ON SOME NEW BASES IN SPACES OF HOLOMO~IC FUNCTIONS M

F. Haslinger

i. Introduction

In this paper we continue the investigation of methods to

construct basic systems due to I.I. Ibraginov, I.S. Arshon and

N.I. Nagnibida (see [7], [8], [9]). Especially we are

concerned with the problem of constructing bases out of a given

Ipml of ~olynomials. If k denotes the degree of sequence

Pm (m=0,I,2,...), we set i o=0 and

m-1 =m+ Z k m=1 2

im j=0 J' ' ' ....

We ask for necessary and sufficient conditions for the system

1 n mpm ( (1.1) {z }n~! u {z z)l (m,n=0,1,2,...)

m

to be a basis in the space ~R of all holomorphic functions on

the open disc Izl <R (0<R_<co) (compare [9], § 7). For

instance it turns out that t~e system (I .I ) constitutes a basis,

if the set IPm: m=Q,l,2,...l is bounded in ~K.

Our constructions of bases are connected with the

Pincherle basis problem, which consists of determining

conditions under which a given Pincherle sequence I C~In__°° 0 forms

a basis in ~R, where

) n (z), n=0,1,2,..., ~n(Z =z Sn

~nr~ and Sn(O)=I for each n=0,I,2,... . A number of papers

deals with the Pincherle basis problem (see [II, [3~, [5], ~67,

[97, [13]), especially with the cases, where the functions Sn

267

are linear or of the form @n(Z)=f(knZ) , where (kn)n=0C° is a

sequence of scalars and f is a suitable holomorphic function.

It is also an object of the present paper to examine the case,

where the functions Sn are polynomials.

Finally, by KSthe's duality principle in function theory

(see F10]), we obtain new polynomial bases, i.e. bases of the

form lqn}na~=0 , whez'e qn are polynomials of degree n.

In the sequel we will use the followin~ method to investigate

complete biorthogonal systems (see [47). It is a method of the

theory of bases in nuclear Fr$chet spaces, which differs in

some essential points from the methods in [97.

We consider the spaces ~ of all holomorphic functions on

the open disc DR= Iz: !zl <R~ (0<R<co) endowed with the

topology of uniform convergence on the compact subsets of

D R . These spaces are nuclear Fr@chet spaces, which can be

identified with the K~the sequence spaces

Go AR= {~=(~n)na°O: It~l'T,r: = Z t~ lrn<m, Vr<Rt

n=0 -n

(see [11], [1 4] and [157).

The isomorphism T between ~ and A R is given by

where

~(f) = (~n) n=0 '

Go f(z)= Z ~n zn.

n-0

The strong dual ~l of ~R can also be identified with a sequence

space

A R = {~ (~n)n~__O: 3r<R with II~!r' ' = r : =sup lqnlr-n<co }, n

where A R' has the inductive limit topology generated by the norms

268

I} II' ( O < r < R ) The d u a l i t y be tween A R L • r •

the formula oo

(~,q> = % gn~n , n=O

and A R' is given by

for ~ ~ A R, q E AR'.

A linear functional L on ~ is continuous if and only if there

exists a number r<R such that

!IT,I1 r ' ' = s u p t l L ( f ) t : ,x r I I f l I _<I, f E ~ l < c o •

If (fn) so ~ n=O is a basis in , then we can define the so-called oo

coefficient functionals by L n ( f ) =c n f o r f = Z Cnf n, f E ~ . n-O

The functionals (Ln)n~=0 are continuous• By the theorem of

Dynin-Mityagin (see [12q [I 41) each basis (fn) ao ' n=O of a nuclear co

Fr@chet space is absolute, i.e. for f= Z Ln(f)f n we have n=0

oo Z I Ln(f)Illfn!Ir < ao

n=O

for each r <R, where

Even more is true: for each r<R there is a number r' <R and a

constant K depending on r and r' such that

co Z I Ln(f) 1 !I fnl r _<_ ~Iflr ,

n-O

for each f E ~K.

Now let (fn,Ln) be a complete biorthogonal sequence for ~, that

is: the set of the finite linear combinations of the elements f n

is dense in ~g and Lm(f n) = &n, m" The question arises as to which

conditions will ensure that a given complete biorthogonal sequence

already constitutes a basis for ~a• The following theorem answers

269

this question.

T~LEOREM A. A complete biorthogonal sequence (fn,Ln) constitutes

a basis in ~ if and only if, for each r<R, there exists a

number r' <R such that

su.~ !i I, n11' < ao . . , ~, i!~dl~ n

For the oroof see F4q.

The corresponding res~.lt for the nuclear K~the sequence

spaces is as follows

~__mHE0~M ~.~ A como!ete~ biortho~onal sequence (~n = (~nk)k=O~° ,

(~nk)k°°=0) constitutes a basi s in A R if and only if, for each

r<R. there exists a number r' <R such thst

!mnkl ~o k ~ , ~ F ( s u o r, I e n<CO

k - ~ - ' . " n _ k - ' " . . . . . . . k = O

2. Construction of basic systems

On the whole we deveioo two different methods of constructing

complete Pincher!e sequences.

(.) T + Iv }~ be a snquence o~ oo!ynomia!s, where Pm(0)=I ,.. J.Je ,, ' - - ~ . 0 - - - -

L~t k denote the degree of ~ (~=n I 2 ) for m=0,I,2, .... m -m - ' ' ' ....

We set ! =0, and o

~-I (2.1) 1 =m+ ~ k. for m= !,2,...

m j=O J

and define c o=O o,

n an(Z) =z for I <n<l I,

l -I %1(z) =z PI(s)'

11+J ~l~+j(z) =z , I ~j ~k I,

and so on- in .~enera!"

(2.2)

270

!

c~ ,(z)=z 2p2(z). -::)

l

~n(z) = z m pro(z),

if n=J. for m=0~1,2:..., and m

(2.3) ~.n( z ) =z n

otherwise.

We set

2 km ~m (z) =pro0 +pmlz +pro2 z +''" +Pmk z

m , m=0,I,2,...,

where PmO = I for each m, then the above defined Pincherele sequence

__~0 0 -L~ __ I~n! n can be writ~.n as _~n infinite u~oer triangular matrix,

if we identify the functions ~n in (~R with the corresponding

sequences in ~R"

It is easily seen that the Pincherle sequence I~nln__°°0 is

complete in each space ~, because the elements of the basis

I~nln~__0 ' where ~n (z) =z n for n=0,I,2,..., are reoresentable

as finite linear combinations of the f~uctions ~n"

(b) For the second method we suppose that the coefficients

Om9 k o f the ~o lynom. io l s Pm s t ' - a l l n o n z e r o We p r o c e e d i n t he

fo!!owing way: we set

(2.4) = ~ and

o "o i (z) =z m pro(z), n

if n=~ , where ! is defined as in (2.1)" m m

271

1 + j ~ Pm, .i÷2 2+ Pm, k m (2.5) e I +j(z) =z m (I + z + z ...+

m Pm j Pm j Pm , , , J

k-j m )

for I _<j_<k -I, and m

~im+k m im+k (2.6) (z) =z m .

In this case we have the following infinite upper triangular

matrix:

Pol Po2 "'' Pok 0 0 0 ... 0 0 0 ... 0

Pok Po2 o

I ... 0 0 0 ... 0 0 0 ... Pol Pol

Pok 0 I .. o 0 0 0 0 0 0 • D o , , o • •

Po2

0 0 0 .•. I 0 0 0 ... 0 0 0 .•.

(2.7) 0 0 0 ... 0 I P11P12 •'" Plk

Plk P12

0 0 0 ... 0 0 I ---- ... P11

P11 Plk I

0 0 0 .•. 0 0 0 I ... ~ 0 0 P12

0 • • •

1

I - - - - - 0 0 • • .

0 0 0 ... 0 0 0 0 ... I 0 0 ...

0 0 0 ..• 0 0 0 0 ... 0 I P21 "'"

GO To show that the Pincherle sequence l~n!n=O is complete, we

o b s e r v e t h a t t h e a b o v e i n f i n i t e m a t r i x c o n s i s t s o f (k m x km)

submatrices along the diagonal, each of which is similar to

the identity matrix•

272

3. ICnln~O and f~nl ~ ' n : O as ba, SP, S

In this section we derive necessary and sufficient conditions

for the complete Pincherle sequences l=nln~__O to be a basis in

~r R. rest~Its are obtsined ~o~ the seouences I~ I ~° Similar . . . . . ~ ' l l lq :O"

PROPOSITION I • (~) Let Ipml ao - m:O be a seauence of polynomials

with degree k m, m=0,I,2,..., and suppose that the sequence

(km)fO is bounded by K>0 Then I~ I ~O = " .... n n=O constitutes a basis

i n ~R (o <R <oo) if ~.~d o~_,7 i ~

I/1. ( 3 . 1 a ) l ira sup IPm i EL<I

r~-e CO * J -

for each j (I ~j _<K).

(b) ! , e t l p m t °°=0 be e s in ( a ) . I~.nln~__o constitutes

a b~sis in ~ao if ~m.d only _if

1 / t (3.1b) iim sup l Om,jl re<co

nTeco

for each j (I <j <K).

P r o o f . (a) To be able to use Theorem A or B, we have

to define a sequence of biortho~onal functionals ILn 1 a°n=0 in '

respectively in AR'. By K~the's duality principle each continuous

linear functional L on ~r R can be represented as an equivalence

class of locally holomorphic functions F L on Iz: !zl->RI

(F L(CO ) -0), where

I (~)f(w)dw (3.2) ~(f) =~-~ / ~, V

for f ~ ~ and a closed rectifiable curve y lying in the

intersection of the domains of ho]omorphy of f and F L, For the

P i n c h e r ! e s e o u e n c e t e n t m ~ n=O we de fine

273

(3-3) F (z)=z n

-(i +I ) m

if n=l m

for m=0,1,2,..., and

(3.4) F I +j(z)=-Pm j m

-(! +I) -(i +j+1) m m

z +z

for 1 ~ j _<k m and m:0,I,2, ....

By formula (3 .2 ) , Ic n,F n!n~__O forms a complete biorthogonal

system. If we identify the functions F with the corresponding n

! sequences in ~R ' we get an infinite lower triangular matrix.

We are now able to apply Theorem B, which yields that the

system l an,Fnlf0 constitutes a basis in ~ if and biorthogonal

only if for each r<R there exists a number s with r<s<R such

that

1 Fnk ~ oo (3.5) sup [( sup k ) ~ I Onk Irk7<e°'

O_<n<~ O_<k<oo s k:O

where ~"n (~nk)k~:0 and F n (Fmk) m = = k=0 are sequences in A R and A R

respectively. Since A R is a nuclear K~the sequence space condition

(3.5) is equivalent to

00 -k O0 (3.6) sup [( Z IFnklS )( Z !a~nklrk)]<oo

O_<n<co k=O k=O

(see r157).

By the definition of the complete biorthogonal system

I~ n,Fnln~_ -0 we obtain that condition (3.6) is equivalent to

the condition: for each r<R there exists a number s with

r<s<R such that

k m i +j -!m7

(3.7a) sup [( T iPmjir m )s <m O_<m<m j =0

and

(3.7b) sup { max [(IPm Is O _ < m < o o I _<j ~k ' ' j

m

2 7 4

-1 -( lm+ j ) m

+s )r Im+j q I < co

(see (2.2), (2.3), (3.3) and (3.4)).

Since the two systems of norms

00

( E [ Ir k , r<R) k=0 k , r<R) and ( sup Igk Irk O<_k<oo

are equivalent in A R (see F15q), condition (3.7a) coincides

with condition (3.7b).

By assumotion,._ the sequence (k m)m=O°° is bounded, for instance

by the integer K>O, therefore it follows from (3.7b) that

'l +j -imp-" m _< M(r,s)

(3.8) IPm,j!s .

for each m=0,I,2,.., and j (I _<j <K), where M(r,s)~0 is a

constant only depending on r and s. From (3.8) we obtain

I/! s Im -J s)q m%_s (3.9) !in sup I~ 1 re<lira sup r(r) ~ N(r, r'

m ~ a o ' -- m , j m~oo J

since i ~ co, as m~oo. Since s can be chosen arbit~ari!y near m r

to I (i.e. for each ~>0 there exists a number r<R such that

I <s<R<1 +e for each s>r with s<R), (3.9) implies (3.1a). r

On the other hand it follows from (3.1a) that

I/l m IP m jl <_S

, -r

for almost all m and each j (I #j %K), which immediately

implies (3.7b), and (a) is proved.

(b) Condition (3.9) implies at once (3.1b), and it follows

from (3.1b) that for a given r<oo there exists a number s>r

such that

275

~ / z

Ip m I m<s ~j - r

for almost all m and each j (I _<j _<K); which again implies (3.7b),

and Proposition I is proved.

R ~.. m a r k . T~__. the sequence (km)m~=0 is not bounded, then

condition (3.7b) remains to be a necessary and sufficient

f condition for !~n! =0 to be a basis in ~R' and one sees that the

behavior of condition (3.7b) strongly depends on the relation

between the two sequences (!)co m m=O and (k) co m m=0' as the following

equivalent form of (3.7b) shows :

!/1_!I +j (m,r,s)/_7._ i ~,_ m~,'-lq<1, ( 3 - 1 0 ) Z im s ~ [ ! ~ m 1

m~oo ,j(m,r,s) ,

where j(m,r,s) is defined by

-!m -(!m+J(m'r's))~ lm+i(m,r,s) (I Pm, j (m,r,s) ! s + s )r

-i -( !m+ j ) im+ j Is m+3 )r 7. max [ ( I Pm

1 % j ~ k ' ' j

In the {eneral case it is possible to derive a simple

sufficient condition for {~}n~=0 to be a basis in ~.

PROPOSITION 2" In ;n l is a basis in ~r ( 0 < R < c O ) , i f one ' .~ ' 0 "

of the fo!!owin~ two equivalent conditions is satisfied: _d ..........................................................................

(3.11) (a) Io • m=0,I,2,... ~ is a bounded set in ~.

~ ~ < ( = o ) (b) lira ~u~ qi ~

where o. - su~o ! .m .! <co for j : 1,2, .... ------ ~ 0_<m~<~ m; ~ ....

Proof . If !Pro m=@,I:2,... I is a bounded set in ,

then

276 k m

Ir j ) <co , (3.12) sup ( T IPm O_.<m<oo j =0 ' ' j

for each r<R, ~hich implies that for each r<R there exists

a number s<R such that

k k m 7 +j -i i m - r m ,'r'i7 sup [( Z IPm Ir m )s mq= sup [(s ) ~ IPm j <GO

O_<m<co ,j =0 ' J O_<m<oo • j=0 ' 8

Therefore (3.7a) is satisfied, and by the proof of Proposition I

I Cnln=O GO forms a basis in ~. Fo~_ the equivalence of (~) and (b)

see [2 J.

Now we turn our attention to the complete Pincherle sequence

n=0 :

I GO PROPOSITION 3: (a) Let {Pm m=0 be a sequence of polynomials

with degree kin, m=0,1,2,..., and suppose that the sequence

( k ) ® m m=O is bounded by K>O Then I~ I constitutes a basis " n 0

in ~g (O<R<eo) if and only if

i/i (3.13a) lim sup IPm-~il m_<1

m~oo Pm, j

for each i__and j __-__with O_<j < i ~ K (we set ~=0).

(b) Let Ipml~O be as in (a). I~nlnGo__O constitutes

a b a s i s i n 3-" i f and o n l y i f

t /J_ (3.13b) !ira su~ ]P~'i] m<co .

- ]0 ' _rq-* O0 m~j

P r o o f . We have again to define a sequence df biorthogonal

function~Is ILn] _ne°O . B.v K~the's duality principle we can again

reduce this auestion_ ~ tO the definition of a secuence~ IGnln=O~° of

locally holomorphic functions on Iz ~ I zl ~RI. We set

277

(3.14) Gn(Z) =z - ( ! +1)

m

if n:! m

for m-0,I,2,..., and

-( im+J ) -( in+J+1 ) (3.15) G] +j Pm

m ,j-1

for I <j _<km and m=0,I,2, ....

By formula (3.2), {~n'Gn]n~-~O forms a complete biorthogonal

system. If we identify the functions G n with the corresponding

sequences in AR', we get an infinite lower triangular matrix.

Applying Theorem B we obtain, that the complete biorthogona!

system {~ ,G In ~ ' n n = 0

forms a basis in ~, if and only if for each

r<R there exists a number s with r<s<R such that

(5.16a)

( 3.1 6b) sup O~m<m

k m i +j -! sup r( r~ I p m jr m )s m

O_<m<oo j =0 ' j ] < oo ,

]. +j km-J P__.., P__. I max [r m (I + Z I m-~-~a~--flri)q[l~Is 1~j<k m i=I Pm, j "' Pm,j-1

-( Zm+j ) + s ~1<oo

-(Zm+j-1 )

and

(3.16c) sup 0.~<m<oo

! +k Pm,k m [ r ~ m( I - - - - - - - - - t

Pm,k,m-1

-(z +k-l) -(l+k m) s ~ m +s )1<~

(see (2 .4 ) , (2 .5 ) , (2 .6 ) , ( 3 . 1 4 ) a n d (3 .15 ) ) .

As in the proof of Proposition I, condition (3.16a) is

equivalent %o

1/ ! , 1 n ~ I . Jim suo I??~ i

r' l-) 09

for I <j s:K (k #K for each m).

278

Condition (3.16b) can also be written in the form

q +j km-J r -m ~ i

sup I max rs(s) ( z I I~ ) + O_<m<oo 1.~j<k i=O Pro, j-1

m

z +j ~-J r m m Pm, j+i r i + (~-) ( 1 + z I 1 ) 1 1 < o o ,

i=I ' Pm,j

since the sequence (k m) ao m=O is bounded, this yields by the same

arguments as in the proof of Proposition I, that (3.16a), (3.16b)

and (3.160) are equivalent to (3.13a) respectively (3.13b).

R e m a r k . Conditions (3.16a), (3.16b) and (3.16c) are also

necessary and sufficient in the general case. It is now also

clear, how to fortune.late an analogous result to Proposition 2.

For this case one has to assume that the set of all polynomials

~,< k m submatrices of (2 7) is bounded in ~. appearing in the k m

4. Polynomial bases

Here we use K~the's duality principle- as already indicated

the stron~ dual space ~K' can be identified with the space of

germs of locally holomorphic functions F on Iz: I zl >-RI with

F(oo )=0, where this space of germs is endowed with the inductive

limit topology determined by the norms

T!F!!r=maxIIF(z)I: !z! =rl,

for r<R, r~R (for details see KSthe [I0], [11]).

Let ~, (0.<p <oo ) denote the space of germs of locally holomorphio

functions on Iz: Iz! _<p}, again endowed with the inductive _limit

topology determined by the norms

iifr!~=m~xll~(z)r : !zl =r!,

279

for r>0, r~ 0.

Now we define a topological isomorphism T between %' and ~IIR

by

( 4 . ~ ) I I (~) (z) =z ~(z)'

f o r F ~ ~r , o nd T z I _<1. ( S i n c e F(oo ) : 0 and F i s h o ! o m o r p h i o in

a neighbourhood of oo ; the right side of (4.1) is also defined

for z : 0. )

We set now

( 4 . 2 ) ~n(z~' = (~~- .... n ) ( Z ) and ~.~n (z ) : (Ton) ( z ) _

for n=~,I,2,.., and obtain in view of (3.3), (3.4) and (3.14), (3.15)

two seauences o ,° ~o!vnomia!s I~ ~ ~ and {~ ~ ~ " ~ " ~ " -n "Ii:O "-z! 'n:O'

where

7

f~(z~..=z m

if n=! m

{or m=r~ I 9 and . . . . . . ' 9 9'" 9 ~ ° ° 9

:I +j z :-n j,~ T e ' l ' " . . . .

_

m

z +j

for ! < ~ < k and m-0,I.2 .... • ~nd

i

..'9_

if n:l for m-,:%,I,2~,..~ and

o 7 +j-1 7 +~ ~ ,.<z)--- ~ z -m +z -m "

m " ~m, j-!

for I <j _<k and m~0.!,2 ..... m

Here we ~J.se ? ~eners 7 method which orJ_~ines from duality

in functional on?lysis and which can s.!so be use@ for a more

~enera! situation (see [67).

280

PROPOSITmON a: Let @Pml co - 'm=O be a sequence of polynomials with

degree km, m=0,I.2,..., and suppose that the sequence

{kin) • m=O is bounded by K> O.

(a) Ifnl co ~ n=O constitutes a basis in (O<R<co) if and

only if

i/i !im sup !Pm 1 m

m-~co ' j ~ I

for each j (I _<j %K).

(b) {~ I ® n n=O constitutes a basis in (O<R<co), if and

only if

I/! f

m

m-,oo Pm, j _<!

0 for each i and j with O<j <i_<K (we set.~=O).

o r o o f . B~ Prooosition I IF ~ constitutes a basis in ~ '~ - • n ~ 0

~K (O<R<oo) if and only if (3.1&) is satisfied; this follows

only if (3.1a) is satisfied. Since T is a topological isomorphism,

!~nl~O fo~m~ ~ b~i~ i~ $~ (O<p<.) ~f ~na on!y if (3.1a) i~

satisfied. _Tn order to show that condition (3.1a) is eilso

necessary and ~'ci_nt fo ! co su_~m ~ r If n n=O to be a basis in , we

choose for a given nu_mber R (O<R%co) a number o (0<0 <co ) such

that 0 <R. If now (3.1a) is satisfied, then I fn!n~__O forms a basis

in ~, which means that for each f ~ there exists a number

0'>0 and a unioue!yN determined sequence (Cn)n °°=O of scalars

such that !!f - ~ Cnfn I * 0 as ~T, oo Since o can be chosen n=O o ' ~" ~ "

arb_~ trarily near to R If nl oo ~- " n=O forms a basis in too.

281

If on the other side If nlnm_O is a basis in ~ then {fn} co ' ' ' n:O

is also a basis in JS for each S>R (see [97 Remark 2.4 or [67).

Let 0~R and f~ then f is in fact holomorohic in

I z: I zl <0'I for a number p' >p. now we have

N 11f- z Cnfnrl

n=O P" -*0, as N~CO,

for each p" with p<0"<o' This implies that if n~ ao ° - n:O

is a basis

in ~ and that (3.1a) is valid. The proof of Dart (b) of

Proposition 4 is analogous.

The case R : Do must be considered separately:

PROPOSITION 5: Under the same assumptions as in ProDosition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

we have :

( a ) - -,~: t f n ] °°0 i s a b a s i s i n j ~ . i f and only i f

I ,/i , m

!im sup IPm ~'l m->GO , o

<GO

for each j (I _<j ~K).

(b) Ign~ oo n=O i s a b a s i s i n ~r" i f and o n l y i f L . . .

I/1 !im sup I ~'I m

.

m~co "m,J <09

for each i and j with 0%j ~i<K (we set O=0).

For the Droof one has only to interchange the roles of the

numbers r and s in (3.7b) and (3.16a), (3.16b), (3.16c)

re sD~ ctive ly.

Finally we derive the corresponding result to Proposition 2.

PROPOSITION 6: I fnl Qo is o basis in ~ (O<R<co), if . .. n=O . . . . . ____

282

I/j < R, (4.3) !im sup qj j-~co

whe~re qj =O~m<co~sup IPm,jl <oo for j =1,2, ....

co forms a P r o o f . Condition (4.3) implies that l~nln= 0

basis in ~I~ ' where q=lim sup qjl/J (see Proposition 2 (b)). j*oo

Thbrefore IF nln~=O is a basis in ~ t and I ~ } co - ~19 ~n n=0 is a basis in

~q, in fact I fnln~°=0 constitutes a basis in ~ for any o>_q.

Now choose p such that q <,o <R and use the same argument as in

the proof of Proposition 4 to show that {fn ~n=0 co is a basis in ~R"

For I~nl ~ n=0 one easily obtains an ana3ogous result.

REFERENCES

[I ] M.G.Arsove, On the Behavior of Pincherle basis functions,

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