THE EXCHANGE PROPERTY
DRAGU ATANASIU AND PIOTR MIKUSI ´NSKI Received 19 June 2005
A simplified construction of tempered Boehmians is presented. The new construction shows that considering delta sequences and convergence arguments is not essential.
1. Introduction
Since Boehmians were introduced, extensions of the Fourier transform to spaces of Boehmians attracted a lot of attention (see [2,3,4,5,6,7, 8,9]). In some cases, the range of the extended Fourier transform is a space of functions. In other constructions, the range is a space of distributions or a space of Boehmians.
In this paper, we would like to consider the space of tempered Boehmians presented in [8]. In this case, the range of the Fourier transform is the space of all distributionsᏰ. This work is motivated by [1].
First we recall briefly the construction of the space of tempered Boehmians. A con-tinuous function f :RN→Cis calledslowly increasingif there is a polynomialponRN such that|f(x)| ≤p(x) for allx∈RN. The space of slowly increasing functions will be denoted byᐃ(RN) or simplyᐃ.
An infinitely differentiable function f :RN→Cis calledrapidly decreasingif sup
|α|≤m sup x∈RN
1 +x21+···+x2N m
Dαf(x)<∞ (1.1)
for every nonnegative integerm, wherex=(x1,. . .,xN),α=(α1,. . .,αN),αn’s are nonneg-ative integers,|α| =α1+···+αN, and
Dα=∂|α|
∂xα=
∂|α| ∂xα1
1 ···∂xNαN
. (1.2)
The space of rapidly decreasing functions is denoted by(RN) or simply. Iff ∈ᐃandϕ∈, then the convolution
f ∗ϕ(x)=
RN f(y)ϕ(x−y)d y (1.3)
is well defined andf ∗ϕ∈ᐃ.
Copyright©2005 Hindawi Publishing Corporation
A sequence (ϕn)∈Nis called adelta sequenceif it satisfies the following conditions: (a)RNϕn(x)dx=1 for alln∈N,
(b)RN|ϕn(x)|dx≤Cfor some constantCand alln∈N, (c) limn→∞x≥εxk|ϕn(x)|dx=0 for everyk∈Nandε >0.
The space oftempered BoehmiansᏮis defined as the space of equivalence classes of pairs of sequences (fn,ϕn), where fn∈ᐃand (ϕn) is a delta sequence as defined above, satisfying
fm∗ϕn= fn∗ϕm ∀m,n∈N, (1.4)
with respect to the equivalence relation defined by
fn,ϕn
∼gn,γn
iffm∗γn=gn∗ϕm,∀m,n∈N. (1.5)
It is shown in [8] that the Fourier transform can be defined for tempered Boehmians and that the range is exactly the space of all distributionsᏰ. Thus, the space of tempered Boehmians can be identified with the space of ultradistributionsᐆ(see, e.g., [10]). In the construction, the particular choice of delta sequences and the fact that the Fourier transform of a delta sequence converges to 1 uniformly on compact subsets ofRNseem to be essential. In this paper, we show that this is not the case. In fact, we give an equivalent construction where convergence plays no role. This approach indicates that the results of [8] follow from a more general principle.
In what follows, we will denote bythe space of tempered distributions, that is, the space of continuous linear functionals on. If f ∈andϕ∈, then the convolution f ∗ϕis defined as (f ∗ϕ)(x)= f(ϕx), whereϕx(z)=ϕ(x−z). It can be shown that, if
f ∈andϕ∈, then f∗ϕ∈ᐃ. The Fourier transform of a tempered distribution f, denoted by f, is the functional ondefined by f(ϕ)= f(ϕ), whereϕis the Fourier transform ofϕ.
2. The exchange property
For a family{ϕj}j∈J= {ϕj}J, whereJis an index set andϕj∈for allj∈J, we define
Mϕj J
=x∈RN:ϕ
j(x)=0∀j∈J . (2.1)
A family of pairs{(fj,ϕj)}J, where fj∈andϕj∈, is said to have theexchange propertyif
fj∗ϕk= fk∗ϕj ∀j,k∈J. (2.2)
Theorem2.1. If{(fj,ϕj)}J has the exchange property andΩ=M({ϕj}J)c (the comple-ment ofM({ϕj}J)inRN), then there exists a uniqueF∈Ᏸ(Ω)such that
fj=Fϕj ∀j∈J. (2.3)
that this definition ofFis independent ofj. Suppose, for someε >0, we have|ϕj(x)|> ε for allx∈Uand|ϕk(x)|> εfor allx∈V. Then, since fj∗ϕk= fk∗ϕj, we have fjϕk=
fkϕjand
fj ϕj =
fk
ϕk (2.4)
onU∩V. Clearly,Fis unique.
We will denote byᏭthe collection of all families of pairs{(fj,ϕj)}J, whereJ is an index set,fj∈andϕj∈for allj∈J, satisfying the exchange property and such that
M({ϕj}J)= ∅.
Note that in the definition ofᏭthe index set is not fixed. If f ∈is arbitrary and ω(x)=e−x·x, then{(f,ω)} ∈Ꮽ. In this case the index set has only one element.
If (ϕj) is a delta sequence, then obviouslyM({ϕj}N)= ∅. However, it is possible that
M({ϕj}J)= ∅and{ϕj}does not contain any subsequence which is a delta sequence. Consider, for example, a sequence{ϕj}N such that{ϕj}Nis a partition of unity. More generally, let{Uj}J be an open covering ofRNand let{ϕj}J be such that|ϕ
j(x)|>0 for
x∈Uj. A family{ϕj}J such thatM({ϕj}J)= ∅will be calledtotal.
Lemma2.2. If{ϕj}J and{γk}Kare total, then{ϕj∗γk}J×Kis total.
Theorem2.3. {(fj,ϕj)}J∈Ꮽif and only if there exists a uniqueF∈Ᏸ(RN)such that
fj=ϕjFfor allj∈J.
Proof. We only need to show that existence of such anF∈Ᏸ(RN) implies the exchange property. Indeed, for any j,k∈Jwe have
fjϕk=Fϕjϕk=Fϕkϕj= fkϕj. (2.5) Definition 2.4. If{(fj,ϕj)}J∈Ꮽ, then the uniqueF∈Ᏸ(RN) such that fj=ϕjFfor all
j∈Jwill be denoted byᏲ({(fj,ϕj)}J) and called theFourier transformof{(fj,ϕj)}J.
Theorem2.5. For everyF∈Ᏸ(RN)there exists{(f
j,ϕj)}J ∈Ꮽsuch thatF=Ᏺ({(fj,
ϕj)}J).
Proof. Let{ϕj}Nbe a total sequence such thatϕj∈Ᏸ(RN) for all j∈N, whereᏰ(RN) denotes the space of smooth functions with compact support. Then, for every j∈N, there is an fj∈such that fj=ϕjF. Clearly,{(fj,ϕj)}N∈ᏭandF=Ᏺ({(fj,ϕj)}N).
Let{(fj,ϕj)}J,{(gk,γk)}K∈Ꮽ. If fj∗γk=gk∗ϕj for all j∈J andk∈K, then we write{(fj,ϕj)}J ∼ {(gk,γk)}K. This relation is clearly symmetric and reflexive. We will show that it is also transitive.
Let{(fj,ϕj)}J,{(gk,γk)}K,{(hl,ψl)}L∈Ꮽ. If{(fj,ϕj)}J∼ {(gk,γk)}Kand{(gk,γk)}K∼ {(hl,ψl)}L, then
for allj∈J,k∈K,l∈L. Hence
fj∗γk∗ψl=gk∗ϕj∗ψl, gk∗ψl∗ϕj=hl∗γk∗ϕj (2.7)
for allj∈J,k∈K,l∈L. Since∗is commutative, we have
fj∗ψl∗γk=hl∗ϕj∗γk (2.8)
for allj∈J,k∈K,l∈L. Now fix j∈J andl∈L. SinceM({γk}K)= ∅and (2.8) holds for everyk∈K, we conclude that fj∗ψl=hl∗ϕj for all j∈J andl∈L, which means that{(fj,ϕj)}J∼ {(hl,ψl)}L.
Note that
fj,ϕj J∼
fj∗ψk,ϕj∗ψk J×K (2.9)
for any total family{ψk}K.
Theorem2.6. Let{(fj,ϕj)}J,{(gk,γk)}K∈Ꮽ. Then
fj,ϕj J∼gk,γk K iff Ᏺfj,ϕj J
=Ᏺgk,γk K
. (2.10)
Proof. LetF=Ᏺ({(fj,ϕj)}J) andG=Ᏺ({(gk,γk)}K). If{(fj,ϕj)}J∼ {(gk,γk)}K, then
Fϕjγk=fjγk=gkϕj=Gγkϕj (2.11)
for allj∈Jandk∈K. HenceF=G, byLemma 2.2. Now assumeF=G. Then
fjγk=Fϕjγk=Gγkϕj=gkϕj (2.12)
for allj∈Jandk∈K. Hence{(fj,ϕj)}J∼ {(gk,γk)}K. Now we defineᏮ=Ꮽ/∼, the space of equivalence classes. In view of Theorems2.5
and2.6, the Fourier transform is a bijection fromᏮtoᏰ. Consequently,Ꮾcan be iden-tified with the space of ultradistributionsᐆ. We will show that, with a properly defined convergence inᏮ, the spaces are isomorphic.
Note that the spacecan be identified with a subspace ofᏮvia f →[{(f∗ω,ω)}], whereω(x)=e−x·x.
Theorem2.7. There exists a delta sequence(ϕn)such that for everyT∈Ꮾ,T=[{(fn,ϕn)}N] for some fn∈ᐃ.
Proof. Let (ψn) be a delta sequence such thatψn∈Ᏸ. Then, for anyT∈Ꮾ, we have
3. Algebraic properties and convergence
Ꮾbecomes a vector space with the operations defined as follows:
λfj,ϕj J
=λ fj,ϕj J
, λ∈C,
fj,ϕj J
+gk,ψk K
=fj∗ψk+gk∗ϕj,ϕj∗ψk J×K
. (3.1)
If [{(fj,ϕj)}J], [{(gk,ψk)}K]∈Ꮾandgk∈for allk∈K, then we can define
fj,ϕj J
∗gk,ψk K
=fj∗gk,ϕj∗ψk J×K
. (3.2)
It is easy to check that these operations are well defined. Note that, in view of
Theorem 2.7, the definition of addition can be simplified.
Definition 3.1. LetT0,T1,T2,. . .∈Ꮾ. It is said that the sequence (Tn) is convergent toT0
and is written asTn→T0if there exists a total family{ϕj}J such that
(a) there exist tempered distributions fj,n, where j∈J andn∈N, such thatTn= [{fj,n,ϕj}J] for alln=0, 1, 2,. . .,
(b) fj,n→fj,0inasn→ ∞for everyj∈J.
Theorem3.2. The Fourier transform is an isomorphism fromᏮtoᏰ.
Proof. Note that, sinceTn→T0inᏮif and only ifTn−T0→0, it suffices to prove
conti-nuity at 0.
AssumeTn→0 inᏮ. Then there exist tempered distributions fj,n, where j∈J and
n∈N, such thatTn=[{(fj,n,ϕj)}J] for alln=1, 2,. . .and fj,n→0 inasn→ ∞for everyj∈J. Ifψ∈Ᏸ, then there arej1,. . .,jksuch that suppψ⊂
k
m=1suppϕjm. Then
lim
n→∞Tnψ=nlim→∞ k
m=1 Tnϕjm
ϕjmψ
k
m=1ϕjm 2
= k
m=1
lim n→∞fjm,n
ϕj
mψ
k
m=1ϕjm 2 =0,
(3.3)
since limn→∞fj,n=0 for everyj∈J, by continuity of the Fourier transform in. This proves continuity ofᏲ:Ꮾ→Ᏸ, because limn→∞Tnψ=0 infor everyψ∈Ᏸimplies limn→∞Tn=0 inᏰ.
Now assume limn→∞Tn=0 inᏰ. ByTheorem 2.7, there exists a delta sequence (ϕj),
j∈N, such that for everyn∈Nwe haveTn=[{(fj,n,ϕj)}N] for somefj,n∈ᐃ. Let (ψk),
k∈N, be a delta sequence such thatψk∈Ᏸfor everyk∈N. Then limn→∞Tnϕjψk=0 in for every j,k∈N. SinceT
nϕj= fj,nfor every j,k∈N, we have limn→∞fj,nψk=0 in , which implies limn→∞fj,n∗ψk=0 in. But
Tn=
fj,n,ϕj J
=fj,n∗ψk,ϕj∗ψk J×K
(3.4)
Acknowledgment
The authors would like to thank Dennis Nemzer for helpful comments.
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Dragu Atanasiu: School of Engineering, Bor˚as University, Bor˚as, SE 501 90, Sweden
E-mail address:dragu.atanasiu@hb.se
Piotr Mikusi ´nski: Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364, USA
