# Unified Treatment of the Krätzel Transformation for Generalized Functions

Authors: S. K. Q. Al-Omari; A. Kılıçman
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We discuss a generalization of the Krätzel transforms on certain spaces of ultradistributions. We have proved that the Krätzel transform of an ultradifferentiable function is an ultradifferentiable function and satisfies its Parseval's inequality. We also provide a complete reading of the transform constructing two desired spaces of Boehmians. Some other properties of convergence and continuity conditions and its inverse are also discussed in some detail.


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## Body

## 1. Introduction

Krätzel, in [1, 2], introduced a generalization of the Meijer transform by the integral:
(1)(Kvpf)(x)=∫R+Zpv(xy)f(y)dy,x>0,
where
(2)Zpv(xy)=∫R+tv-1e-tp-xy/tdt,p>0(∈N), v∈ℂ. Then this generalization is known as Krätzel transform.Letx be in R+. Denote by 𝒮+, or 𝒮(R+), the space of all complex-valued smooth functions ϕ(t) on R+ such that
(3)supx∈K|𝒟kϕ(x)|<∞,x∈R+, where K runs through compact subsets of R+; see [3]. The strong dual 𝒮+′ of 𝒮+ consists of distributions of compact supports.Later, the authors in [4] have studied the Kvp transformation in a space of distributions of compact support inspired by known kernel method. They, also, have obtained its properties of analyticity and boundedness and have established its inversion theorem. In the sense of classical theory, the Meijer transformation and the Laplace transformation in [5] are presented as special forms of the cited transform for p=1 and p=1, v=±1/2, respectively.It is worth mentioning in this note that a suitable motivation of the cited transform has thoroughly been discussed in [6] by the aid of a Fréchet space of constituted functions of infinitely differentiable functions over (0,∞).This paper is a continuation of the work obtained in [4]. We are concerned with a general study of the transform in the space of ultradistributions and further discuss its extension to Boehmian spaces in some detail. We are employing the adjoint method and method of kernels for our purpose to extend the classical integral transform to generalized functions and hence ultradistributions.


## 2. Ultradistributions

The theory of ultradistributions is one of generalizations of the theory of Schwartz distributions; see [3, 7]. Since then, in the recent past and even earlier, it was extensively studied by many authors such as Roumieu [8, 9], Komatsu [10], Beurling [11], Carmichael et al. [12], Pathak [13, 14], and Al-Omari [15, 16].By an ultradifferentiable function we mean an infinitely smooth function whose derivatives satisfy certain growth conditions as the order of the derivatives increases. Unlike sequences presented in [15, 16], ai,i=0,1,…, wherever it appears, denotes a sequence of positive real numbers. Such omission of constraints may ease the analysis.Letα be a real number but fixed and ℒr be the space of Lebesgue integrable functions on R+. Denote by 𝒮+(ℒr,α,(ai),a) (resp., 𝒮+(ℒr,α,{ai},a)), 1≤r≤∞, the subsets of 𝒮+ of all complex valued infinitely smooth functions on R+ such that, for some constant m1(>0),
(4)supα,x∈K∥𝒟kφ(x)∥ℒr≤m1aαaα
for all a>0 (for some a>0), where K is a compact set traverses R+.The elements of the dual spaces,𝒮+′(ℒr,α,(ai),a)  (𝒮+′(ℒr,α,{ai},a)), are the Beurling-type (Roumieu-type) ultradistributions. It may be noted that 𝒮+(ℒr,α,(ai),a)⊂𝒮+(ℒr,α,{ai},a)⊂𝒮+. Thus, every distribution of compact support is an ultradistribution of Roumieu type and further, and an ultradistribution of Roumieu type is of Beurling-type. Natural topologies on 𝒮+(ℒr,α,(ai),a) (resp., 𝒮+(ℒr,α,{ai},a)) can be generated by the collection of seminorms:
(5)∥φ∥r,a=supα,x∈K∥𝒟kφ(x)∥ℒraαaα,a>0.
A sequence (φn)→φ∈𝒮+(ℒr,α,(ai),a) (resp., 𝒮+(ℒr,α,{ai},a)) if
(6)limn→∞supα,x∈K∥𝒟k(φn(x)-φ(x))∥ℒr=0,
and there is a constant m1>0 independent of n such that
(7)limn→∞supα,x∈K∥𝒟k(φn(x)-φ(x))∥ℒr≤m1aαaα
for all a>0.  𝒮+(ℒr,α,(ai),a)  (𝒮+(ℒr,α,{ai},a))  is dense in 𝒮+, convergence in 𝒮+(ℒr,α,(ai),a)  (𝒮+(ℒr,α,{ai},a)) implies convergence in 𝒮+, and consequently a restriction of any f∈𝒮+′ to 𝒮+(ℒr,α,(ai),a)(𝒮+(ℒr,α,{ai},a)) is in 𝒮+′(ℒr,α,(ai),a)(𝒮+′(ℒr,α,{ai},a)).


## 3. The Krätzel Transform of Tempered Ultradistributions

In this section of this paper we define the Krätzel transform of tempered ultradistributions by using both of kernel and adjoint methods. We restrict our investigation to the case of Beurling type since the other investigation for the Roumieu-type tempered ultradistributions is almost similar.Lemma 1.
Let
ϕ
∈
𝒮
+
(
ℒ
r
,
α
,
(
a
i
)
,
a
)
and then
K
v
p
ϕ
∈
𝒮
+
(
ℒ
r
,
α
,
(
a
i
)
,
a
).Proof.
Letϕ∈𝒮+(ℒr,α,(ai),a) and x be fixed, and then Kvpϕ certainly exists. By differentiation with respect to x we get
(8)𝒟x(Kvpϕ)(x)=∬R+𝒟xtv-1e-tp-xy/tdtϕ(y)dy=∬R+-yttv-1e-tp-xy/tdtϕ(y)dy=∫R+‍-y∫R+‍t(v-1)-1e-tp-xy/tdtϕ(y)dy=∫R+‍-yZpv-1(xy)ϕ(y)dy.
Hence the principle of mathematical induction on the kth derivative gives
(9)𝒟xk(Kvpϕ)(x)=(-1)k∫R+‍Zpv-k(xy)ykϕ(y)dy.
From [4], we deduce that
(10)|𝒟xk(Kvpϕ)(x)|≤α1∫R+|(y)(2v-p)/(2p+2)e-(xy)p/(p+1)|ykϕ(y)dy
for some constant α1. The assumption that ϕ∈𝒮+(ℒr,α,(ai),a) implies that the products under the integral sign, Ψ=ykϕ(y) and (xy)(2v-p)/(2p+2)e-(xy)p/(p+1)Ψ, are also in 𝒮+(ℒr,α,(ai),a). Moreover, ϕ∈𝒮+(ℒr,α,(ai),a) ensures that the integral:
(11)Ϝ(x)=α1∫R+‍|(xy)(2v-p)/(2p+2)e-(xy)p/(p+1)|ykϕ(y)dy
belongs to 𝒮+(ℒr,α,(ai),a). Hence
(12)∥𝒟xkϜ(x)∥ℒr≤m1aαaα
for some constant m1. Therefore, from the above inequality we get
(13)∥𝒟xk(Kvpϕ)(x)∥ℒr≤∥𝒟xkϜ(x)∥ℒr≤m1aαaα,
for certain positive constant m1. This proves the lemma.From Lemma1 we deduce that the Krätzel transform is bounded and closed from 𝒮+(ℒr,α,(ai),a) into itself. Next, we establish the Parseval’s relation for the Krätzel transform.Theorem 2.
Letf and g be absolutely integrable functions over R+ and then
(14)∫R+‍f(x)(Kvpg)(x)dx=∫R+‍(Kvpf)(x)g(x)dx,
where Kvpf and Kvpg are the Krätzel transforms of f and g, respectively.Proof.
It is clear thatKvpf and Kvpg are continuous and bounded on R+. Moreover, the Fubini’s theorem allows us to interchange the order of integration:
(15)∫R+‍f(x)(Kvpg)(x)dx=∫R+‍(∫R+‍f(x)Zpv(xy)dx)g(y)dy.
Equation (15) follows since the Krätzel kernel Zpv(xy) applies for the functions f and g, when the order of integration is interchanged. This completes the proof of the theorem.Now, in consideration of Theorem2, the adjoint method of extending the Krätzel transform can be read as
(16)〈Kvpf,ϕ〉=〈f,Kvpϕ〉,
where f∈𝒮+′(ℒr,α,(ai),a) and ϕ∈𝒮+(ℒr,α,(ai),a).Theorem 3.
Given that
f
∈
𝒮
+
′
(
ℒ
r
,
α
,
(
a
i
)
,
a
)
then
K
v
p
f
∈
𝒮
+
′
(
ℒ
r
,
α
,
(
a
i
)
,
a
).Proof.
Consider a zero convergent sequence(ϕn) in 𝒮+(ℒr,α,(ai),a) then certainly (Kvpϕn) is a zero-convergent sequence in the same space. It follows from (16) that
(17)〈Kvpf,ϕn〉=〈f,Kvpϕn〉⟶0asn⟶∞.
Linearity is obvious. This completes the proof.From the above theorem we deduce that the Krätzel transform of a tempered ultradistribution is a tempered ultradistribution. Moreover, the boundedness property ofKvpf,f∈𝒮+′(ℒr,α,(ai),a) follows from the following theorem.Theorem 4.
Letf∈𝒮+′(ℒr,α,(ai),a) and then Kvpf is bounded.Proof.
See [4, Proposition  2.3].It is interesting to know that the Krätzel transform can be defined in an alternative way, namely, by the kernel method. Letf∈𝒮+′(ℒr,α,(ai),a), and then
(18)(Kvpf)(x)=〈f(y),Zpv(xy)〉.
In fact, (18) is a straightforward consequence of Lemma 1.Theorem 5.
Letf∈𝒮+′(ℒr,α,(ai),a) then Kvpf is infinitely differentiable and
(19)𝒟xk(Kvpf)(x)=〈f(t),𝒟xkZpv(xt)〉
for every k∈N and x∈R+.Proof.
See [4, Proposition  2.2].


## 4. Boehmian Spaces

Boehmians were first constructed as a generalization of regular Mikusinski operators [17]. The minimal structure necessary for the construction of Boehmians consists of the following elements:(i)
a nonempty setA,(ii)
a commutative semigroup(B,*),(iii)
an aperation⊙:A×B→A such that for each x∈A and s1,s2∈B,x⊙(s1*s2)=(x⊙s1)⊙s2,(iv)
a collectionΔ⊂BN such that
(a)
ifx,y∈A,(sn)∈Δ,x⊙sn=y⊙sn for all n, then x=y,(b)
if(sn),(tn)∈Δ, then (sn*tn)∈Δ.Elements ofΔ are called delta sequences. Consider
(20)g={(xn,sn):xn∈A,(sn)∈Δ,xn⊙sm=xm⊙sn,∀m,n∈N}.
If (xn,sn),(yn,tn)∈g, xn⊙tm=ym⊙sn, for all m,n∈N, then we say (xn,sn)~(yn,tn). The relation ~ is an equivalence relation in g. The space of equivalence classes ing is denoted by β. Elements of β are called Boehmians. Between A and β there is a canonical embedding expressed as
(21)x⟶x⊙snsn.
The operation ⊙ can be extended to β×A by
(22)xnsn⊙t=xn⊙tsn.
In β, there are two types of convergence:(δ convergence)  a sequence (hn) in β is said to be δ convergent to h in β, denoted by hn→δh, if there exists a delta sequence (sn) such that (hn⊙sn),(h⊙sn)∈A, for all k,n∈N, and (hn⊙sk)→(h⊙sk) as n→∞, in A, for every k∈N,(Δ convergence) a sequence (hn) in βis said to be Δ convergent to h in β, denoted by hn→Δh, if there exists a (sn)∈Δ such that (hn-h)⊙sn∈A, for all n∈N, and (hn-h)⊙sn→0 as n→∞ in A. For further discussion see [17–21].


## 5. The Ultra-Boehmian Spaceβs+

Denote byD+, or D(R+), the Schwartz space of ℂ∞ functions of bounded support. Let Δ+ be the family of sequences (sn)∈D(R+) such that the following holds:(Δ1)∫R+sn(x)dx=1, for all n∈N,(Δ2)sn(x)≥0, for all n∈N,(Δ3)suppsn⊂(0,εn),εn→0 as n→∞.It is easy to see that each (sn) in Δ+ forms a delta sequence.Letf∈𝒮+′(ℒr,α,(ai),a) and σ∈D(R+) be related by the expression:
(23)(f·σ)υ=f(σ⊛υ),
where
(24)(σ⊛υ)(x)=∫R+σ(t)υ(xt)dt
for every υ∈𝒮+(ℒr,α,(ai),a).Lemma 6.
Let
f
∈
𝒮
+
′
(
ℒ
r
,
α
,
(
a
i
)
,
a
)
and
σ
∈
D
+
and then
f
·
σ
∈
𝒮
+
′
(
ℒ
r
,
α
,
(
a
i
)
,
a
).Proof.
Using the weak topology of𝒮+′(ℒr,α,(ai),a), we write
(25)|(f·σ)υ|=|f(σ⊛υ)|≤C∥σ⊛υ∥r,a,
where υ∈𝒮+(ℒr,α,(ai),a). Hence, to complete the proof, we are merely required to show that σ⊛υ∈𝒮+(ℒr,α,(ai),a).
First, ifσ∈D+ and υ∈𝒮+(ℒr,α,(ai),a), then choosing a compact set K containing the support of σ yields
(26)(σ⊛υ)(x+Δx)-(σ⊛υ)(x)Δx=∫R+‍σ(t)υ((x+Δx)t)-υ(xt)Δxdt
which is dominated by σ(t)|𝒟xυ(x)|. The dominated convergence theorem and the principle of mathematical induction implies
(27)𝒟xk(σ⊛υ)=σ⊛𝒟xkυ.
Finally
(28)∫R+‍|𝒟k(σ⊛υ)(x)|rdx=∫R+‍|(σ⊛𝒟kυ)(x)|rdx≤∫R+‍∫K‍|σ(t)dt𝒟kυ(xt)|rdx≤M∫R+‍|𝒟kυ(xt)|rdx.
Therefore
(29)∥σ⊛υ∥r,a≤d∥υ∥r,a<daαaαforsomeconstantd.
Thus σ⊛υ∈𝒮+(ℒr,α,(ai),a). This completes the proof of the lemma.Lemma 7.
Let
f
1
,
f
2
∈
𝒮
+
′
(
ℒ
r
,
α
,
(
a
i
)
,
a
)
and
σ
∈
D
+
and then
(i)
α
f
·
σ
=
α
(
f
·
σ
), α∈ℂ,
(ii)
(
f
1
+
f
2
)
·
σ
=
f
1
·
σ
1
+
f
2
·
σ
2.Proof of the above Lemma is obvious.Lemma 8.
Let
f
n
,
f
∈
𝒮
+
′
(
ℒ
r
,
α
,
(
a
i
)
,
a
)
and
σ
∈
D
+
and then
(30)
f
n
·
σ
⟶
f
·
σ
.Proof.
Letυ∈𝒮+′(ℒr,α,(ai),a) and then σ⊛υ∈𝒮+′(ℒr,α,(ai),a). Therefore
(31)(fn·σ-f·σ)υ=((fn-f)·σ)υ=(fn-f)(σ⊛υ).
Hence fn·σ→f·σ as n→∞.Lemma 9.
Let
f
∈
𝒮
+
′
(
ℒ
r
,
α
,
(
a
i
)
,
a
)
and
(
s
n
)
∈
Δ
+
then
(32)
f
·
s
n
⟶
f
i
n
𝒮
+
′
(
ℒ
r
,
α
,
(
a
i
)
,
a
)
a
s
n
⟶
∞
.Proof.
Letυ∈𝒮+(ℒr,α,(ai),a) and suppsn⊂(0,εn), n∈N and then
(33)(f·sn)υ=f(sn⊛σ).
It is sufficient to establish that sn⊛υ→υ as n→∞. By using (27) and Δ1 imply that
(34)∫R+‍|𝒟xk(sn⊛υ-υ)(x)|rdx=∫R+|(sn⊛𝒟xkυ-𝒟xkυ)(x)|rdx=∫R+‍|∫0εnsn(t)(𝒟xkυ(xt)-𝒟xkυ(x))dt|rdx.
Hence, the mean value theorem implies
(35)∫R+|𝒟xk(sn⊛υ-υ)(x)|rdx≤∫R+|∫0εnξsn(t)𝒟xk+1υ(xξ)dt|rdx,ξ∈(0,t). Let A1=sups∈K|𝒟xk+1ψ(s)|, where K is certain compact set. The calculations show that
(36)∥sn⊛υ-υ∥r,a≤Fεn⟶0asn⟶∞,
where F is certain constant. Hence Lemma 9. The Boehmian space βs+ is therefore constructed.


## 6.βZ+ and the Krätzel Transform of Ultra-Boehmians

Denote byZ(R+), or Z+, the space of functions which are Krätzel transforms of ultradistributions in 𝒮+′(ℒr,α,(ai),a), and then convergence on Z+ can be defined in such away that En→E* in Z+ if fn→f∈𝒮+′(ℒr,α,(ai),a) as n→∞, where En=Kvpfn and E*=Kvpf. Let E∈Z+ and σ∈D+ and then it is proper to define
(37)(E⊚σ)(u)=∫R+‍E(ut)σ(t)dt
for each u∈R+.Lemma 10.
Letf∈𝒮+′(ℒr,α,(ai),a) and σ∈D+ and then Kvp(f·σ)=Kvpf⊚σ.Proof.
LetK be a compact set containing the support of σ, and then from (18) it follows that
(38)Kvp(f·σ)(u)=〈(f·σ)(y),Zpv(uy)〉=〈f(y),〈σ(t),Zpv((ut)y)〉〉=∫R+〈f(y),Zpv((ut)y)〉σ(t)dt=Kvpf⊚σ.
Hence the lemma follows.Lemma 11.
LetE∈Z+ and σ∈D+ and then E⊚σ∈Z+.Proof.
E
∈
Z
+ implies Kvpf=E, for some f∈𝒮+′(ℒr,α,(ai),a). Hence E⊚σ=Kvpf⊚σ=Kvp(f·σ)∈Z+.The following are lemmas which can be easily proved by the aid of the corresponding lemmas from the previous section. Detailed proof is avoided. First, ifKv-1p is the inverse Krätzel transform of Kvp, and thenLemma 12.
LetE∈Z+ and σ∈D+ and then
(39)Kv-1p(E⊚σ)=Kv-1pE·σ.Lemma 13.
Let
E
1
,
E
2
∈
Z
+
and then for all
σ
1
,
σ
2
∈
D
+ we have(1)
(E1+E2)⊚σ1=E1⊚σ1+E2⊚σ1,(2)
(aE)⊚σ1=a(E⊚σ1).Lemma 14.
LetEn→E and (sn)∈Δ+ and then En⊚sn→E.Lemma 15.
LetEn→E and σ∈D+ and then En⊚σ→E⊚σ.With the previous analysis, the Boehmian spaceβZ+ is constructed. The sum of two Boehmians and multiplication by a scalar in βZ+ is defined in a natural way [fn/ϕn]+[gn/ψn]=[((fn⊚ψn)+(gn⊚ϕn))/(ϕn⊚ψn)] and α[fn/ϕn]=[α(fn/ϕn)],α∈ℂ.The operation⊚ and the differentiation are defined by
(40)[fnϕn]⊚[gnψn]=[fn⊚gnϕn⊚ψn],𝒟k[fnϕn]=[𝒟kfnϕn].With the aid of Lemma10 we define the extended Krätzel transform of a Boehmian [fn/sn]∈βs+ to be a Boehmian in βz+ expressed by the relation:
(41)K→vp[fnsn]=[Kvpfnsn].Lemma 16.
K
→
v
p
:
β
s
+
→
β
Z
+ is well defined and linear mapping.Proof is a straightforward conclusion of definitions.Definition 17.
Let[En/sn]∈βZ+ and then the inverse of K→ is defined as follows:
(42)K→v-1p[Ensn]=[Kv-1pEnsn],for each(sn)∈Δ+.Lemma 18.
K
→
v
p is an isomorphism from βs+ into βZ+.Proof.
AssumeK→vp[fn/sn]=K→vp[gn/tn], then it follows from (41) and the concept of quotients of two sequences Kvpfn⊚tm=Kvpgm⊚sn. Therefore, Lemma 10 implies Kvp(fn·tm)=Kvp(gm·ϕn). Employing properties of Kvp implies fn·tm=gm·sn. Thus, [fn/sn]=[gn/tn]. Next we establish that K→vp is onto. Let [Kvpfn/sn]∈βZ+ be arbitrary and then Kvpfn⊚sm=Kvpfm⊚sn for every m,n∈N. Hence fn,fm∈𝒮+′(ℒr,α,(ai),a) are such that Kvp(fn·sm)=Kvp(fm·sn). Hence the Boehmian [fn/sn]∈βs+ satisfies the equation K→vp[fn/sn]=[Kvpfn/sn].
This completes the proof of the lemma.Lemma 19.
Let[En/sn]∈βZ+, En=Kvpfn, and ϕ∈D+ and then
(43)K→v-1p([Ensn]⊚ϕ)=[En-1sn]·ϕ,K→vp([fnsn]•ϕ)=[Ensn]⊚ϕ.Proof.
It follows from (42) that
(44)K→v-1p([Ensn]⊚ϕ)=K→v-1p([En⊚ϕsn])=[Kv-1p(En⊚ϕ)sn].
Applying Lemma 12 leads to
(45)K→v-1p([Ensn]⊚ϕ)=[Kv-1pEnsn]·ϕ=[En-1sn]·ϕ.
Proof of the second part is similar. This completes the proof of the lemma.Theorem 20.
K
→
v
p
:
β
s
+
→
β
Z
+ and K→v-1p:βZ+→βs+ are continuous with respect to δ and Δ convergences.Proof.
First of all, we show thatK→vp:βs+→βZ+ and K→v-1p:βZ+→βs+ are continuous with respect to δ convergence. Let βn→δβ in βs+ as n→∞ and then we establish that K→vpβn→K→vpβ as n→∞. In view of [10], there are fn,k and fk in 𝒮+′(ℒr,α,(ai),a) such that
(46)βn=[fn,ksk],β=[fksk]
such that fn,k→fk as n→∞ for every k∈N. The continuity condition of the Krätzel transform implies En,k→Ek as n→∞, En,k=Kvpfn,k, and Ek=Kvpfk in the space Z+. Thus, [En,k/sk]→[Ek/sk] as n→∞ in βZ+.
To prove the second part of the lemma, letgn→δg∈βZ+ as n→∞. From [10], we have gn=[En,k/sk] and g=[Ek/sk] for some En,k,Ek∈Z+ where En,k→Ek as n→∞. Hence Kv-1pEn,k→Kv-1pEk in βs+ as n→∞. That is, [En,k-1/sk]→[Ek-1/sk] as n→∞. Using (42) we get K→v-1p[En,k/sk]→K→v-1p[Ek/sk] as n→∞.
Now, we establish continuity ofK→vp and K→v-1p with respect to Δ+ convergence. Let βn→Δβ in βs+as n→∞. Then, we find fn∈𝒮+′(ℒr,α,(ai),a), and (sn)∈Δ+ such that (βn-β)·sn=[(fn·sk)/sk] and fn→0 as n→∞. Employing (41) we get
(47)K→vp((βn-β)·sn)=[Kvp(fn·sk)sk].
Hence, from (41) and Lemma 19 we have K→vp((βn-β)·sn)=[(En⊚sk)/sk]=Ek→0 as n→∞ in Z+. Therefore
(48)K→vp((βn-β)·sn)=(K→vpβn-K→vpβ)⊚sn⟶0asn⟶∞.
Hence,K→vpβn→ΔK→vpβ as n→∞. Finally, let gn→Δg in βZ+ as n→∞ and then we find Ek∈Z+ such that (gn-g)⊚sn=[(Ek⊚sk)/sk] and Ek→0 as n→∞ for some (sn)∈Δ+ and Ek=Kvpfn.
Next, using (42), we obtain K→v-1p((gn-g)⊚sn)=[Kv-1p(Ek⊚sk)/sk]. Lemma 19 implies that
(49)K→v-1p((gn-g)⊚sn)=[fn·sksk]=fn⟶0asn⟶∞in𝒮+′(ℒr,α,(ai),a).
Thus K→v-1p((gn-g)⊚sn)=(K→v-1pgn-K→v-1pg)·sn→0 as n→∞. Hence, we have K→v-1pgn→ΔK→v-1pg as n→∞ in βs+.
This completes the proof of the theorem.



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*Source: 750524-2013-05-13.xml*