Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 750524, 7 pages
http://dx.doi.org/10.1155/2013/750524
Research Article
Unified Treatment of the Krätzel Transformation for
Generalized Functions
S. K. Q. Al-Omari1 and A. KJlJçman2
1
2
Department of Applied Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11134, Jordan
Department of Mathematics and Institute of Mathematical Research, Universiti Putra Malaysia (UPM),
43400 Serdang, Selangor, Malaysia
Correspondence should be addressed to A. Kılıçman; akilicman@putra.upm.edu.my
Received 26 March 2013; Accepted 18 April 2013
Academic Editor: Mustafa Bayram
Copyright © 2013 S. K. Q. Al-Omari and A. Kılıçman. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
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.
1. Introduction
Krätzel, in [1, 2], introduced a generalization of the Meijer
transform by the integral:
(𝐾V𝑝 𝑓) (𝑥) = ∫ ZV𝑝 (𝑥𝑦) 𝑓 (𝑦) 𝑑𝑦,
R+
𝑥 > 0,
(1)
where
ZV𝑝 (𝑥𝑦) = ∫ 𝑡V−1 𝑒−𝑡
𝑝
−𝑥𝑦/𝑡
R+
𝑑𝑡,
(2)
𝑝 > 0(∈ N), V ∈ C. Then this generalization is known as
Krätzel transform.
Let 𝑥 be in R+ . Denote by S+ , or S(R+ ), the space of all
complex-valued smooth functions 𝜙(𝑡) on R+ such that
󵄨
󵄨
sup 󵄨󵄨󵄨󵄨D𝑘 𝜙 (𝑥)󵄨󵄨󵄨󵄨 < ∞,
𝑥∈K
(3)
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].
2
Abstract and Applied Analysis
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.
Let 𝛼 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),
󵄩
󵄩
sup 󵄩󵄩󵄩󵄩D𝑘 𝜑 (𝑥)󵄩󵄩󵄩󵄩L𝑟 ≤ 𝑚1 𝑎𝛼 𝑎𝛼
𝛼,𝑥∈K
A sequence (𝜑𝑛 ) →
S+ (L𝑟 , 𝛼, {𝑎𝑖 }, 𝑎)) if
𝜑
∈
𝑎 > 0.
󵄩
󵄩
lim sup 󵄩󵄩D𝑘 (𝜑𝑛 (𝑥) − 𝜑 (𝑥))󵄩󵄩󵄩󵄩L𝑟 = 0,
(5)
(6)
−𝑥𝑦/𝑡
R+
=∬
𝑑𝑡𝜙 (𝑦) 𝑑𝑦
−𝑦 V−1 −𝑡𝑝 −𝑥𝑦/𝑡
𝑑𝑡𝜙 (𝑦) 𝑑𝑦
𝑡 𝑒
R+ 𝑡
(V−1)−1 −𝑡𝑝 −𝑥𝑦/𝑡
= ∫ −𝑦 ∫ 𝑡
R+
R+
𝑒
(8)
𝑑𝑡𝜙 (𝑦) 𝑑𝑦
= ∫ −𝑦ZV−1
𝑝 (𝑥𝑦) 𝜙 (𝑦) 𝑑𝑦.
Hence the principle of mathematical induction on the 𝑘th
derivative gives
𝑘
D𝑘𝑥 (𝐾V𝑝 𝜙) (𝑥) = (−1)𝑘 ∫ ZV−𝑘
𝑝 (𝑥𝑦) 𝑦 𝜙 (𝑦) 𝑑𝑦.
R+
(9)
From [4], we deduce that
󵄨
󵄨󵄨 𝑘 𝑝
󵄨󵄨D𝑥 (𝐾V 𝜙) (𝑥)󵄨󵄨󵄨
󵄨
󵄨
(10)
󵄨󵄨 (2V−𝑝)/(2𝑝+2) −(𝑥𝑦)𝑝 /(𝑝+1) 󵄨󵄨 𝑘
󵄨󵄨 𝑦 𝜙 (𝑦) 𝑑𝑦
≤ 𝛼1 ∫ 󵄨󵄨󵄨(𝑦)
𝑒
󵄨󵄨
R+ 󵄨
∈
for some constant 𝛼1 . The assumption that 𝜙
S+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) implies that the products under the integral
𝑝
sign, Ψ = 𝑦𝑘 𝜙(𝑦) and (𝑥𝑦)(2V−𝑝)/(2𝑝+2) 𝑒−(𝑥𝑦) /(𝑝+1) Ψ, are
also in S+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎). Moreover, 𝜙 ∈ S+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎)
ensures that the integral:
󵄨󵄨
(2V−𝑝)/(2𝑝+2) −(𝑥𝑦)𝑝 /(𝑝+1) 󵄨󵄨󵄨 𝑘
𝑒
ϝ (𝑥) = 𝛼1 ∫ 󵄨󵄨󵄨(𝑥𝑦)
󵄨󵄨 𝑦 𝜙 (𝑦) 𝑑𝑦 (11)
󵄨
R+ 󵄨
belongs to S+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎). Hence
and there is a constant 𝑚1 > 0 independent of 𝑛 such that
󵄩
󵄩
lim sup 󵄩󵄩󵄩󵄩D𝑘 (𝜑𝑛 (𝑥) − 𝜑 (𝑥))󵄩󵄩󵄩󵄩L𝑟 ≤ 𝑚1 𝑎𝛼 𝑎𝛼
𝑛→∞
𝛼,𝑥∈K
𝑝
R+
S+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) (resp.,
󵄩󵄩
𝑛→∞
𝛼,𝑥∈K
D𝑥 (𝐾V𝑝 𝜙) (𝑥) = ∬ D𝑥 𝑡V−1 𝑒−𝑡
(4)
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 Beurlingtype (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:
󵄩󵄩 𝑘
󵄩
󵄩󵄩D 𝜑 (𝑥)󵄩󵄩󵄩 𝑟
󵄩󵄩 󵄩󵄩
󵄩
󵄩L ,
󵄩󵄩𝜑󵄩󵄩𝑟,𝑎 = sup
𝑎𝛼 𝑎𝛼
𝛼,𝑥∈K
Proof. Let 𝜙 ∈ S+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) and 𝑥 be fixed, and then
𝐾V𝑝 𝜙 certainly exists. By differentiation with respect to 𝑥 we
get
󵄩󵄩 𝑘
󵄩
󵄩󵄩D𝑥 ϝ (𝑥)󵄩󵄩󵄩 𝑟 ≤ 𝑚1 𝑎𝛼 𝑎𝛼
󵄩
󵄩L
(7)
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
(12)
for some constant 𝑚1 . Therefore, from the above inequality
we get
󵄩
󵄩
󵄩
󵄩󵄩 𝑘 𝑝
󵄩󵄩D𝑥 (𝐾V 𝜙) (𝑥)󵄩󵄩󵄩 𝑟 ≤ 󵄩󵄩󵄩D𝑘𝑥 ϝ (𝑥)󵄩󵄩󵄩 𝑟 ≤ 𝑚1 𝑎𝛼 𝑎𝛼 ,
󵄩L
󵄩L 󵄩
󵄩
(13)
for certain positive constant 𝑚1 . This proves the lemma.
From Lemma 1 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.
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 Roumieutype tempered ultradistributions is almost similar.
Theorem 2. Let 𝑓 and 𝑔 be absolutely integrable functions over
R+ and then
Lemma 1. Let 𝜙 ∈ S+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) and then 𝐾V𝑝 𝜙 ∈
S+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎).
where 𝐾V𝑝 𝑓 and 𝐾V𝑝 𝑔 are the Krätzel transforms of 𝑓 and 𝑔,
respectively.
∫ 𝑓 (𝑥) (𝐾V𝑝 𝑔) (𝑥) 𝑑𝑥 = ∫ (𝐾V𝑝 𝑓) (𝑥) 𝑔 (𝑥) 𝑑𝑥,
R+
R+
(14)
Abstract and Applied Analysis
3
Proof. It is clear that 𝐾V𝑝 𝑓 and 𝐾V𝑝 𝑔 are continuous and
bounded on R+ . Moreover, the Fubini’s theorem allows us to
interchange the order of integration:
∫ 𝑓 (𝑥) (𝐾V𝑝 𝑔) (𝑥) 𝑑𝑥
R+
(15)
= ∫ (∫
R+
R+
𝑓 (𝑥) ZV𝑝
(𝑥𝑦) 𝑑𝑥) 𝑔 (𝑦) 𝑑𝑦.
Equation (15) follows since the Krätzel kernel ZV𝑝 (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 Theorem 2, the adjoint method
of extending the Krätzel transform can be read as
⟨𝐾V𝑝 𝑓, 𝜙⟩ = ⟨𝑓, 𝐾V𝑝 𝜙⟩ ,
(16)
Theorem 3. Given that 𝑓 ∈ S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) then 𝐾V𝑝 𝑓 ∈
S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎).
Proof. Consider a zero convergent sequence (𝜙𝑛 ) in
S+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) then certainly (𝐾V𝑝 𝜙𝑛 ) is a zero-convergent
sequence in the same space. It follows from (16) that
=
⟨𝑓, 𝐾V𝑝 𝜙𝑛 ⟩
󳨀→ 0
as 𝑛 󳨀→ ∞.
and then
𝐾V𝑝 𝑓
is
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.
Let 𝑓 ∈ S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎), and then
(𝐾V𝑝 𝑓) (𝑥) = ⟨𝑓 (𝑦) , ZV𝑝 (𝑥𝑦)⟩ .
Theorem 5. Let 𝑓 ∈ S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) then 𝐾V𝑝 𝑓 is infinitely
differentiable and
for every 𝑘 ∈ N and 𝑥 ∈ R+ .
Proof. See [4, Proposition 2.2].
Elements of Δ are called delta sequences. Consider
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 in g is denoted
by 𝛽. Elements of 𝛽 are called Boehmians. Between A and 𝛽
there is a canonical embedding expressed as
𝑥 󳨀→
𝑥 ⊙ 𝑠𝑛
.
𝑠𝑛
(21)
𝑥𝑛
𝑥 ⊙𝑡
⊙𝑡= 𝑛
.
𝑠𝑛
𝑠𝑛
(22)
In 𝛽, there are two types of convergence:
(𝛿 convergence) a sequence (ℎ𝑛 ) in 𝛽 is said to be
𝛿
𝛿 convergent to ℎ in 𝛽, denoted by ℎ𝑛 →
󳨀 ℎ, 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 𝛽 is said to be
Δ
Δ convergent to ℎ in 𝛽, denoted by ℎ𝑛 󳨀
→ ℎ, 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].
(18)
In fact, (18) is a straightforward consequence of Lemma 1.
D𝑘𝑥 (𝐾V𝑝 𝑓) (𝑥) = ⟨𝑓 (𝑡) , D𝑘𝑥 ZV𝑝 (𝑥𝑡)⟩
(a) if 𝑥, 𝑦 ∈ A, (𝑠𝑛 ) ∈ Δ, 𝑥 ⊙ 𝑠𝑛 = 𝑦 ⊙ 𝑠𝑛 for all 𝑛,
then 𝑥 = 𝑦,
(b) if (𝑠𝑛 ), (𝑡𝑛 ) ∈ Δ, then (𝑠𝑛 ∗ 𝑡𝑛 ) ∈ Δ.
The operation ⊙ can be extended to 𝛽 × A by
From the above theorem we deduce that the Krätzel
transform of a tempered ultradistribution is a tempered ultradistribution. Moreover, the boundedness property of 𝐾V𝑝 𝑓,
𝑓 ∈ S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) follows from the following theorem.
Theorem 4. Let 𝑓 ∈
bounded.
(i) a nonempty set 𝐴,
(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 collection Δ ⊂ B𝑁 such that
(17)
Linearity is obvious. This completes the proof.
S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎)
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:
g = {(𝑥𝑛 , 𝑠𝑛 ) : 𝑥𝑛 ∈ A, (𝑠𝑛 ) ∈ Δ, 𝑥𝑛 ⊙𝑠𝑚 = 𝑥𝑚 ⊙𝑠𝑛 , ∀𝑚, 𝑛 ∈ N} .
(20)
where 𝑓 ∈ S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) and 𝜙 ∈ S+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎).
⟨𝐾V𝑝 𝑓, 𝜙𝑛 ⟩
4. Boehmian Spaces
(19)
5. The Ultra-Boehmian Space 𝛽𝑠+
Denote by D+ , 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:
(Δ 1 ) ∫R 𝑠𝑛 (𝑥)𝑑𝑥 = 1, for all 𝑛 ∈ N,
+
(Δ 2 ) 𝑠𝑛 (𝑥) ≥ 0, for all 𝑛 ∈ N,
(Δ 3 ) supp 𝑠𝑛 ⊂ (0, 𝜀𝑛 ), 𝜀𝑛 → 0 as 𝑛 → ∞.
It is easy to see that each (𝑠𝑛 ) in Δ + forms a delta sequence.
4
Abstract and Applied Analysis
Let 𝑓 ∈ S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) and 𝜎 ∈ D (R+ ) be related by
the expression:
(𝑓 ⋅ 𝜎) 𝜐 = 𝑓 (𝜎 ⊛ 𝜐) ,
(23)
(𝜎 ⊛ 𝜐) (𝑥) = ∫ 𝜎 (𝑡) 𝜐 (𝑥𝑡) 𝑑𝑡
(24)
where
R+
Lemma 8. Let 𝑓𝑛 , 𝑓 ∈ S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) and 𝜎 ∈ D+ and then
𝑓𝑛 ⋅ 𝜎 󳨀→ 𝑓 ⋅ 𝜎.
(30)
Proof. Let 𝜐 ∈ S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) and then 𝜎 ⊛ 𝜐
S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎). Therefore
(𝑓𝑛 ⋅ 𝜎 − 𝑓 ⋅ 𝜎) 𝜐 = ((𝑓𝑛 − 𝑓) ⋅ 𝜎) 𝜐 = (𝑓𝑛 − 𝑓) (𝜎 ⊛ 𝜐) .
(31)
for every 𝜐 ∈ S+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎).
Hence 𝑓𝑛 ⋅ 𝜎 → 𝑓 ⋅ 𝜎 as 𝑛 → ∞.
Lemma 6. Let 𝑓 ∈ S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) and 𝜎 ∈ D+ and then
𝑓 ⋅ 𝜎 ∈ S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎).
Lemma 9. Let 𝑓 ∈ S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) and (𝑠𝑛 ) ∈ Δ + then
Proof.
Using the weak topology of S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎), we write
󵄨
󵄨 󵄨
󵄨󵄨
󵄨󵄨(𝑓 ⋅ 𝜎) 𝜐󵄨󵄨󵄨 = 󵄨󵄨󵄨𝑓 (𝜎 ⊛ 𝜐)󵄨󵄨󵄨 ≤ 𝐶‖𝜎 ⊛ 𝜐‖𝑟,𝑎 ,
(25)
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, if 𝜎 ∈ D+ and 𝜐 ∈ S+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎), then choosing
a compact set K containing the support of 𝜎 yields
(𝜎 ⊛ 𝜐) (𝑥 + Δ𝑥) − (𝜎 ⊛ 𝜐) (𝑥)
Δ𝑥
in S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ) , 𝑎)
𝑓 ⋅ 𝑠𝑛 󳨀→ 𝑓
as 𝑛 󳨀→ ∞.
(26)
Proof. Let 𝜐 ∈ S+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) and supp 𝑠𝑛 ⊂ (0, 𝜀𝑛 ), 𝑛 ∈ N
and then
(𝑓 ⋅ 𝑠𝑛 ) 𝜐 = 𝑓 (𝑠𝑛 ⊛ 𝜎) .
D𝑘𝑥 (𝜎 ⊛ 𝜐) = 𝜎 ⊛ D𝑘𝑥 𝜐.
(27)
It is sufficient to establish that 𝑠𝑛 ⊛ 𝜐 → 𝜐 as 𝑛 → ∞. By
using (27) and Δ 1 imply that
󵄨
󵄨𝑟
∫ 󵄨󵄨󵄨󵄨D𝑘𝑥 (𝑠𝑛 ⊛ 𝜐 − 𝜐) (𝑥)󵄨󵄨󵄨󵄨 𝑑𝑥
R
󵄨𝑟
󵄨
= ∫ 󵄨󵄨󵄨󵄨(𝑠𝑛 ⊛ D𝑘𝑥 𝜐 − D𝑘𝑥 𝜐) (𝑥)󵄨󵄨󵄨󵄨 𝑑𝑥
R
Hence, the mean value theorem implies
󵄨𝑟
󵄨
∫ 󵄨󵄨󵄨󵄨D𝑘𝑥 (𝑠𝑛 ⊛ 𝜐 − 𝜐) (𝑥)󵄨󵄨󵄨󵄨 𝑑𝑥
R
󵄨󵄨𝑟
󵄨󵄨 𝜀𝑛
󵄨󵄨
󵄨
≤ ∫ 󵄨󵄨󵄨∫ 𝜉𝑠𝑛 (𝑡) D𝑘+1
𝜐
𝑑𝑡
(𝑥𝜉)
󵄨󵄨 𝑑𝑥,
𝑥
󵄨󵄨
R+ 󵄨󵄨 0
󵄨𝑟
󵄨
󵄨
󵄨𝑟
∫ 󵄨󵄨󵄨󵄨D𝑘 (𝜎 ⊛ 𝜐) (𝑥)󵄨󵄨󵄨󵄨 𝑑𝑥 = ∫ 󵄨󵄨󵄨󵄨(𝜎 ⊛ D𝑘 𝜐) (𝑥)󵄨󵄨󵄨󵄨 𝑑𝑥
R
R
+
󵄨
≤ ∫ ∫ 󵄨󵄨󵄨󵄨𝜎 (𝑡) 𝑑𝑡D𝑘 𝜐 (𝑥𝑡)󵄨󵄨󵄨󵄨 𝑑𝑥
R
K
󵄨𝑟
+
󵄨𝑟
󵄨
≤ 𝑀 ∫ 󵄨󵄨󵄨󵄨D𝑘 𝜐 (𝑥𝑡)󵄨󵄨󵄨󵄨 𝑑𝑥.
R
+
(28)
Therefore
‖𝜎 ⊛ 𝜐‖𝑟,𝑎 ≤ 𝑑‖𝜐‖𝑟,𝑎 < 𝑑𝑎𝛼 𝑎𝛼 for some constant 𝑑.
(29)
Thus 𝜎 ⊛ 𝜐 ∈ S+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎). This completes the proof of
the lemma.
Lemma 7. Let 𝑓1 , 𝑓2 ∈ S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) and 𝜎 ∈ D+ and
then
Proof of the above Lemma is obvious.
(34)
󵄨󵄨𝑟
󵄨󵄨 𝜀𝑛
󵄨
󵄨
= ∫ 󵄨󵄨󵄨∫ 𝑠𝑛 (𝑡) (D𝑘𝑥 𝜐 (𝑥𝑡) − D𝑘𝑥 𝜐 (𝑥)) 𝑑𝑡󵄨󵄨󵄨 𝑑𝑥.
󵄨󵄨
R+ 󵄨󵄨 0
+
Finally
(ii) (𝑓1 + 𝑓2 ) ⋅ 𝜎 = 𝑓1 ⋅ 𝜎1 + 𝑓2 ⋅ 𝜎2 .
(33)
+
which is dominated by 𝜎(𝑡)|D𝑥 𝜐(𝑥)|. The dominated convergence theorem and the principle of mathematical induction
implies
(i) 𝛼𝑓 ⋅ 𝜎 = 𝛼(𝑓 ⋅ 𝜎), 𝛼 ∈ C,
(32)
+
𝜐 ((𝑥 + Δ𝑥) 𝑡) − 𝜐 (𝑥𝑡)
= ∫ 𝜎 (𝑡)
𝑑𝑡
Δ𝑥
R+
+
∈
(35)
𝜉 ∈ (0, 𝑡). Let 𝐴 1 = sup𝑠∈K |D𝑘+1
𝑥 𝜓(𝑠)|, where K is certain
compact set. The calculations show that
󵄩
󵄩󵄩
󵄩󵄩𝑠𝑛 ⊛ 𝜐 − 𝜐󵄩󵄩󵄩𝑟,𝑎 ≤ 𝐹𝜀𝑛 󳨀→ 0 as 𝑛 󳨀→ ∞,
(36)
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 by Z(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 = 𝐾V𝑝 𝑓𝑛 and 𝐸∗ = 𝐾V𝑝 𝑓. Let 𝐸 ∈ Z+ and 𝜎 ∈ D+ and
then it is proper to define
(𝐸 ⊚ 𝜎) (𝑢) = ∫ 𝐸 (𝑢𝑡) 𝜎 (𝑡) 𝑑𝑡
R+
for each 𝑢 ∈ R+ .
(37)
Abstract and Applied Analysis
5
Lemma 10. Let 𝑓 ∈ S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) and 𝜎 ∈ D+ and then
𝐾V𝑝 (𝑓 ⋅ 𝜎) = 𝐾V𝑝 𝑓 ⊚ 𝜎.
Lemma 16. K⃗ 𝑝V : 𝛽𝑠+ → 𝛽Z+ is well defined and linear
mapping.
Proof. Let K be a compact set containing the support of 𝜎,
and then from (18) it follows that
Proof is a straightforward conclusion of definitions.
𝐾V𝑝 (𝑓 ⋅ 𝜎) (𝑢) = ⟨(𝑓 ⋅ 𝜎) (𝑦) , ZV𝑝 (𝑢𝑦)⟩
Definition 17. Let [𝐸𝑛 /𝑠𝑛 ] ∈ 𝛽Z+ and then the inverse of K⃗ is
defined as follows:
−1
= ⟨𝑓 (𝑦) , ⟨𝜎 (𝑡) , ZV𝑝 ((𝑢𝑡) 𝑦)⟩⟩
= ∫
R+
⟨𝑓 (𝑦) , ZV𝑝
𝑝
((𝑢𝑡) 𝑦)⟩ 𝜎 (𝑡) 𝑑𝑡 = 𝐾V𝑝 𝑓⊚𝜎.
(38)
Lemma 11. Let 𝐸 ∈ Z+ and 𝜎 ∈ D+ and then 𝐸 ⊚ 𝜎 ∈ Z+ .
Proof. 𝐸 ∈ Z+ implies 𝐾V𝑝 𝑓 = 𝐸, for some 𝑓 ∈
S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎). Hence 𝐸 ⊚ 𝜎 = 𝐾V𝑝 𝑓 ⊚ 𝜎 = 𝐾V𝑝 (𝑓 ⋅ 𝜎) ∈
Z+ .
The following are lemmas which can be easily proved
by the aid of the corresponding lemmas from the previous
−1
𝑝
section. Detailed proof is avoided. First, if 𝐾V is the inverse
Krätzel transform of 𝐾V𝑝 , and then
Lemma 12. Let 𝐸 ∈ Z+ and 𝜎 ∈ D+ and then
𝑝
−1
𝑝
𝐾V (𝐸 ⊚ 𝜎) = 𝐾V 𝐸 ⋅ 𝜎.
for each (𝑠𝑛 ) ∈ Δ + .
(39)
Proof. Assume K⃗ 𝑝V [𝑓𝑛 /𝑠𝑛 ] = K⃗ 𝑝V [𝑔𝑛 /𝑡𝑛 ], then it follows from
(41) and the concept of quotients of two sequences 𝐾V𝑝 𝑓𝑛 ⊚
𝑡𝑚 = 𝐾V𝑝 𝑔𝑚 ⊚ 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 K⃗ 𝑝V
is onto. Let [𝐾V𝑝 𝑓𝑛 /𝑠𝑛 ] ∈ 𝛽Z+ be arbitrary and then 𝐾V𝑝 𝑓𝑛 ⊚
𝑠𝑚 = 𝐾V𝑝 𝑓𝑚 ⊚ 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
K⃗ 𝑝V [𝑓𝑛 /𝑠𝑛 ] = [𝐾V𝑝 𝑓𝑛 /𝑠𝑛 ].
This completes the proof of the lemma.
Lemma 19. Let [𝐸𝑛 /𝑠𝑛 ] ∈ 𝛽Z+ , 𝐸𝑛 = 𝐾V𝑝 𝑓𝑛 , and 𝜙 ∈ D+ and
then
−1
(1) (𝐸1 + 𝐸2 ) ⊚ 𝜎1 = 𝐸1 ⊚ 𝜎1 + 𝐸2 ⊚ 𝜎1 ,
Lemma 14. Let 𝐸𝑛 → 𝐸 and (𝑠𝑛 ) ∈ Δ + and then 𝐸𝑛 ⊚ 𝑠𝑛 →
𝐸.
Lemma 15. Let 𝐸𝑛 → 𝐸 and 𝜎 ∈ D+ and then 𝐸𝑛 ⊚𝜎 → 𝐸⊚𝜎.
With the previous analysis, the Boehmian space 𝛽Z+ 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
D𝑘 [
𝑓𝑛
D𝑘 𝑓𝑛
]=[
].
𝜙𝑛
𝜙𝑛
(40)
With the aid of Lemma 10 we define the extended Krätzel
transform of a Boehmian [𝑓𝑛 /𝑠𝑛 ] ∈ 𝛽𝑠+ to be a Boehmian in
𝛽z+ expressed by the relation:
𝐾𝑝 𝑓
𝑓
K⃗ 𝑝V [ 𝑛 ] = [ V 𝑛 ] .
𝑠𝑛
𝑠𝑛
(43)
𝑓
𝐸
K⃗ 𝑝V ([ 𝑛 ] ∙ 𝜙) = [ 𝑛 ] ⊚ 𝜙.
𝑠𝑛
𝑠𝑛
(2) (𝑎𝐸) ⊚ 𝜎1 = 𝑎(𝐸 ⊚ 𝜎1 ).
𝑓𝑛
𝑔
𝑓 ⊚ 𝑔𝑛
]⊚[ 𝑛] = [ 𝑛
],
𝜙𝑛
𝜓𝑛
𝜙𝑛 ⊚ 𝜓𝑛
−1
𝐸
𝐸
𝑝
K⃗ V ([ 𝑛 ] ⊚ 𝜙) = [ 𝑛 ] ⋅ 𝜙,
𝑠𝑛
𝑠𝑛
Lemma 13. Let 𝐸1 , 𝐸2 ∈ Z+ and then for all 𝜎1 , 𝜎2 ∈ D+ we
have
[
(42)
Lemma 18. K⃗ 𝑝V is an isomorphism from 𝛽𝑠+ into 𝛽Z+ .
Hence the lemma follows.
−1
−1 𝐸
𝐾V 𝐸𝑛 ]
𝑝
K⃗ V [ 𝑛 ] = [
,
𝑠𝑛
𝑠𝑛
[
]
(41)
Proof. It follows from (42) that
−1
𝑝
−1
−1
𝐾V (𝐸𝑛 ⊚ 𝜙) ]
𝐸 ⊚𝜙
𝐸
𝑝
𝑝
K⃗ V ([ 𝑛 ] ⊚ 𝜙) = K⃗ V ([ 𝑛
]) = [
.
𝑠𝑛
𝑠𝑛
𝑠𝑛
[
]
(44)
Applying Lemma 12 leads to
−1
−1
𝐸−1
𝐾V 𝐸𝑛 ]
𝐸
𝑝
⋅ 𝜙 = [ 𝑛 ] ⋅ 𝜙.
K⃗ V ([ 𝑛 ] ⊚ 𝜙) = [
𝑠𝑛
𝑠𝑛
𝑠𝑛
[
]
𝑝
(45)
Proof of the second part is similar. This completes the proof
of the lemma.
−1
𝑝
Theorem 20. K⃗ 𝑝V : 𝛽𝑠+ → 𝛽Z+ and K⃗ V : 𝛽Z+ → 𝛽𝑠+ are
continuous with respect to 𝛿 and Δ convergences.
−1
𝑝
Proof. First of all, we show that K⃗ 𝑝V : 𝛽𝑠+ → 𝛽Z+ and K⃗ V :
𝛽Z+ → 𝛽𝑠+ are continuous with respect to 𝛿 convergence.
𝛿
Let 𝛽𝑛 →
󳨀 𝛽 in 𝛽𝑠+ as 𝑛 → ∞ and then we establish that
6
Abstract and Applied Analysis
K⃗ 𝑝V 𝛽𝑛 → K⃗ 𝑝V 𝛽 as 𝑛 → ∞. In view of [10], there are 𝑓𝑛,𝑘 and
𝑓𝑘 in S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ), 𝑎) such that
𝛽𝑛 = [
𝑓𝑛,𝑘
],
𝑠𝑘
𝛽=[
𝑓𝑘
]
𝑠𝑘
(46)
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 = 𝐾V𝑝 𝑓𝑛,𝑘 , and 𝐸𝑘 = 𝐾V𝑝 𝑓𝑘 in the space Z+ .
Thus, [𝐸𝑛,𝑘 /𝑠𝑘 ] → [𝐸𝑘 /𝑠𝑘 ] as 𝑛 → ∞ in 𝛽Z+ .
𝛿
󳨀 𝑔 ∈
To prove the second part of the lemma, let 𝑔𝑛 →
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
−1
−1
𝑝
𝑝
𝑛 → ∞. Hence 𝐾V 𝐸𝑛,𝑘 → 𝐾V 𝐸𝑘 in 𝛽𝑠+ as 𝑛 → ∞.
−1
That is, [𝐸𝑛,𝑘
/𝑠𝑘 ] → [𝐸𝑘−1 /𝑠𝑘 ] as 𝑛 → ∞. Using (42) we
−1
−1
𝑝
𝑝
get K⃗ V [𝐸𝑛,𝑘 /𝑠𝑘 ] → K⃗ V [𝐸𝑘 /𝑠𝑘 ] as 𝑛 → ∞.
−1
𝑝
Now, we establish continuity of K⃗ 𝑝V and K⃗ V with respect
Δ
→ 𝛽 in 𝛽𝑠+ as 𝑛 → ∞. Then, we
to Δ + convergence. Let 𝛽𝑛 󳨀
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
𝐾𝑝 (𝑓 ⋅ 𝑠 )
K⃗ 𝑝V ((𝛽𝑛 − 𝛽) ⋅ 𝑠𝑛 ) = [ V 𝑛 𝑘 ] .
𝑠𝑘
(47)
Hence, from (41) and Lemma 19 we have K⃗ 𝑝V ((𝛽𝑛 − 𝛽) ⋅ 𝑠𝑛 ) =
[(𝐸𝑛 ⊚ 𝑠𝑘 )/𝑠𝑘 ] = 𝐸𝑘 → 0 as 𝑛 → ∞ in Z+ . Therefore
K⃗ 𝑝V ((𝛽𝑛 − 𝛽) ⋅ 𝑠𝑛 )
(48)
= (K⃗ 𝑝V 𝛽𝑛 − K⃗ 𝑝V 𝛽) ⊚ 𝑠𝑛 󳨀→ 0 as 𝑛 󳨀→ ∞.
Δ
Δ
Hence, K⃗ 𝑝V 𝛽𝑛 󳨀
→ K⃗ 𝑝V 𝛽 as 𝑛 → ∞. Finally, let 𝑔𝑛 󳨀
→ 𝑔 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 = 𝐾V𝑝 𝑓𝑛 .
−1
−1
𝑝
𝑝
Next, using (42), we obtain K⃗ V ((𝑔𝑛 −𝑔)⊚𝑠𝑛 ) = [𝐾V (𝐸𝑘 ⊚
𝑠𝑘 )/𝑠𝑘 ]. Lemma 19 implies that
−1
𝑝
K⃗ V ((𝑔𝑛 − 𝑔) ⊚ 𝑠𝑛 )
=[
𝑓𝑛 ⋅ 𝑠𝑘
] = 𝑓𝑛 󳨀→ 0 as 𝑛 󳨀→ ∞ in S󸀠+ (L𝑟 , 𝛼, (𝑎𝑖 ) , 𝑎) .
𝑠𝑘
(49)
−1
−1
−1
𝑝
𝑝
𝑝
Thus K⃗ V ((𝑔𝑛 −𝑔)⊚𝑠𝑛 ) = (K⃗ V 𝑔𝑛 − K⃗ V 𝑔)⋅𝑠𝑛 → 0 as 𝑛 → ∞.
−1
−1
Δ
𝑝
𝑝
Hence, we have K⃗ V 𝑔𝑛 󳨀
→ K⃗ V 𝑔 as 𝑛 → ∞ in 𝛽𝑠+ .
This completes the proof of the theorem.
Acknowledgment
The authors express their sincere thanks to the referees for the
careful and detailed reading of the earlier version of paper
and the very helpful suggestions that improved the paper
substantially. The second author also acknowledges that this
project was partially supported by University Putra Malaysia
under the ERGS Grant Scheme with Project no. 5527068.
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