Existence of basis in some Whitney function spaces

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Existence of Basis in Some Whitney Function Spaces Mustafa Kesir and Mefharet Kocatepe

Abstract

It has been shown that some spaces of Whitney functions on Cantor-type sets have bases. 2000 Mathematics Subject Classification: 46A45, 46EIO, 46A35, 46A04

Key words: Whitney Function space, basis

I Introduction

In this paper we will be interested in the existence of basis in certain nuclear Frechet spaces. By a basis we mean a Schauder basis, and whether every nuclear Frechet space has a basis had been a long standing open problem. After Mityagin and Zobin [?] answered this question in the negative, the related open problem now is the following: are there nuclear Frechet function spaces without basis? It is believed that if there is such a space one of the candidates is the space of Whitney functions E(K). In this article we consider two cases of E(K) and show that they have basis. However our results are existence results. We do not have any results about the form of the basis. But in [?] Goncharov has given the actual construction of a basis in the space, where the Cantor-type set is such that Nn

=

N for all n.

II Preliminaries

E(K) and Extension Property

Let E(K) denote the space of Whitney functions on a perfect compact set K

c

lR with the topology defined by the norms

(2)

where

Iflq

=

sup{lf(j)(x)1 : x E K,j

S

q} and R~f(x)

=

f(x) - T3f(x)

is the Taylor remainder. Each function

f

E E(K) is extendable to a COO-function on the line. These spaces are Frechet and quotients of s, the space of rapidly decreasing sequences, hence they are also nuclear. If there exists a linear continuous extension operator L : E(K) -+ coo(lR), then we say

that the compact set K has the extension property. In [7) Tidten applied Vogt's characterization for the splitting of exact sequences of Frechet spaces and showed that the extension property of K and the dominated norm property (property DN) of the space E(K) are equivalent.

A Fn\chet space X with a fundamental system of seminorrns

(II . Ilq)

is said to have the property (DN) ([7)) if

C

3piiq3r,C> 0:

1! . l!q::;tll·I!P+TI! ·I!"

t>O.

Here p, q, rENo

=

{O, 1,2, . . . }.

Cantor Type Sets. We consider the following generalization of the classical Cantor set. Let 10 = [0,1), (Nn)'(' be a sequence of positive integers with Nn ~ 2, and (in);;" be a sequence of

positive real numbers such that

lo

=

1, Nn · In

<

In-l for all n ~ 1.

At the nth step we delete Nl . N2 · ·· Nn-1 . Nn open intervals of length hn

=

In-fv:N,(ln

from each interval in such a way that all the remaining closed intervals have length In . We call the compact set obtained at nth step In. Then we set

00 K

=

K ((In), (Nn))

=

n

In.

n=l

The resulting compact set is called a Cantor type set. Let an be defined by

h

=

l/eOlI and In

=

_l~~l

for n ~ 2. Then an

>

1 for n ~ 2 and In

=

e-OI, ... OI

n

for all n ~ 1.

We will consider mainly the following special cases for Kj Nn

=

N for all n for some fixed N ~ 2 in Nand Nn / 00 as n -+ 00. Then corresponding compact sets will be denoted by KN and Koo respectively.

The following theorem has been proved in [7). Theorem 1.

a) If lim sup an

<

N then the space E(KN) has the property (DN) .

b) The space E(Koo) has property (DN) if and only if there is a constant M such that In ~

h,:!

for all n .

Diametral Dimension of E(K). Approximative and diametral dimensions, introduced by Kol-mogorov [7), Pelczyllski [7) and Bessaga, Pelczynski and Rolewicz [7), were the first linear topolog-ical invariants applicable to isomorphic classification of nonnormed Frechet spaces. We follow the notation of [7).

Let X be a Frechet space with a fundamental system of neighborhoods (Uq), let

(3)

denote the n-th Kolmogorov diameter of Uq with respect to Up. Then

We will consider the counting function corresponding to the diametral dimension

(3(t)

=

(3(Up, Uq, t)

=

min{dimL: t· Uq C Up

+

L}, t> O.

Clearly, the diametral dimension can be characterized in terms of (3 in the following way. Proposition. h'n) E r(X) if and only if

The following two theorems are of crucial importance in this paper. The first one is due to Ar-sian, Goncharov and Kocatepe [?] and it allows us to calculate the counting function (3 for the neighborhoods in the space X

=

E(K) with K

=

K«ln), (Nn)).

Theorem 2. Let X

=

E(K) with K

=

K«ln}, (Nn)), let p and q, p

<

q be fixed natural numbers. If t ~ ~ l~-q, then (3(Up , Uq, t) ~ (q+1)N1 ... Nn . If t ~ 5(q - p)! l~-q, then (3(Up , Uq, t) ~ Nl ... Nn . The second theorem is due to Aytuna, Krone and Terzioglu and it gives a sufficient condition for the existence of a basis. First we recall the definition of infinite type power series space Aco({3). Given a sequence {3

=

({3n) of real numbers such that 0

<

(3n / 00, Aoo(3) is defined as

00

Aoo({3)

=

{x

=

(xn) : Ilxllk

=

L

IXnlek}3n

<

00, k

=

0,1,2, ... }.

n=l

We note that it is well-known that

f(Aoo({3))

=

{h'n) : :3k:3C

>

0: h'nl

~

Cek}3n, all

n}.

Theorem 3. Let X be a Frechet space with property (DN) and assume that X is isomorphic to a quotient space of s. Also assume f(X)

=

f(Aoo({3)) for some {3

=

((3n) where Aoo({3) is stable,

i.e. supn ~:

<

00. Then the space X has a basis.

The proof follows immediately from Lemma 2.3 and Corollary 1.5 in [?].

III Main Result

We note that the second part of the following theorem gives a partial answer to the question in [?].

Theorem 4. (a) If limsuPn-+co Qn

<

N, then the space E(KN) has a basis. (b) If E(Koo) has property (DN) and limsupn .... coQn

<

00, then the space E(Koo) has a basis.

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Proof. The proofs of the two parts are similar. So we shall prove the first part in detail and indicate the changes in the second part.

Define A(n)

=

al ... an and extend A to the interval (n, n

+

1) in such a way such that A(x) is

strictly increasing and continuous on the interval [1,00). Let S(x) = x (in the second part S(x)

will be different). Let p

<

q be fixed. Then

1 1 1 1 1

o

< -

< ... < -- < -

< ...

and - -+ 00 as n -+ 00

- lrp - lrp - - l~::::~ - l~-P - l~-P

Since In+l/1n

<

l/N::; 1/2, we have In+m/ln

<

1/2m. So I _{im 1m sup}I· _-1-In+m

₌

_o.

m-oo n-+oo n

We fix m such that limsuPn-->oo(ln+m/ln)

<

l/(lO(q - p)!) and find no such that for all n ;::: no we have In-J/1n-l-m ::; 1/(5(q - p)!). Then

1 1 1 1 1

5(q - p)! - -_lq-P

< -

_{- lq-P}and -_IP-q

< - - -

_{- 5 lq-p' -}n> no. n-l-m n-l n n+m

Let t

>

0 be large enough (say t ;::: IQ:P =: to). Find a unique n ;::: no such that no-l Then by (??) 1 1 < t < -l~::::~ - l~-q 1 1 1 5(q_p)I_- <t< _{. lq-P} _- _{- 5 lq-P} -n-l-m n+m

On the other hand, (??) holds if and only if

which is equivalent to

A-l(~) <n::;A-l(~)+l

_q-p _q-p Now by Theorem 2, (??) implies

NS(n-l-m) ::; f3 (Up, Uq, t) ::; (q

+

1) NS(n+m).

This last inequality combined with (??) gives

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(2)

(3)

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NS(A-l(~)-I-m) ::;f3(Up,Uq,t)::;(q+1).NS(A-l(~)+Hm)

fort;::: to. (6)

Define f3

=

(f3n) by f3n

=

A (S-1 U~J:r)). Now we show that r(£(KN))

=

r(Aoo(f3)). For every n,

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Let now (--yn) E

r

(E(KN )). Then

Vp :Jq VC :Jn1 "In ~ n1: (3 (Up, Uq,

Chn

I) :S n.

Assume n ~ n1 and

Chn\

~ to. Then the left hand inequality in (??) implies

i..From k :S 8-1 _{U~ ~)}

<

_k

+

_{1, it follows that}

( -1 (Inn) )

a1 ... ak+l+m ::; A 8 In N

+

1

+

m

<

a1 ... ak+2+m· Thus;

A (8-1

(tiN)

+

₁

+

m)

1

<

(3n

<

ak+l ... ak+2+m·

Since (an) is bounded, the sequences

(A

(8-

1 _U~~

+

₁

+

m)))

_{and ((3n)}_{are equivalent. Thus there} exist M and D

>

0 such that hn\ :S DeMf3n for all n, that is (--yn) E r(Aoo((3)).

Conversely assume (--yn) E r(Aoo((3)). Then there exists M

>

0 such that hn\ ::; eMf3n for large n.

We show that (--yn) E r(E(KN )), i.e.,

Vp:Jq VC: (3(Up , Uq, C\'I'nl) :S n, large n.

Given n, as before let k

=

k(n) be such that k ::; 8-1 _U~~)

<

_k

+

_1._{We find a such that} lim sup an

<

a

<

N.

n

Then an ::; a for large n. Since limq---+oo ~

=

+00, we have an integer qo such that

(q+l) InN

2(M

+

1)am+3 _:S _qIn,,' q

~

qO·

(q

+

l)InN

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(8) Given p, choose q such that q ~ 2p and q ~ go. Then q/2 :S q - p and (??) holds. Let C> 0 be

Ina In(q+l)

given. Since (q

+

l)InN

=

a InN , (??) is equivalent to

3 In(o+l) q

(M

+

l)am_{+ +}InN :S

2.

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Choose an integer J-L such that Nil :S q

+

1

<

NIl+l. Then J-L ::; Inl~tP. So (??) implies

and therefore

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which implies (M

+

I)Ql ... Qk+l ::; (q - p)Ql ... Qk-p.-m-2 that is (M

+

I)A(k

+

1) ::; (q - p)A(k - J-t - m - 2). Since lnl~tl)

<

J-t + 1, we have k-J-t-m-2=k-(J-t+l)-I-m::;S -1 ( In - n ) - In(q + 1) -1-m. InN InN Thus (M+l)A(S-1

G:;)) ::;

(M+l)A(k+l) ::;

(q-P)A(8-1G:~)_lnl~;I-I-m)

Also C ::; ef3n for large n, so CI'Ynl ::; e(M+l),Bn from which it follows that

Chnl ::; e(q-P) ·A(S-l(~ )_lnl~tlLl-m)

A-I (In(Chnl)) + 1 + m

<

8-1 (lnn) _ In(q + 1) .

q-p - InN InN

Since 8(x - a) ::; 8(x) - a for all large x, we have

8 (8-1 (Inn) _ In(q + 1))

<

Inn _In(q+ 1)

InN InN - InN InN

for all large n. So

8 (A

-l (In(Chnl)) )

<

Inn _In(q+ 1)

+1+m -1 N IN' q-p n n which implies S(A-l(~) +l+m) l!!.!L1n(q+l) n N q-p

<

NlnN InN

= --.

- q+ 1 (10)

So by (??) we get

f3

(Up, Uqj CI'Ynl) ::; n for large n, which means by the Proposition that bn) E

r(E(KN)).

Next we show that

(f32n/f3n)

is bounded. Since S-I(1 + x) ::; 1 + S-I(x) for all large x, with

k

=

k(n),

8-1 (In(2n))::; S-1 (1+ Inn)::; 1+8-1 (Inn)

<

1+(k+l).

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Since (O:n) is bounded, (f32n/f3n) is also bounded. By Theorem 1, limsupO:n

<

N implies £(KN) has (DN). So by Theorem 3 the space £(KN) has a basis.

Next we consider the case of £(Koo) and indicate the places where the proof differs from the previous one. In this case clearly limn--->oo In+dln

=

0, so we have m

=

1.

Let N := min{Ni : Ni

>

limsuPn--->ooO:n} and let Tn be defined by Nn = NTn. Then Tn /00, and N1· N2··· Nn = N T,+T2+·+Tn. Then we define Sen) = T1

+

T2

+ ... +

Tn, n EN. We have

Sen) /00, and S(k

+

1) - S(k) = Tk+l /00.

We extend S to (n ,n

+

1) in such a way that Sex) is strictly increasing and continuous on the

interval [1,00) and

Sex

+

1) - Sex) /00 as x / 00.

Then for a constant c, Sex - c) :::; Sex) - c and S-1(1

+

x) :::; 1

+

S-1(X) both hold for all large

x. N, as chosen above, satisfies limsuPn--->oo O:n

<

N , thus (??) can also be satisfied. Then the previous proof can be repeated.

D

References

[1] B. Arslan, A. P. Goncharov, M. Kocatepe, Spaces of Whitney functions on Cantor-type sets,

Canadian J. Math. 54 (2002), 225-238.

[2] A. Aytuna, J. Krone, T. Terzioglu, Complemented infinite type power series spaces of nuclear Frechet spaces, Math. Ann. 283(1989), 193-202.

[3] C. Bessaga, A. Pelczynski, S. Rolewicz, On diametral approximative dimension and, linear homogeneity of F-spaces, Bull. Acad. Pol. Sci. 9 (1961), 677-683.

[4] A. Goncharov, Bases in the spaces of COO-functions on Cantor-type sets, to appear in

Con-structive Approximation.

[5] A.N. Kolmogorov, On the linear dimension of topological vector spaces, DAN USSR. 120 (1958), 239-341. (in Russian).

[6] B.S. Mitiagin, Approximate dimension and bases in nuclear spaces, Russ. Math. SU1'Veys 16

(4) (1961) , 59-127 (English translation).

[7] B. S. Mitjagin, N. M. Zobin, Contre-example

a

l'existence d'une base dans un espace de Frechet nucleaire, C. R. Acad. Sci. Paris Ser. A 279 (1974), 325-327.

[8] A. Pelczynski, On the approximation of S-spaces by finite dimensional spaces, Bull. Acad.

Polon. Sci. 5 (1957) , 879-881.

[9]

M. Tidten, Fortsetzungen von Coo-Funktionen, welche auf einer abgeschlossenen Menge in lR definiert sind, manuscripta math. 27 (1979), 291-312.

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Mustafa Ke~ir

Bilkent University

Department of Mathematics 06800 Bilkent, Ankara, Thrkey Current address: Northeastern University Department of Mathematics 360 Huntington Avenue, Boston, MA 02115, USA e-mail: kesir.m@neu.edu Mefharet Kocatepe Bilkent University Department of Mathematics 06800 Bilkent, Ankara, Thrkey e-mail: kocatepe@fen.bilkent.edu.tr