# On the Generalized Krätzel Transform and Its Extension to Bohemian Spaces

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

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## Abstract

We investigate the Krätzel transform on certain class of generalized functions. We propose operations that lead to the construction of desired spaces of generalized functions. The Krätzel transform is extended and some of its properties are obtained.


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## Body

## 1. Introduction

In recent years, integral transforms of Bohemian have comprised an active area of research. Several integral transforms are extended to various spaces of Bohemian, especially, that permit a factorization property of Fourier convolution type.On the other hand, several integral transforms that have not permitted a factorization property of Fourier convolution type are also extended to various spaces of Bohemian. In the sequence of these integral transforms, the Krätzel transform [1]
(1)(𝒦vρf)(x)=∫ℝ+𝒵ρv(xy)f(y)dy,
where
(2)𝒵ρv(xy)=∫ℝ+tv-1e-tρ-xy/tdt,ρ>0(∈ℕ), v∈ℂ, x>0 is extended to certain space of ultra-Bohemian, denoted by 𝒮+(lr,α,(αi),a) and 𝒮+(lr,α,{αi},a), 1≤r≤∞, respectively.Another form of the Krätzel transform was initially introduced in [2], and defined as a generalization of the Laplace transform in [3] as
(3)(lv(ρ)f)(x)=∫ℝ+λv(ρ)(xy)f(y)dy,
where
(4)λv(ρ)(x)=(2π)((ρ-1)/2)ρ(1/2)T(v+1-(1/ρ))×(xρ)ρv∫1∞(tρ-1)v-(1/ρ)e-xtdt,
for ρ∈ℕ and Rev>-1+(1/ρ),x>0. The lv(ρ) transform was extended to generalized functions in [3] and to distributions in [4]. By lρ(X), we denote the space of equivalence classes of measurable functions f:X→ℝ such that
(5)∫X|f|ρdμ<∞,
where two measurable functions are equivalent when they are equal to μ a.e.Forlρ(X), the lρ-norm is defined as
(6)∥f∥lρ=(∫X|f|ρdμ)1/ρ.
The Banach space lρ,μ of Lebesgue measurable functions is defined by [3, page 446]
(7)∥f∥ρ,μ=∥x-μf∥ρ<∞.
Due to [3, Proposition 2.1], we state the following theorem.Theorem 1.
Let1≤ρ≤∞, μ,v∈ℂ, ρ>0, (1/ρ)+(1/ρ′)=1 and Reμ>-(1/ρ′)-min{0,ρRev} and then lv(ρ) is a continuous linear mapping from lρ,μ into lρ,2/(ρ-μ-1).Note that we always assume that the hypothesis of Theorem1 is satisfied. By ⋎ we denote the Mellin-type convolution product of first order defined by [5]
(8)(f⋎φ)(y)=∫ℝ+f(yt-1)t-1φ(t)dt.
By 𝒟(ℝ+), or simply 𝒟, denote the Schwartz space of test functions of compact support defined on ℝ+. Then we have the following definition.Definition 2.
Letf,g∈lρ,μ. For f and g, we define the operation ⊗ given by
(9)(f⊗g)(x)=∫ℝ+f(xt)g(t)dt.Then we easily see that the operations in (8) and (9) are very basic for the next construction of the desired Bohemian spaces.Theorem 3.
Letf∈lρ,μ and φ∈𝒟, then lv(ρ)(f⋎φ)(x)=((lv(ρ)f)⊗φ)(x).Proof.
Letx>0. For f∈lρ,μ and φ∈𝒟 we, by (3) and (8), get that
(10)lv(ρ)(f⋎φ)(x)=∫ℝ+(∫ℝ+f(yt-1)t-1φ(t)dt)λv(ρ)(xy)dy=∫ℝ+(∫ℝ+f(yt-1)t-1φ(t)dt).
A change of variables y=tz in (10) implies
(11)lv(ρ)(f⋎φ)(x)=∫ℝ+(∫ℝ+f(z)λv(ρ)(xtz)dz)φ(t)dt=∫ℝ+(lv(ρ)f)(xt)φ(t)dt.
Hence, by (9), we have
(12)lv(ρ)(f⋎φ)(x)=((lv(ρ)f)⊗φ)(x).
This completes the proof of theorem.


## 2. General Bohemian

The structure necessary for the construction of Bohemian consists of the following:(1)
a groupX and a commutative semigroup (Y,*);(2)
a operation⊙:X×Y→X such that x⊙(υ1*υ2)=(x⊙υ1)⊙υ2, for all x∈X and υ1,υ2,∈Y;(3)
a collectionΔ⊂Yℕ such that
(i)
for all(υn),(σn)∈Δ, we have (υn*σn)∈Δ,(ii)
Ifx⊙υn=y⊙υn, then x=y where x,y∈X,(υn)∈Δ, and n∈ℕ.Then the setΔ that satisfies (i) and (ii) is called the set of all delta sequences.LetA={(xn,υn):xn∈X,(υn)∈Δ,xn⊙υm=xm⊙υn,forallm,n∈ℕ}. Then we say (xn,υn)~(yn,σn) if there are (xn,υn),(yn,σn)∈A, such that xn⊙σm=ym⊙υn, for all m,n∈ℕ. The relation ~ is an equivalence relation in A. The space of equivalence classes in A is called the space of Bohemian and denoted by 𝔹ℍ. Each element of 𝔹ℍ is called Bohemian. Then the convergence in 𝔹ℍ is defined as follows.(1)
(hn)∈𝔹ℍ is said to be δ-convergent to h∈𝔹ℍ, denoted by hn→δh as n→∞, if ∃ a delta sequence (υn) such that (hn⊙υn),(h⊙υn)∈X, and (hn⊙υk)→(h⊙υk)∈X as n→∞, for all k,n∈ℕ.(2)
(hn)∈𝔹ℍ is said to be Δ-convergent to h∈𝔹ℍ, denoted by hn→Δh as n→∞, if ∃ a (υn)∈Δ such that (hn-h)⊙υn∈X, for all n∈ℕ, and (hn-h)⊙υn→0∈X as n→∞.The following theorem is equivalent to the statement of δ-convergence.Theorem 4.
h
n
→
δ
h
∈
𝔹
ℍ as n→∞ if and only if there are fn,k,fk∈X and υk∈Δ such that hn=[fn,k/υk],h=[fk/υk] and, for all k∈ℕ, fn,k→fk∈X as n→∞. See [6–17] for more details.


## 3. The Spaces𝔹ℍ1(lρ,μ) and 𝔹ℍ2(lρ,2/(ρ-μ-1))

In this section we construct the space𝔹ℍ(lρ,μ,(𝒟,⋎),⋎,Δ)(or𝔹ℍ1(lρ,μ)) and the space 𝔹ℍ(lρ,2/(ρ-μ-1),(𝒟,⋎),⊗,Δ)(or𝔹ℍ2(lρ,2/(ρ-μ-1))) of Bohemian and give their properties.At the first step, we prove the following connecting theorem.Theorem 5.
Letf∈lρ,μ and φ,ψ∈𝒟; then, f⊗(φ⋎ψ)=(f⊗φ)⊗ψ.Proof.
Letf∈lρ,μ and φ,ψ∈𝒟; then, applying the Fubinitz theorem yields
(13)(f⊗(φ⋎ψ))(x)=∫ℝ+f(xt)(φ⋎ψ)(t)dt=∫ℝ+f(xt)(∫ℝ+‍φ(ty)ψ(y)y-1dy)dt=∫ℝ+(∫ℝ+f(xt)φ(ty)y-1dy)ψ(y)dy.
The change of variables t=yz implies dt=ydz and further
(14)(f⊗(φ⋎ψ))(x)=∫ℝ+(∫ℝ+f(xyz)φ(z)dz)ψ(y)dy.
Therefore,
(15)(f⊗(φ⋎ψ))(x)=∫ℝ+(f⊗φ)(xy)ψ(y)dy.
Hence, (15) implies
(16)(f⊗(φ⋎ψ))(x)=((f⊗φ)⊗ψ)(x).
This completes the proof of the theorem.Next, forthcoming theorems prove the existence of the space𝔹ℍ1(lρ,μ).Theorem 6.
Letf∈lρ,μ and φ∈𝒟 and then f⋎φ∈lρ,μ.Proof.
For eachf∈lρ,μ and φ∈𝒟, we have
(17)∥xμ(f⋎φ)(x)∥ρ,μρ=∫ℝ+|xμ∫ℝ+‍f(xt)φ(t)dt|ρdx.
By Jensen’s theorem, we write
(18)∥xμ(f⋎φ)(x)∥ρ,μρ≤∫ℝ+∫ℝ+|xμ∫ℝ+f(xt)φ(t)dt|ρdx≤∫ℝ+∫ℝ+|xμf(xt)|ρ|φ(t)|dtdx≤∥f∥ρ,μρ∫K|φ(t)|dt,
where K=[a,b], b>a>0 is a compact subset containing the support of φ. Hence, from (18), we get
(19)∥xμ(f⋎φ)(x)∥ρ,μρ≤∥f∥ρ,μρM(b-a),
where M is certain positive real number. This completes the proof of the theorem.Theorem 7.
Letf∈lρ,μ and φ,ψ∈𝒟 and then(i)
f⋎(φ+ψ)=f⋎φ+f⋎ψ;(ii)
(αf)⋎φ=α(f⋎φ);(iii)
let(fn)∈lρ,μ and fn→f as n→∞ then for φ∈𝒟, and we have fn⋎φ→f⋎φ as n→∞;(iv)
f⋎(φ⋎ψ)=(f⋎φ)⋎ψ.Proof of Part (i), (ii), and (iii) follows from properties of integral operators∫. Similarly, the proof of Part (iv) is straightforward from the properties of ⋎ proved in [5].Following theorem is straightforward, and detailed proof is omitted.Theorem 8.
Letf∈lρ,μ and (μn)∈Δ then f⋎μn→f as n→∞.Proof.
Since𝒟 is a dense subspace of lρ then it is a dense subspace of lρ,μ. Hence, there can be found ψ∈𝒟 such that
(20)∥f-ψ∥ρ,μ<ε
for ε>0. Also, from (18), we have
(21)∥(f-ψ)⋎μn∥ρ,μρ≤∥f∥ρ,μρ∥μn∥l1.
Let g(t)=yμψ(yt-1)t-1 then; g(t)∈𝒟 and hence uniformly continuous on ℝ+.
Therefore, there isδ>0 such that
(22)|g(y)-g(x)|<εwhenever|y-x|<δ.
Thus, using (22), we get
(23)∥(ψ×μn-ψ)(y)∥ρ,μρ=∫ℝ+|yμ(ψ×μn-ψ)(y)|ρdy≤∫ℝ+∫ℝ+|-ψ(y))yμ(-ψ(y)ψ(yt-1)t-1μn(t)-ψ(y)(yt-1))μn(t)|ρdtdy≤∫ℝ+∫ℝ+|yμ(ψ(yt-1)t-1-ψ(y))|ρ×|μn(t)|dtdy=∫ℝ+∫ℝ+|g(y)-g(1)|ρ|μn(t)|dtdy=∫ℝ+∫ℝ+ερ|μn(t)|dtdy.
By (22), suppμn(t)→0 as n→∞ implies that there can be found N∈ℕ, such that suppμn⊆[0,δ], for all n≥N. Further, the fact that the function ψ is of compact support thus this implies that suppψ(y)⊆K=[a,b], where K is a compact subset of ℝ+. Thus, from (23), we write
(24)∥ψ⋎μn-ψ∥ρ,μρ≤ερ∫ab∫0δMdt=ερ(b-a)(2δ)M.
Now, we have
(25)∥f×μn-f∥ρ,μ≤∥(f-ψ)⋎μn∥ρ,μ+∥ψ×μn-ψ∥ρ,μ+∥(f-ψ)∥ρ,μ.
On using (20), (21) and (24) prove that
(26)∥f×μn-f∥ρ,μ⟶0asϵ⟶0.
Thus the theorem is proved. Then the Bohemian space 𝔹ℍ1(lρ,μ) is therefore constructed. The operations such as sum and multiplication by a scalar of two Bohemian in 𝔹ℍ1(lρ,μ) are defined in a natural way
(27)[(fn)(μn)]+[(gn)(τn)]=[(fn⋎τn+gn⋎μn)(μn⋎τn)],α[(fn)(μn)]=[(afn)(μn)],
where α  is a complex number.
Similarly, the operation⋎ and differentiation are defined by
(28)[(fn)(μn)]⋎[(gn)(τn)]=[(fn⋎gn)(μn⋎τn)],𝒟α[(fn)(μn)]=[(𝒟αfn)(μn)].
Now constructing the space 𝔹ℍ2(lρ,2/(ρ-μ-1)) follows from theorems which were used for constructing the space 𝔹ℍ1(lρ,μ). Therefore, the corresponding proofs of Theorems 10 and 11 are omitted.Theorem 9.
Letf∈lρ,2/(ρ-μ-1) and φ∈𝒟 and then f⊗φ∈lρ,2/(ρ-μ-1).Theorem 10.
Letf∈lρ,2/(ρ-μ-1), φ,ψ∈𝒟 and then the following hold:(i)
f⊗(φ+ψ)=f⊗φ+f⊗ψ;(ii)
(αf)⊗φ=α(f⊗φ);(iii)
iffn→f in lρ,2/(ρ-μ-1), as n→∞ then fn⊗φ→f⊗φ as n→∞;(iv)
if(μn)∈Δ, then f⊗μn→f as n→∞.Theorem 11.
Letf∈lρ,2/(ρ-μ-1) and φ,ψ∈𝒟 and then f⊗(φ⋎ψ)=(f⊗φ)⊗ψ.Proof of this theorem is similar to that of Theorem6. Thus the space 𝔹ℍ2(lρ,2/(ρ-μ-1)) can be regarded as Bohemian space.


## 4. Krätzel Transform of Bohemian

By aid of Theorem4, we have the right to define the Krätzel transform of [(fn)/(μn)]∈𝔹ℍ1(lρ,μ) as the Bohemian
(29)lv(ρ)^[(fn)(μn)]=[(lv(ρ)fn)(μn)]
is embedded in the space 𝔹ℍ2(lρ,2/(ρ-μ-1)).Theorem 12.
The mappinglv(ρ)^:𝔹ℍ1(lρ,μ)→𝔹ℍ2(lρ,2/(ρ-μ-1)) is(i)
well defined,(ii)
linear.Proof.
Let[(fn)/(μn)],[(gn)/(ψn)]∈𝔹ℍ1(lρ,μ) be such that [(fn)/(μn)]=[(gn)/(ψn)], and then fn⋎ψm=gm⋎μn=gn⋎μm. Using Theorem 4 implies lv(ρ)fn⊗ψm=lv(ρ)gn⊗μm, for all n,m. The idea of quotient of sequences in 𝔹ℍ2(lρ,2/(ρ-μ-1)) implies that
(30)lv(ρ)fnμnisequivalenttolv(ρ)gnψn.
That is,
(31)[(lv(ρ)fn)(μn)]=[(lv(ρ)gn)(ψn)].
To prove part (ii) of the theorem, if [(fn)/(μn)],[(gn)/(ψn)]∈𝔹ℍ1(lρ,μ), then
(32)lv(ρ)^([(fn)(μn)]+[(gn)(ψn)])=[(lv(ρ)fn⊗ψn+lv(ρ)gn⊗μn)(μn⊗ψn)].
Hence
(33)lv(ρ)^([(fn)(μn)]+[(gn)(ψn)])=lv(ρ)^[(fn)(μn)]+lv(ρ)^[(gn)(ψn)].
Also, if α∈ℂ, then
(34)αlv(ρ)^[(fn)(μn)]=α[(lv(ρ)fn)(μn)]=[(lv(ρ)(αfn))(μn)].
Hence
(35)αlv(ρ)^[(fn)(μn)]=lv(ρ)^(α[(fn)(μn)]).
This completes the proof.Definition 13.
Let[(lv(ρ)fn)/(μn)]∈𝔹ℍ2(lρ,2/(ρ-μ-1)). We define the inverse of lv(ρ)^ transform of the Bohemian [(lv(ρ)fn)/(μn)] as
(36)lv(ρ)^-1[(lv(ρ)fn)(μn)]=[(lv(ρ))-1(lv(ρ)fn)(μn)]=[(fn)(μn)],
for each (μn)∈Δ.Theorem 14.
l
v
(
ρ
)
^
:
𝔹
ℍ
1
(
l
ρ
,
μ
)
→
𝔹
ℍ
2
(
l
ρ
,
2
/
(
ρ
-
μ
-
1
)
) is an isomorphism.Proof.
Letlv(ρ)^[(fn)/(μn)]=lv(ρ)^[(gn)/(ψn)]∈𝔹ℍ2(lρ,2/(ρ-μ-1)) and then by (29) we get lv(ρ)fn⊗ψm=lv(ρ)gm⊗μn. Therefore, Theorem 4 implies
(37)lv(ρ)(fn⋎ψm)=lv(ρ)(gm⋎μn).
Thus fn⋎ψm=gm⋎μn, for all m,n∈ℕ. The concept of quotients of equivalent classes of 𝔹ℍ1(lρ,μ) then gives
(38)[(fn)(μn)]=[(gn)(ψn)].
This proves that lv(ρ)^ is injective.
To show thatlv(ρ) is surjective, let [(lv(ρ)fn)/(μn)]∈𝔹ℍ2(lρ,2/(ρ-μ-1)). Then (lv(ρ)fn)/(μn) is a quotient of sequences in 𝔹ℍ2(lρ,2/(ρ-μ-1)).Hence, lv(ρ)fn⊗μm=lv(ρ)fm⊗μn, for all m,n∈ℕ. Once again, Theorem 4 implies that lv(ρ)(fn⋎μm)=lv(ρ)(fm⋎μn). Hence [(fn)/(μn)]∈𝔹ℍ1(lρ,μ) satisfies
(39)lv(ρ)[(fn)(μn)]=[(lv(ρ)fn)(μn)].
This completes the proof of the theorem.Theorem 15.
Letψ∈𝒟 and[(lv(ρ)fn)/(μn)]∈𝔹ℍ2(lρ,2/(ρ-μ-1)) then one has(i)
lv(ρ)^-1([(lv(ρ)fn)/(μn)]⊗ψ)=[(fn)/(μn)]⋎ψ;(ii)
lv(ρ)^([(fn)/(μn)]⋎ψ)=[(lv(ρ)fn)/(μn)]⊗ψ.Proof.
By using (29), we have
(40)lv(ρ)^-1([(lv(ρ)fn)(μn)]⊗ψ)=[(lv(ρ))-1((lv(ρ)fn)⊗ψ)(μn)].
Theorem 4 then gives
(41)lv(ρ)^-1([(lv(ρ)fn)(μn)]⊗ψ)=[(fn)⋎ψ(μn)]=[(fn)(μn)]⋎ψ.
Hence the part (i) of the theorem is proved. The proof of part (ii) is similar thus we omit the details. This completes the proof of the theorem.Theorem 16.
The mappings(i)
lv(ρ)^:𝔹ℍ1(lρ,μ)→𝔹ℍ2(lρ,2/(ρ-μ-1)) are continuous with respect to δ and Δ-convergence.(ii)
lv(ρ)^-1:𝔹ℍ2(lρ,2/(ρ-μ-1))→𝔹ℍ1(lρ,μ) are continuous with respect to δ and Δ-convergence.Proof.
At first, let us show thatlv(ρ)^ and lv(ρ)^-1 are continuous with respect to δ-convergence.
Letβn→δβ in 𝔹ℍ1(lρ,μ) as n→∞ and then we show that lv(ρ)^βn→lv(ρ)^β as n→∞. By virtue of Theorem 5, we can find fn,k and fk in lρ,μ such that
(42)βn=[fn,kμk],β=[fkμk]
such that fn,k→fk as n→∞ for every k∈ℕ. Hence, lv(ρ)fn,k→lv(ρ)fk∈lρ,2/(ρ-μ-1) as n→∞. Thus,
(43)[lv(ρ)fn,kμk]⟶[lv(ρ)fkμk]∈𝔹ℍ2(lρ,2/(ρ-μ-1))
as n→∞.
To prove the Part (ii), Letgn,g∈𝔹ℍ2(lρ,2/(ρ-μ-1)) be such that gn→δg as n→∞. Then, once again, by Theorem 5, gn=[lv(ρ)fn,k/μk] and g=[lv(ρ)fk/μk] for some fn,k,fk∈lρ,μ and lv(ρ)fn,k→lv(ρ)fk as n→∞. Hence [fn,k/μk]→[fk/μk] as n→∞.
Using (36), we get
(44)lv(ρ)^-1[lv(ρ)fn,kμk]⟶lv(ρ)^-1[lv(ρ)fkμk]asn→∞.
Now, we establish that lv(ρ)^ and lv(ρ)^-1 are continuous with respect to Δ-convergence.
Letβn→Δβ in 𝔹ℍ1(lρ,μ) as n→∞. Then, there exist fn∈lρ,μ and (μn)∈Δ such that (βn-β)⋎μn=[((fn)⋎μk)/μk] and fn→0 as n→∞. By applying (29) we get
(45)lv(ρ)^((βn-β)⋎μn)=[lv(ρ)((fn)⋎μk)μk].
Hence, we have lv(ρ)^((βn-β)⋎μn)=[((lv(ρ)fn)⊗μk)/μk]=lv(ρ)fn→0 as n→∞ in lρ,2/(ρ-μ-1).
Therefore(46)lv(ρ)^((βn-β)⋎μn)=(lv(ρ)^βn-lv(ρ)^β)⊗μn⟶0asn⟶∞.
Hence, lv(ρ)^βn→Δlv(ρ)^β as n→∞.
Finally, letgn→Δg∈𝔹ℍ2(lρ,2/(ρ-μ-1)) as n→∞ and then we find lv(ρ)fk∈lρ,2/(ρ-μ-1) such that (gn-g)⊗μn=[(lv(ρ)fk⊗μk)/μk] and lv(ρ)fk→0 as n→∞ for some (μn)∈Δ.
Now, using (36), we obtain
(47)lv(ρ)^-1((gn-g)⊗μn)=[(lv(ρ))-1(lv(ρ)fk⊗μk)μk].
Theorem 4 implies
(48)lv(ρ)^-1((gn-g)⊗μn)=[(fn)⋎μkμk]=fn⟶0asn⟶∞inlρ,μ.
Thus
(49)lv(ρ)^-1((gn-g)⊗μn)=(lv(ρ)^-1gn-lv(ρ)^-1g)⋎μn⟶0asn⟶∞.
From this, we find that lv(ρ)^-1gn→Δlv(ρ)^-1g as n→∞ in 𝔹ℍ1(lρ,μ).
This completes the proof of the theorem.Theorem 17.
Thelv(ρ)^ transform is consistent with the classical transform lv(ρ).Proof.
For everyf∈lρ,μ, let β be its representative in 𝔹ℍ1(lρ,μ) and then β=[(f⋎(μn))/(μn)] where (μn)∈Δ, for all n∈ℕ. Since (μn) is independent from the representative, for all n∈ℕ, therefore
(50)lv(ρ)^(β)=lv(ρ)^([f⋎(μn)(μn)])=[lv(ρ)f⊗(μn)(μn)],
which is the representative of lv(ρ)f∈lρ,2/(ρ-μ-1). Thus this completes the proof.Theorem 18.
Let[(gn)/(ψn)]∈𝔹ℍ2(lρ,2/(ρ-μ-1)) and then a necessary and sufficient condition for [(gn)/(ψn)] to be in the range of lv(ρ)^ is that gn belongs to range of lv(ρ), for all n∈ℕ.Proof.
Let[(gn)/(ψn)] be in the range of lv(ρ)^ and then it is clear that gn belongs to the range of lv(ρ), for all n∈ℕ.
To establish the converse, letgn be in the range of lv(ρ), for all n∈ℕ. Then there is fn∈lρ,μ such that lv(ρ)fn=gn, n∈ℕ. Since [(gn)/(ψn)]∈𝔹ℍ2(lρ,2/(ρ-μ-1)),
(51)gn⊗ψm=gm⊗ψn,
for all m,n∈ℕ, therefore,
(52)lv(ρ)(fn⋎φn)=lv(ρ)(fm⋎φn),∀m,n∈ℕ,
where fn∈lρ,μ and φn∈Δ, for all n∈ℕ. Then it follows that we get fn⋎φm=fm⋎φn, for all m,n∈ℕ. Thus fn/φn is a quotient of sequences in 𝔹ℍ1(lρ,μ). Therefore, we have [(fn)/(φn)]∈𝔹ℍ1(lρ,μ) and
(53)lv(ρ)^([(fn)(φn)])=[(gn)(ψn)].
Hence the theorem is proved.Theorem 19.
Ifβ=[(fn)/(φn)]∈𝔹ℍ1(lρ,μ) and γ=[(κn)/(μn)]∈𝔹ℍ1(lρ,μ), then one has
(54)lv(ρ)^([(fn)(φn)]⋎[(κn)(μn)])=[(lv(ρ)fn)(φn)]⊗[(κn)(μn)].Proof.
Let the hypothesis of the theorem be satisfied for some[(fn)/(φn)] and [(κn)/(μn)]. Therefore,
(55)lv(ρ)^([(fn)(φn)]⋎[(κn)(μn)])=lv(ρ)^([(fn⋎κn)(φn⋎μn)])=[((lv(ρ)fn)⊗κn)(φn⊗μn)]=[(lv(ρ)fn)(φn)]⊗[(κn)(μn)].
This completes the proof.



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