A power series with a finite domain of convergence

ItOCZNIKI POLSKIEGO TOW ARZYSTW A MATEMATYCZNE GO Séria I: PEACE MATEMATY CZNE X V (1971)

A N N A L E S SOCÏETATIS МДТНЕМДТ1САЕ POLONAE Series I: COMMENTATIONES MATHEMATICAE X V (1971)

W.

(Warszawa)

A power series with a finite domain of convergence Let X be a complex F-spaee ( = a complete linear metric space, cf. [1]). Consider for a sequence (xn) с: X the power series

where t is a complex scalar. Denote by В thq> domain of convergence of series (1), i.e. the set of all complex numbers t such that series (1) is convergent in X. In some cases the closure В must be a proper or improper disc on the complex plane, i.e. a set {teC: |tf| < r}, where 0 < r < oo. It is so if X is a Banach space, or a locally bounded space (cf. [2]) or more generally a pseudoconvex F-space, i.e. a projective limit of a sequence of locally bounded spaces. S. Bolewicz asked (oral communication) what is the shape of the set В for a power series in a general F-space, in particular, whether В must be connected. In this note we give a negative answer to the former question. We exhibit examples of power series having finite, a priori prescribed domains of convergence. More exactly we prove that in the space S (the space of all Lebesgue measurable functions on the unit interval [0, 1], with identification of two functions equal almost everywhere, and with topology of convergence in the measure) for any given finite subset B c G containing the zero, there exists series (1) convergent exactly on B.

Le m m a 1 .

Let X be an F -space and let there exist a sequence (yn) cz X, n = 0 , 1 , . . . , such that

Let A = (cq ..., aA) be a finite non-void subset of the complex plane G and щ Ф 0. Then there exists in X a sequence (xn) such that series (1) conver­

ges in X if teA и {0} and diverges if tiA и {0}.

OO

(

)

(i) limyn = 0,

(ii) the series yntn diverges for any complex t Ф 0.

П

116 W. Z e l a z k o

Proof. Assume first

(2) |<*г1 i = 1, 2, ..., by

and let /^o ? be complex numbers such that the polynomial

(3) p(t) = +

has as its roots the numbers a15 ..., ak. We define xn as the w-th term in the sequence

А>2Л)? ^i^o) •••» fiicVoi РоУп РхУи •**> PkVit РоУ2> •••?

i.e. writing

(4) n = (fc+l)* + r,

where 0 < r < Jc + 1 , we set xn = pry8. Series (1) may be written now in the form

(б» 2 V ” =

w s r= 0

Thus if teD, i.e. t is a root of polynomial (3), then, in view of (5), the n-th partial sum of series (1) is given by

n r

s n = £ XptP = yst(k+1)s £

p—0 q=0

where s and r are related to n by formula (4). So in view of (i) and (2) we have lim $n = 0 and series (1) converges in X.

Assume now that t il ) . We can rewrite (5) as

= P ( t ) ] ? y sTs,

n s

where T = tk+1 and p{t) is given by formula (3). We have p(t) Ф 0 and, by (ii), the right-hand series diverges, so diverges series (1).

To obtain our lemma in the general case we set a = (max ja^j)-1 and cis = a-1 • a8. Thus |cq| < 1, i = 1 and so we have a sequence xi such that the series converges exactly on {ax, ..., a№ }u {0}. It is evident that if we put xn = anxn the corresponding series (1) is convergent exactly on A и {0}.

L

2. In the space 8 there exists a sequence (yn) satisfying con­

ditions (i) and (ii) of the previous lemma.

Proof. Consider a sequence of partitions of the unit segment onto,

say, 2n disjoint segments of equal length: s\, ..., s2™ , where n is the index

of a partition. Write now the sequence of segments , s?, s i ,

4 , ... and

F i n i t e d o m a i n o f c o n v e r g e n c e 117

denote its n-th term by s(n). Put yn to be the characteristic function of s{n\ i.e.

<Рп(*)

I 1 for

[О for te[0, l ] \ S (w).

Thus there is a sequence of integers in such that

%+i

(в) У <pp(t) = i

P=in+l

on [ 0 ,

]. We put now yn = nncpn(t), it is a sequence of elements of the space 8. We shall show that this sequence satisfies (i) and (ii) of Lemma

.

Ad (i). Since y{s{n))

, where p is the Lebesgue measure on the unit interval, an(pn(t) tends to

его in 8 whatever are the complex coef­

ficients an.

Ad (ii). Let < be a complex scalar different from zero; choose an integer nQ satisfying

(7) я

>1«Г1-

П

Take in > n0 and consider

8in, where 8n = fryptp ■ Thefunc- p

tions yptp, in+ l < p < in+i, have disjoint supports and by (7) we have

\yptp\ > 1 on the support for p > in > n0. Consequently

%+i

*>=*»+1

on [

,

] and the sequence 8n cannot be convergent in 8.

From Lemmas la n d 2 follows our main result:

Th e o r e m.

Let D be a finite subset of the complex plane containing zero. There exists in the space 8 a sequence (xn) such that for the domain of convergence of the power series ]?xntn, taC, equals precisely D.

B em ark. If in construction of Lemma 2 we put cn instead of nn7 then the closure of the domain of convergence of a suitable power series consists of a disc of radius c~l and of a finite number of isolated points. It is not clear what sets can be domains of convergence for power series with coefficients from an F-space (they must be Gd-sets) and for which spaces there are series with domains of convergence different from discs.

It would be interesting to give suitable characterizations.

References

[1] S. B a n a c h , Théorie des opérations linéaires, Warszawa 1932.

[2] S. R o le w ic z , On a certain class of topological linear spaces, Bull. Acad. Polon.

Sci. (1957).

IN ST IT U T E OF MATHEMATICS PO L ISH ACADEMY OF SCIENCES