
j
)=
c
j
and
j
j
j
2
e
;


C
j
c
j
j
2
e
;

(
z
j
)
in 
j
. We nex combine he
j
:s using a pa i ion o uni y.
Le
g
2
C
1
c
(
C
) b e such ha
g
= 1 o
j
z
j
<
0
=
2,
g
= 0 o
j
z
j
>
0
and
j
@g
j
C

0
. Pu
(
z
)=
X
g
(
z
;
z
j
)
j
:
Then
(
z
j
) =
c
j
so
in e p ola es he igh alues. Finally we shall mo di y
o ge a
holomo phic in e p ola ing unc ion by sol ing a

@
-equa ion. No e  s ha
j

@
j
2
e
;


C
X
j
c
j
j
2
e
;

(
z
j
)
j

@g
(
z
;
z
j
)
j
2
:
We hen apply Ho mande s

@
- heo em H], which implies ha we can nd a solu ion
U
o

@U
=

@
sa is ying
Z
j
U
j
2
e
;


Z
j

@
j
2
e
;



:
Since 



and

@
anishes when
z
2
(
z
j

0
=
2) o some
z
j
he igh hand side is
domina ed by some cons an imes
Z
j

@
j
2
e
;


C
0
X
j
c
j
j
2
e
;

(
z
j
)
j

@g
(
z
;
z
j
)
j
2

C
00
A:
Consequen ly
Z
j
U
j
2
e
;


Z
j
U
j
2
e
;


C
00
A:
ON INTERPOLATION AND SAMPLING 7
Mo eo e
U
(
z
j
)=0 o each
z
j
since
e
;


1
=
j
z
;
z
j
j
2
nea
z
j
. Le
h
=
;
U:
Then
h
(
z
j
)=
c
j
, and
R
j
h
j
2
e
;

<
1
since b o h
and
U
sa is y his es ima e. The p o o
o Theo em 1 a is he e o e comple e.
We now u n o he sampling pa o Theo em 1. Fo a momen , le

be an a bi a y
p osi i e measu e on
C
. Assume




C
and ha



>


+

which is he analog o he densi y condi ion o gene al measu es. Pu
=(

;



)

E
and

=(

+

)
:
As b e o e



bu his ime we ha e




;
:
Le
h
2
F
2

,and le
U
=
j
h
j
2
e
;

. Since
log
U
;


i ollows ha

U
;
U

:
Mo eo e ,
U
2
L
1
(
C
). Le
g
be a cu -o unc ion such ha
g

0,
g
= 1 o
j
z
j
<
1 and
g
=0 o
j
z
j
>
2. Then
lim
R
!1
Z
g
(
z=R
)
U
=lim
R
!1
Z
1
=R
2
(
g
)(
z=R
)
U
=0
:
By he die en ial inequali y o
U
Z
U



0
so i ollows ha
(3)

Z
j
h
j
2
e
;


Z
j
h
j
2
e
;

d:
This is al eady an inequali y o sampling yp e. Wewould like ocho ose as b e o e

=
P

j
bu we canno do ha di ec ly since

would hen be iden ically equal o
;1
on he
supp o o

,so he inequali y would be o no alue.
8 BO BERNDTSSON AND JOAQUIM ORTEGA
Ins ead we shall ake

o b e a smo o hed e sion o asumo Di ac measu es. Le ; b e
a sequence which is dense wi h esp ec o 

. Le

be dened as

=
X
1

2

(0

)
(
z
;
z
j
)
whe e 0
< <
1 ( we make no dis inc ion be ween an absolu ely con inuous measu e and
i s densi y wi h esp ec o Leb esgue measu e). Since ; is dense we can cho ose
so close
o 1 ha



>


+
=
2

o some la ge
. Then

=

+

will sa is y he es ima es
(i')

;
C





and
(ii')
j

;
log

2
;

j
C
in
j
z
;
z
j
j

.
Take
h
2
F
2

. Then
h
2
F
2

so (3) holds. This in conjunc ion wi h (i') and (ii') gi es
(4)
=
2
Z
j
h
j
2
e
;


C
X

;
2

2
Z
j
z
;
z
j
j
<
j
h
j
2
e
;

:
Now we ha e o use ha
h
is holomo phic (so a we ha e used only ha log
j
h
j
2
is
subha monic.) Fix
j
o he momen and w i e
(5)
Z
j
z
;
z
j
j
<
j
h
j
2
e
;

=
Z
j
z
;
z
j
j
<
j
he
;
G
j
j
2
e
;

e
2
<
G
j

Z
j
z
;
z
j
j
<
j
g
j
j
2
e
;

(
z
j
)

whe e
g
j
=
he
;
G
j
is holomo phic in
j
z
;
z
j
j
<
1 and
g
j
(
z
j
)=
h
(
z
j
). Clea ly
1

2
Z
j
z
;
z
j
j
<
j
g
j
j
2
e
;

(
z
j
)

2
j
h
(
z
j
)
j
2
e
;

(
z
j
)
+
C
2
e
;

(
z
j
)
sup
j
z
;
z
j
j
<
j
g
0
j
j
2
:
Bu , by Cauchy's es ima e
sup
j
z
;
z
j
j
<
j
g
0
j
j
2
e
;

(
z
j
)

C
Z
j
z
;
z
j
j
<
1
j
g
j
j
2
e
;

(
z
j
)

Z
j
z
;
z
j
j
<
1
j
h
j
2
e
;

:
Summing o e
j
we nd
=
2
Z
j
h
j
2
e
;


C
;
2
X
j
h
(
z
j
)
j
2
e
;

(
z
j
)
+
C
2
;
2
Z
C
j
h
j
2
e
;

:
Cho osing

small enough we can abso b he second e m on he igh in he le hand side.
This p o es ha ; sa ises he le o he inequali ies in he sampling condi ion. The
o he inequali y is i ial since we ha e assumed ha ; is sepa a ed, so Theo em 1b is
now comple ely p o ed.
ON INTERPOLATION AND SAMPLING 9
3. In e pola ion and sampling in
D
.
Le

be a subha monic unc ion in
D
, and le
F
2

(
D
)be dened by
F
2

(
D
) =
h
2
H
(
D
)
k
k
2

=:
Z
D
j
j
2
e
;

1
;j
z
j
2
<
1g
:
A sequence ;=
z
j
g
D
is
sampling
o
F
2

i he e a e cons an s
A
and
B
such ha o
any
h
2
F
2

(
D
)
A
k
h
k
2


X
j
h
(
z
j
)
j
2
e
;

(
z
j
)
(1
;j
z
j
j
2
)

B
k
h
k
2

:
The sequence ; is called
in e pola ing
i o any sequence
c
j
g
such ha
X
j
c
j
j
2
e
;

(
z
j
)
(1
;j
z
j
j
2
)
<
1
we can nd an
h
2
F
2

(
D
) such ha
h
(
z
j
)=
c
j
.
A sequence ; is called uni o mly disc e e o sepa a ed i
in
j
6
=
k

(
z
j
z
k
)
>
0

whe e

is he pseudo-hyp e b olic dis ance in
D
,

(
z 
)=




z
;

1
;

z




:
Following S2] we shall dene he no ion o upp e and lowe densi y o a uni o mly sepa a ed
sequence in
D
. No ice howe e ha ou deni ion die s om he one in S2] by a ac o
o wo.
D
+
(;) = lim sup
!
1
sup
'
2
Au (
D
)
X
z
n
2
'
(;)
1
=
2
<
j
z
n
j
<
log
1
j
z
n
j
2
log
1
1
;
and
D
;
(;) = lim in
!
1
in
'
2
Au (
D
)
X
z
n
2
'
(;)
1
=
2
<
j
z
n
j
<
log
1
j
z
n
j
2
log
1
1
;
The main esul s o S2] conce ning weigh ed Be gman spaces in he disk a e he ollowing
Theo em C.
A sequence
;
is in e pola ing o
F
2

(
D
)
wi h

=

log 1
=
(1
;j
z
j
2
)
i and
only i i is uni o mly sepa a edand
D
+
(;)
<:
16 BO BERNDTSSON AND JOAQUIM ORTEGA
Theo em F.
Le

be a (plu i)subha monic unc ion in
C
n
such ha
i@

@ >  >
0
.
Then i
is a

@
-closed
(0

1)
- o m in
C
n
and
u
is he solu ion o

@u
=
which is o
minimal no m in
L
2
(
C
n
e
;

)
,
u
sa ises
(*) sup
j
u
j
e
;
e
=
2

C
sup
j
j
e
;
=
2
:
This Theo em has as a consequence a mo e p ecise s a emen ha we will need. Le
h
be any ha monic unc ion in
C
and w i e
h
=
<
H
whe e
H
is en i e. Then
=
ue
;
H=
2
is he canonical solu ion o

@
=
e
;
H=
2
in
L
2
(
e
;

+
h
). Applying Theo em F o his
si ua ion ins ead we see ha
e

(
z
) can be eplaced by
^
(

;
h
)(
z
)+
h
(
z
)

whe e
h
is any ha monic unc ion in
C
. I is e en enough o assume ha
h
is ha monic
in 
z
=: (
z 
0
=
2), since any such
h
can be app oxima ed by unc ions ha a e globally
dened. Cho osing
h
o be he ha monic ex ension o

om
@

z
o he in e io o 
z
,
we see ha
e

(
z
) can be eplaced by

(
z
)
;
G
(
z
) whe e
G
is he G een's p o en ial o 

o e 
z
. In pa icula , i he Laplacian o

is uni o mly b ounded, he G een's po en ial
will be b ounded, so Theo em F holds wi h
e

eplaced by

.
In ou case whe e

=

+
one sees in a simila way ha we can eplace
e

by

in (*).
Le us now o a momen supp ose ha ou sequence ; is ni e. Then

@
lies in
L
2
so
Theo em F applies and we see ha he canonical solu ion o

@u
=

@
sa ises
sup
j
u
j
2
e
;


C
sup
j

@
j
2
e
;
=
2

C
sup
j
c
j
j
2
e
;

:
Mo eo e he canonical solu ion also sa ises he
L
2
es ima e om sec ion 2, so i ollows
ha
u
(
z
j
) = 0. Le ing
h
=
;
u
we ge an in e p ola ing unc ion in
BF

.Since he
no m do es no dep end on he numb e o p oin s in he sequence, we can also p e mi inni e
sequences by a no mal amily a gumen .
We can also ea he disk case in asimila manne . Le

b e subha monic in he disk,
and supp ose ha he in a ian Laplacian o

is uni o mly b ounded. Pu
BF

(
D
) =
2
H
(
D
) sup
z
2
D
j
(
z
)
j
2
e
;


C
g
 :
A subsequence ; o he disk is in e p ola ing o
BF

i o any sequence
c
j
g
such ha
sup
j
c
j
j
2
e
;

(
z
j
)
(1
;j
z
j
j
2
)

C
he e is a unc ion
in
BF

such ha
(
z
j
)=
c
j
. We hen ha e
Theo em 6.
Le
;
beasequence in he disk which sa ises he hypo hesis o Theo em 4
(a). Then
;
is in e pola ing o
BF

(
D
)
.
The p o o again ollows he same pa e n as in he
L
2
case he only die ence b eing
ha weneed o eplace he
L
2
-es ima es o

@
by uni o m es ima es. In Be3] he e is also
a heo em analogous o Theo em F o he case o he disk (o ball in
C
n
)bu i equi es
ha
e

>
4. In one dimension one can a oid his es ic ion by ins ead app ealing o he
ollowing heo em om Be2].

ON INTERPOLATION AND SAMPLING 17
Theo em G.
Le

be a subha monic unc ion and

=: min ((1
;j
z
j
)

1
=
(1
;j
z
j
))
:
Le
be a unc ion in
D
such ha
sup
j
j

e
;
=
2
<
+
1
:
Le
u
be he canonical solu ion o

@u
=
in
L
2

.Then
sup
j
u
j
e
;
e
=
2

C
sup
j
j

e
;
=
2

whe e
e

(
z
)= sup

(
z
)
<
1
=
2

(

)
.
In ou cons uc ion om sec ion 3 i holds (a e sub ac ion o a ha monic unc ion
like in he case o en i e space), ha
e

;

is uni o mly b ounded, and mo eo e he
co esp onding

will be o size 1
=
(1
;j
z
j
2
). The e o e Theo em 6 ollows in he same way
as Theo em 5.
Finally, i may be wo h ema king ha since we use he canonical solu ion o he

@
-equa ion, we can in e p ola e be ween
L
2
and
L
1
and ge simila esul s in
L
p
o
p
be ween 2and inni y.
Appendix: A p oo o Theo em E
The one dimensional case o Ho mande 's heo em ha we used in sec ion 2 says ha
i

is subha monic in a domain in
C
we can sol e he equa ion

@u
=
g
wi h he es ima e
(8)
Z
j
u
j
2
e
;


Z
j
g
j
2


e
;

in .
Tha

@u
=
g
in he sense o dis ibu ions means ha i

is any unc ion in
C
1
c
( ) i
holds ha
Z
g



=
;
Z
u
@
@z
:
Taking he sup emum o he no m o he igh hand side o e all
g
such ha
R
j
g
j
2


e
;


1,
we ge om (8)
(9)
Z


j

j
2
e


Z




@
@z




2
e


o all

2
C
1
c
( ). Con e sely (9) implies Ho mande 's heo em, and he usual p o o
consis s p ecisely in es ablishing (9) using in eg a ion by pa s.
18 BO BERNDTSSON AND JOAQUIM ORTEGA
Top o e Theo em E we le =
D
b e he uni disk and pu as b e o e

0
= log 1
=
(1
;j
z
j
2
).
Pu

=

+

0
in (9). The c ucial p oin is ha

0
sa ises
(10) 

0
j
@
0
j
2
:
Subs i u e

=
e
;

0
in (9), and assume 





0
.Then we ge
(

+1)
Z


0
j

j
2
e

;

0

Z




@
@z
;

@
0
@z




2
e

;

0


(1 + 2
=
)
Z




@
@z




2
e

;

0
+(1+
=
2)
Z
j

j
2




@
0
@z




2
e

;

0
:
Rea anging and using (10) we ob ain
(
=
2)
Z


0
j

j
2
e

;

0

(1 + 2
=
)
Z




@
@z




2
e

;

0

o mo e explici ly
Z
j

j
2
e

=
(1
;j
z
j
2
)

C=
2
Z




@
@z




2
(1
;j
z
j
2
)
e

:
A s anda d unc ional analysis a gumen hen shows ha o a gi en
g
wemay nd
u
such
ha
Z
g



=
;
Z
u

@
@z
o all

2
C
1
c
(
D
), and
Z
j
u
j
2
e
;

1
;j
z
j
2

C

Z
j
g
j
2
e
;

(1
;j
z
j
2
)
:
This comple es he p o o o Theo em E.
ON INTERPOLATION AND SAMPLING 19
Re e ences
Be1] Be nd sson B.,
A simple p oo o an
L
2
-es ima e o

@
on comple e Kahle mani olds
, Rep o
(1992).
Be2] ,
Weigh ed es ima es o

@
in
C
, Duke Ma h. Jou nal
66
(1992), 239{255.
Be3] ,
Uni o m es ima es wi h weigh s o he

@
-equa ion
, p ep in (1994).
B] Beu ling A.,
The col lec ed wo ks o A ne Beu ling Vol 2
, Bi khause , Bos on, 1989, pp. 341{365.
D-F] Donelly H. and Fee man C.,
L
2
-cohomology and index heo em o he Be gman me ic
,Ann.o
Ma h.
118
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Bo Be nd sson: Depa men o Ma hema ics CTH S-412 96 G

o ebo g Sweden.
e-mail: bob@ma h.chalme s.se
Joaquim O ega: Depa men o Ma hema ics, UPC (ETSEIB) 08028 Ba celona, Spain.
e-mail: jo [email p o ec ed]