Unified Treatment of the Krätzel Transformation for Generalized Functions

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.

1. Introduction

Krätzel, in [1, 2], introduced a generalization of the Meijer transform by the integral: 𝐾^𝑝_𝑣𝑓(𝑥)=_𝐑+𝚉^𝑣_𝑝𝑥𝑦𝑓𝑦𝑑𝑦,𝑥>0, (1) where 𝚉^𝑣_𝑝𝑥𝑦=_𝐑+𝑡^𝑣−1𝑒^{−𝑡𝑝−𝑥𝑦/𝑡}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_𝑥∈𝐊𝒟^𝑘𝜙(𝑥)<∞, (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 𝐾^𝑝_𝑣 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], 𝑎_𝑖,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 ℒ^𝑟 be the space of Lebesgue integrable functions on 𝐑₊. Denote by 𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) (resp., 𝒮₊(ℒ^𝑟,𝛼,{𝑎_𝑖},𝑎)), 1≤𝑟≤∞, the subsets of 𝒮₊ of all complex valued infinitely smooth functions on 𝐑₊ such that, for some constant 𝑚₁(>0), sup_{𝛼,𝑥∈𝐊}𝒟^𝑘𝜑(𝑥)_ℒ𝑟≤𝑚₁𝑎^𝛼𝑎_𝛼 (4) for all 𝑎>0 (for some 𝑎>0), where 𝐊 is a compact set traverses 𝐑₊.

The elements of the dual spaces, 𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎)  (𝒮^′₊(ℒ^𝑟,𝛼,{𝑎_𝑖},𝑎)), are the Beurling-type (Roumieu-type) ultradistributions. It may be noted that 𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎)⊂𝒮₊(ℒ^𝑟,𝛼,{𝑎_𝑖},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₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) (resp., 𝒮₊(ℒ^𝑟,𝛼,{𝑎_𝑖},𝑎)) can be generated by the collection of seminorms: 𝜑_𝑟,𝑎=sup_{𝛼,𝑥∈𝐊}𝒟^𝑘𝜑(𝑥)_ℒ𝑟𝑎^𝛼𝑎_𝛼,𝑎>0. (5) A sequence (𝜑_𝑛)→𝜑∈𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) (resp., 𝒮₊(ℒ^𝑟,𝛼,{𝑎_𝑖},𝑎)) if lim_𝑛→∞sup_{𝛼,𝑥∈𝐊}𝒟^𝑘𝜑_𝑛(𝑥)−𝜑(𝑥)_ℒ𝑟=0, (6) and there is a constant 𝑚₁>0 independent of 𝑛 such that lim_𝑛→∞sup_{𝛼,𝑥∈𝐊}𝒟^𝑘𝜑_𝑛(𝑥)−𝜑(𝑥)_ℒ𝑟≤𝑚₁𝑎^𝛼𝑎_𝛼 (7) for all 𝑎>0.  𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎)  (𝒮₊(ℒ^𝑟,𝛼,{𝑎_𝑖},𝑎))  is dense in 𝒮₊, convergence in 𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎)  (𝒮₊(ℒ^𝑟,𝛼,{𝑎_𝑖},𝑎)) implies convergence in 𝒮₊, and consequently a restriction of any 𝑓∈𝒮^′₊ to 𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎)(𝒮₊(ℒ^𝑟,𝛼,{𝑎_𝑖},𝑎)) is in 𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎)(𝒮^′₊(ℒ^𝑟,𝛼,{𝑎_𝑖},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^𝑝_𝑣𝜙∈𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎).

Proof. Let 𝜙∈𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) and 𝑥 be fixed, and then 𝐾^𝑝_𝑣𝜙 certainly exists. By differentiation with respect to 𝑥 we get 𝒟_𝑥𝐾^𝑝_𝑣𝜙(𝑥)=_𝐑+𝒟_𝑥𝑡^𝑣−1𝑒^{−𝑡𝑝−𝑥𝑦/𝑡}𝑦=𝑑𝑡𝜙𝑑𝑦_𝐑+−𝑦𝑡𝑡^𝑣−1𝑒^{−𝑡𝑝−𝑥𝑦/𝑡}𝑑𝑡𝜙𝑦𝑑𝑦=_𝐑+−𝑦_𝐑+𝑡^{(𝑣−1)−1}𝑒^{−𝑡𝑝−𝑥𝑦/𝑡}𝑦=𝑑𝑡𝜙𝑑𝑦_𝐑+−𝑦𝚉_𝑝^𝑣−1𝑥𝑦𝜙𝑦𝑑𝑦. (8) Hence the principle of mathematical induction on the 𝑘th derivative gives 𝒟^𝑘_𝑥𝐾^𝑝_𝑣𝜙(𝑥)=(−1)^𝑘_𝐑+𝚉_𝑝^𝑣−𝑘𝑥𝑦𝑦^𝑘𝜙𝑦𝑑𝑦. (9) From [4], we deduce that 𝒟^𝑘_𝑥𝐾^𝑝_𝑣𝜙(𝑥)≤𝛼_1𝐑+𝑦^{(2𝑣−𝑝)/(2𝑝+2)}𝑒^{−(𝑥𝑦)𝑝/(𝑝+1)}𝑦^𝑘𝜙𝑦𝑑𝑦 (10) for some constant 𝛼₁. The assumption that 𝜙∈𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) implies that the products under the integral sign, Ψ=𝑦^𝑘𝜙(𝑦) and (𝑥𝑦)^{(2𝑣−𝑝)/(2𝑝+2)}𝑒^{−(𝑥𝑦)𝑝/(𝑝+1)}Ψ, are also in 𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎). Moreover, 𝜙∈𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) ensures that the integral: ϝ(𝑥)=𝛼_1𝐑+𝑥𝑦^{(2𝑣−𝑝)/(2𝑝+2)}𝑒^{−(𝑥𝑦)𝑝/(𝑝+1)}𝑦^𝑘𝜙𝑦𝑑𝑦 (11) belongs to 𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎). Hence 𝒟^𝑘_𝑥ϝ(𝑥)_ℒ𝑟≤𝑚₁𝑎^𝛼𝑎_𝛼 (12) for some constant 𝑚₁. Therefore, from the above inequality we get 𝒟^𝑘_𝑥𝐾^𝑝_𝑣𝜙(𝑥)_ℒ𝑟≤𝒟^𝑘_𝑥ϝ(𝑥)_ℒ𝑟≤𝑚₁𝑎^𝛼𝑎_𝛼, (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₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) into itself. Next, we establish the Parseval’s relation for the Krätzel transform.

Theorem 2. Let 𝑓 and 𝑔 be absolutely integrable functions over 𝐑₊ and then _𝐑+𝑓(𝑥)𝐾^𝑝_𝑣𝑔(𝑥)𝑑𝑥=_𝐑+𝐾^𝑝_𝑣𝑓(𝑥)𝑔(𝑥)𝑑𝑥, (14) where 𝐾^𝑝_𝑣𝑓 and 𝐾^𝑝_𝑣𝑔 are the Krätzel transforms of 𝑓 and 𝑔, respectively.

Proof. It is clear that 𝐾^𝑝_𝑣𝑓 and 𝐾^𝑝_𝑣𝑔 are continuous and bounded on 𝐑₊. Moreover, the Fubini’s theorem allows us to interchange the order of integration: _𝐑+𝑓(𝑥)𝐾^𝑝_𝑣𝑔(𝑥)𝑑𝑥=_𝐑+𝐑+𝑓(𝑥)𝚉^𝑣_𝑝𝑥𝑦𝑑𝑥𝑔𝑦𝑑𝑦. (15) Equation (15) follows since the Krätzel kernel 𝚉^𝑣_𝑝(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 𝐾^𝑝_𝑣𝑓,𝜙=𝑓,𝐾^𝑝_𝑣𝜙, (16) where 𝑓∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) and 𝜙∈𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎).

Theorem 3. Given that 𝑓∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) then 𝐾^𝑝_𝑣𝑓∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎).

Proof. Consider a zero convergent sequence (𝜙_𝑛) in 𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) then certainly (𝐾^𝑝_𝑣𝜙_𝑛) is a zero-convergent sequence in the same space. It follows from (16) that 𝐾^𝑝_𝑣𝑓,𝜙_𝑛=𝑓,𝐾^𝑝_𝑣𝜙_𝑛⟶0as𝑛⟶∞. (17)
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 of 𝐾^𝑝_𝑣𝑓,𝑓∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) follows from the following theorem.

Theorem 4. Let 𝑓∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) and then 𝐾^𝑝_𝑣𝑓 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. Let 𝑓∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎), and then 𝐾^𝑝_𝑣𝑓(𝑥)=𝑓𝑦,𝚉^𝑣_𝑝𝑥𝑦. (18) In fact, (18) is a straightforward consequence of Lemma 1.

Theorem 5. Let 𝑓∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) then 𝐾^𝑝_𝑣𝑓 is infinitely differentiable and 𝒟^𝑘_𝑥𝐾^𝑝_𝑣𝑓(𝑥)=𝑓(𝑡),𝒟^𝑘_𝑥𝚉^𝑣_𝑝(𝑥𝑡) (19) 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 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⊂𝙱^𝑁 such that(a)if 𝑥,𝑦∈𝙰,(𝑠_𝑛)∈Δ,𝑥⊙𝑠_𝑛=𝑦⊙𝑠_𝑛 for all 𝑛, then 𝑥=𝑦,(b)if (𝑠_𝑛),(𝑡_𝑛)∈Δ, then (𝑠_𝑛∗𝑡_𝑛)∈Δ.

Elements of Δ are called delta sequences. Consider 𝚐=𝑥_𝑛,𝑠_𝑛∶𝑥_𝑛∈𝙰,𝑠_𝑛∈Δ,𝑥_𝑛⊙𝑠_𝑚=𝑥_𝑚⊙𝑠_𝑛,∀𝑚,𝑛∈𝐍. (20) If (𝑥_𝑛,𝑠_𝑛),(𝑦_𝑛,𝑡_𝑛)∈g, 𝑥_𝑛⊙𝑡_𝑚=𝑦_𝑚⊙𝑠_𝑛, for all 𝑚,𝑛∈𝐍, then we say (𝑥_𝑛,𝑠_𝑛)∼(𝑦_𝑛,𝑡_𝑛). 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 𝑥⟶𝑥⊙𝑠_𝑛𝑠_𝑛. (21) The operation ⊙ can be extended to 𝛽×𝙰 by 𝑥_𝑛𝑠_𝑛𝑥⊙𝑡=^𝑛⊙𝑡𝑠_𝑛. (22) In 𝛽, there are two types of convergence: (𝛿 convergence)  a sequence (ℎ_𝑛) in 𝛽 is said to be 𝛿 convergent to ℎ in 𝛽, denoted by ℎ→−_𝑛^𝛿ℎ, if there exists a delta sequence (𝑠_𝑛) such that (ℎ_𝑛⊙𝑠_𝑛),(ℎ⊙𝑠_𝑛)∈𝙰, for all 𝑘,𝑛∈𝐍, and (ℎ_𝑛⊙𝑠_𝑘)→(ℎ⊙𝑠_𝑘) as 𝑛→∞, in 𝙰, for every 𝑘∈𝐍, (Δ convergence) a sequence (ℎ_𝑛) in 𝛽is said to be Δ convergent to ℎ in 𝛽, denoted by ℎ−→_𝑛^Δℎ, if there exists a (𝑠_𝑛)∈Δ such that (ℎ_𝑛−ℎ)⊙𝑠_𝑛∈𝙰, for all 𝑛∈𝐍, and (ℎ_𝑛−ℎ)⊙𝑠_𝑛→0 as 𝑛→∞ in 𝙰. For further discussion see [17–21].

5. The Ultra-Boehmian Space 𝛽_𝑠+

Denote by 𝙳₊, or 𝙳(𝐑₊), the Schwartz space of ℂ^∞ functions of bounded support. Let Δ₊ be the family of sequences (𝑠_𝑛)∈𝙳(𝐑₊) such that the following holds: (Δ₁)∫_𝐑+𝑠_𝑛(𝑥)𝑑𝑥=1, for all 𝑛∈𝐍, (Δ₂)𝑠_𝑛(𝑥)≥0, for all 𝑛∈𝐍, (Δ₃)supp𝑠_𝑛⊂(0,𝜀_𝑛),𝜀_𝑛→0 as 𝑛→∞.It is easy to see that each (𝑠_𝑛) in Δ₊ forms a delta sequence.

Let 𝑓∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) and 𝜎∈𝙳(𝐑₊) be related by the expression: 𝑓⋅𝜎𝜐=𝑓(𝜎⊛𝜐), (23) where (𝜎⊛𝜐)(𝑥)=_𝐑+𝜎(𝑡)𝜐(𝑥𝑡)𝑑𝑡 (24) for every 𝜐∈𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),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 of 𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎), we write 𝑓⋅𝜎𝜐=𝑓(𝜎⊛𝜐)≤𝐶‖𝜎⊛𝜐‖_𝑟,𝑎, (25) where 𝜐∈𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎). Hence, to complete the proof, we are merely required to show that 𝜎⊛𝜐∈𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎).
First, if 𝜎∈𝙳₊ and 𝜐∈𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎), then choosing a compact set 𝐊 containing the support of 𝜎 yields (𝜎⊛𝜐)(𝑥+Δ𝑥)−(𝜎⊛𝜐)(𝑥)=Δ𝑥_𝐑+𝜎(𝑡)𝜐((𝑥+Δ𝑥)𝑡)−𝜐(𝑥𝑡)Δ𝑥𝑑𝑡 (26) which is dominated by 𝜎(𝑡)|𝒟_𝑥𝜐(𝑥)|. The dominated convergence theorem and the principle of mathematical induction implies 𝒟^𝑘_𝑥(𝜎⊛𝜐)=𝜎⊛𝒟^𝑘_𝑥𝜐. (27) Finally _𝐑+𝒟^𝑘(𝜎⊛𝜐)(𝑥)^𝑟𝑑𝑥=_𝐑+𝜎⊛𝒟^𝑘𝜐(𝑥)^𝑟≤𝑑𝑥_𝐑+𝐊𝜎(𝑡)𝑑𝑡𝒟^𝑘𝜐(𝑥𝑡)^𝑟𝑑𝑥≤𝑀_𝐑+𝒟^𝑘𝜐(𝑥𝑡)^𝑟𝑑𝑥. (28) Therefore ‖𝜎⊛𝜐‖_𝑟,𝑎≤𝑑‖𝜐‖_𝑟,𝑎<𝑑𝑎^𝛼𝑎_𝛼forsomeconstant𝑑. (29) Thus 𝜎⊛𝜐∈𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎). This completes the proof of the lemma.

Lemma 7. Let 𝑓₁,𝑓₂∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) and 𝜎∈𝙳₊ and then(i)𝛼𝑓⋅𝜎=𝛼(𝑓⋅𝜎), 𝛼∈ℂ,(ii)(𝑓₁+𝑓₂)⋅𝜎=𝑓₁⋅𝜎₁+𝑓₂⋅𝜎₂.

Proof of the above Lemma is obvious.

Lemma 8. Let 𝑓_𝑛,𝑓∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) and 𝜎∈𝙳₊ and then𝑓_𝑛⋅𝜎⟶𝑓⋅𝜎. (30)

Proof. Let 𝜐∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) and then 𝜎⊛𝜐∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎). Therefore 𝑓_𝑛⋅𝜎−𝑓⋅𝜎𝜐=𝑓_𝑛−𝑓⋅𝜎𝜐=𝑓_𝑛−𝑓(𝜎⊛𝜐). (31) Hence 𝑓_𝑛⋅𝜎→𝑓⋅𝜎 as 𝑛→∞.

Lemma 9. Let 𝑓∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) and (𝑠_𝑛)∈Δ₊ then𝑓⋅𝑠_𝑛⟶𝑓𝑖𝑛𝒮^′₊ℒ^𝑟,𝛼,𝑎_𝑖,𝑎𝑎𝑠𝑛⟶∞. (32)

Proof. Let 𝜐∈𝒮₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) and supp𝑠_𝑛⊂(0,𝜀_𝑛), 𝑛∈𝐍 and then 𝑓⋅𝑠_𝑛𝜐=𝑓𝑠_𝑛⊛𝜎. (33) It is sufficient to establish that 𝑠_𝑛⊛𝜐→𝜐 as 𝑛→∞. By using (27) and Δ₁ imply that _𝐑+𝒟^𝑘_𝑥𝑠_𝑛⊛𝜐−𝜐(𝑥)^𝑟=𝑑𝑥_𝐑+𝑠_𝑛⊛𝒟^𝑘_𝑥𝜐−𝒟^𝑘_𝑥𝜐(𝑥)^𝑟=𝑑𝑥_{𝐑+𝜀𝑛0}𝑠_𝑛(𝑡)𝒟^𝑘_𝑥𝜐(𝑥𝑡)−𝒟^𝑘_𝑥𝜐(𝑥)𝑑𝑡^𝑟𝑑𝑥. (34) Hence, the mean value theorem implies _𝐑+𝒟^𝑘_𝑥𝑠_𝑛⊛𝜐−𝜐(𝑥)^𝑟𝑑𝑥≤_𝐑+^𝜀𝑛₀𝜉𝑠_𝑛(𝑡)𝒟_𝑥^𝑘+1𝜐(𝑥𝜉)𝑑𝑡^𝑟𝑑𝑥, (35)𝜉∈(0,𝑡). Let 𝐴₁=sup_𝑠∈𝐊|𝒟_𝑥^𝑘+1𝜓(𝑠)|, where 𝐊 is certain compact set. The calculations show that 𝑠_𝑛⊛𝜐−𝜐_𝑟,𝑎≤𝐹𝜀_𝑛⟶0as𝑛⟶∞, (36) where 𝐹 is certain constant. Hence Lemma 9. The Boehmian space 𝛽_𝑠+ is therefore constructed.

6. 𝛽_𝚉+ 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^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎), and then convergence on 𝚉₊ can be defined in such away that 𝐸_𝑛→𝐸_∗ in 𝚉₊ if 𝑓_𝑛→𝑓∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) as 𝑛→∞, where 𝐸_𝑛=𝐾^𝑝_𝑣𝑓_𝑛 and 𝐸_∗=𝐾^𝑝_𝑣𝑓. Let 𝐸∈𝚉₊ and 𝜎∈𝙳₊ and then it is proper to define (𝐸⊚𝜎)(𝑢)=_𝐑+𝐸(𝑢𝑡)𝜎(𝑡)𝑑𝑡 (37) for each 𝑢∈𝐑₊.

Lemma 10. Let 𝑓∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) and 𝜎∈𝙳₊ and then 𝐾^𝑝_𝑣(𝑓⋅𝜎)=𝐾^𝑝_𝑣𝑓⊚𝜎.

Proof. Let 𝐊 be a compact set containing the support of 𝜎, and then from (18) it follows that 𝐾^𝑝_𝑣𝑓⋅𝜎(𝑢)=𝑓⋅𝜎𝑦,𝚉^𝑣_𝑝=𝑓𝑦,𝜎(𝑡)𝑢𝑦,𝚉^𝑣_𝑝()𝑦=𝑢𝑡_𝐑+𝑓𝑦,𝚉^𝑣_𝑝(𝑢𝑡)𝑦𝜎(𝑡)𝑑𝑡=𝐾^𝑝_𝑣𝑓⊚𝜎. (38) Hence the lemma follows.

Lemma 11. Let 𝐸∈𝚉₊ and 𝜎∈𝙳₊ and then 𝐸⊚𝜎∈𝚉₊.

Proof. 𝐸∈𝚉₊ implies 𝐾^𝑝_𝑣𝑓=𝐸, for some 𝑓∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎). Hence 𝐸⊚𝜎=𝐾^𝑝_𝑣𝑓⊚𝜎=𝐾^𝑝_𝑣(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, if 𝐾^−1𝑝_𝑣 is the inverse Krätzel transform of 𝐾^𝑝_𝑣, and then

Lemma 12. Let 𝐸∈𝚉₊ and 𝜎∈𝙳₊ and then 𝐾^−1𝑝_𝑣(𝐸⊚𝜎)=𝐾^−1𝑝_𝑣𝐸⋅𝜎. (39)

Lemma 13. Let 𝐸₁,𝐸₂∈𝚉₊ and then for all 𝜎₁,𝜎₂∈𝙳₊ we have(1)(𝐸₁+𝐸₂)⊚𝜎₁=𝐸₁⊚𝜎₁+𝐸₂⊚𝜎₁, (2)(𝑎𝐸)⊚𝜎₁=𝑎(𝐸⊚𝜎₁).

Lemma 14. Let 𝐸_𝑛→𝐸 and (𝑠_𝑛)∈Δ₊ and then 𝐸_𝑛⊚𝑠_𝑛→𝐸.

Lemma 15. Let 𝐸_𝑛→𝐸 and 𝜎∈𝙳₊ and then 𝐸_𝑛⊚𝜎→𝐸⊚𝜎.

With the previous analysis, the Boehmian space 𝛽_𝚉+ is constructed. The sum of two Boehmians and multiplication by a scalar in 𝛽_𝚉+ is defined in a natural way [𝑓_𝑛/𝜙_𝑛]+[𝑔_𝑛/𝜓_𝑛]=[((𝑓_𝑛⊚𝜓_𝑛)+(𝑔_𝑛⊚𝜙_𝑛))/(𝜙_𝑛⊚𝜓_𝑛)] and 𝛼[𝑓_𝑛/𝜙_𝑛]=[𝛼(𝑓_𝑛/𝜙_𝑛)],𝛼∈ℂ.

The operation ⊚ and the differentiation are defined by 𝑓_𝑛𝜙_𝑛⊚𝑔_𝑛𝜓_𝑛=𝑓_𝑛⊚𝑔_𝑛𝜙_𝑛⊚𝜓_𝑛,𝒟^𝑘𝑓_𝑛𝜙_𝑛=𝒟^𝑘𝑓_𝑛𝜙_𝑛. (40)

With the aid of Lemma 10 we define the extended Krätzel transform of a Boehmian [𝑓_𝑛/𝑠_𝑛]∈𝛽_𝑠+ to be a Boehmian in 𝛽_𝚣+ expressed by the relation: 𝐊^𝑝_𝑣𝑓_𝑛𝑠_𝑛=𝐾^𝑝_𝑣𝑓_𝑛𝑠_𝑛. (41)

Lemma 16. 𝐊^𝑝_𝑣∶𝛽_𝑠+→𝛽_𝚉+ is well defined and linear mapping.

Proof is a straightforward conclusion of definitions.

Definition 17. Let [𝐸_𝑛/𝑠_𝑛]∈𝛽_𝚉+ and then the inverse of 𝐊 is defined as follows: 𝐊^−1𝑝_𝑣𝐸_𝑛𝑠_𝑛=𝐾^−1𝑝_𝑣𝐸_𝑛𝑠_𝑛, (42)for each (𝑠_𝑛)∈Δ₊.

Lemma 18. 𝐊^𝑝_𝑣 is an isomorphism from 𝛽_𝑠+ into 𝛽_𝚉+.

Proof. Assume 𝐊^𝑝_𝑣[𝑓_𝑛/𝑠_𝑛]=𝐊^𝑝_𝑣[𝑔_𝑛/𝑡_𝑛], then it follows from (41) and the concept of quotients of two sequences 𝐾^𝑝_𝑣𝑓_𝑛⊚𝑡_𝑚=𝐾^𝑝_𝑣𝑔_𝑚⊚𝑠_𝑛. Therefore, Lemma 10 implies 𝐾^𝑝_𝑣(𝑓_𝑛⋅𝑡_𝑚)=𝐾^𝑝_𝑣(𝑔_𝑚⋅𝜙_𝑛). Employing properties of 𝐾^𝑝_𝑣 implies 𝑓_𝑛⋅𝑡_𝑚=𝑔_𝑚⋅𝑠_𝑛. Thus, [𝑓_𝑛/𝑠_𝑛]=[𝑔_𝑛/𝑡_𝑛]. Next we establish that 𝐊^𝑝_𝑣 is onto. Let [𝐾^𝑝_𝑣𝑓_𝑛/𝑠_𝑛]∈𝛽_𝚉+ be arbitrary and then 𝐾^𝑝_𝑣𝑓_𝑛⊚𝑠_𝑚=𝐾^𝑝_𝑣𝑓_𝑚⊚𝑠_𝑛 for every 𝑚,𝑛∈𝐍. Hence 𝑓_𝑛,𝑓_𝑚∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) are such that 𝐾^𝑝_𝑣(𝑓_𝑛⋅𝑠_𝑚)=𝐾^𝑝_𝑣(𝑓_𝑚⋅𝑠_𝑛). Hence the Boehmian [𝑓_𝑛/𝑠_𝑛]∈𝛽_𝑠+ satisfies the equation 𝐊^𝑝_𝑣[𝑓_𝑛/𝑠_𝑛]=[𝐾^𝑝_𝑣𝑓_𝑛/𝑠_𝑛].
This completes the proof of the lemma.

Lemma 19. Let [𝐸_𝑛/𝑠_𝑛]∈𝛽_𝚉+, 𝐸_𝑛=𝐾^𝑝_𝑣𝑓_𝑛, and 𝜙∈𝙳₊ and then 𝐊^−1𝑝_𝑣𝐸_𝑛𝑠_𝑛=𝐸⊚𝜙^𝑛−1𝑠_𝑛𝐊⋅𝜙,_𝑝𝑣𝑓_𝑛𝑠_𝑛=𝐸•𝜙^𝑛𝑠_𝑛⊚𝜙. (43)

Proof. It follows from (42) that 𝐊^−1𝑝_𝑣𝐸_𝑛𝑠_𝑛⊚𝜙=𝐊^−1𝑝_𝑣𝐸_𝑛⊚𝜙𝑠_𝑛=𝐾^−1𝑝_𝑣𝐸_𝑛⊚𝜙𝑠_𝑛. (44) Applying Lemma 12 leads to 𝐊^−1𝑝_𝑣𝐸_𝑛𝑠_𝑛=𝐾⊚𝜙^−1𝑝𝑣𝐸_𝑛𝑠_𝑛𝐸⋅𝜙=^𝑛−1𝑠_𝑛⋅𝜙. (45) Proof of the second part is similar. This completes the proof of the lemma.

Theorem 20. 𝐊^𝑝_𝑣∶𝛽_𝑠+→𝛽_𝚉+ and 𝐊^−1𝑝_𝑣∶𝛽_𝚉+→𝛽_𝑠+ are continuous with respect to 𝛿 and Δ convergences.

Proof. First of all, we show that 𝐊^𝑝_𝑣∶𝛽_𝑠+→𝛽_𝚉+ and 𝐊^−1𝑝_𝑣∶𝛽_𝚉+→𝛽_𝑠+ are continuous with respect to 𝛿 convergence. Let 𝛽→−_𝑛^𝛿𝛽 in 𝛽_𝑠+ as 𝑛→∞ and then we establish that 𝐊^𝑝_𝑣𝛽_𝑛→𝐊^𝑝_𝑣𝛽 as 𝑛→∞. In view of [10], there are 𝑓_𝑛,𝑘 and 𝑓_𝑘 in 𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎) such that 𝛽_𝑛=𝑓_𝑛,𝑘𝑠_𝑘𝑓,𝛽=^𝑘𝑠_𝑘 (46) such that 𝑓_𝑛,𝑘→𝑓_𝑘 as 𝑛→∞ for every 𝑘∈𝐍. The continuity condition of the Krätzel transform implies 𝐸_𝑛,𝑘→𝐸_𝑘 as 𝑛→∞, 𝐸_𝑛,𝑘=𝐾^𝑝_𝑣𝑓_𝑛,𝑘, and 𝐸_𝑘=𝐾^𝑝_𝑣𝑓_𝑘 in the space 𝚉₊. Thus, [𝐸_𝑛,𝑘/𝑠_𝑘]→[𝐸_𝑘/𝑠_𝑘] as 𝑛→∞ in 𝛽_𝚉+.
To prove the second part of the lemma, let 𝑔→−_𝑛^𝛿𝑔∈𝛽_𝚉+ as 𝑛→∞. From [10], we have 𝑔_𝑛=[𝐸_𝑛,𝑘/𝑠_𝑘] and 𝑔=[𝐸_𝑘/𝑠_𝑘] for some 𝐸_𝑛,𝑘,𝐸_𝑘∈𝚉₊ where 𝐸_𝑛,𝑘→𝐸_𝑘 as 𝑛→∞. Hence 𝐾^−1𝑝_𝑣𝐸_𝑛,𝑘→𝐾^−1𝑝_𝑣𝐸_𝑘 in 𝛽_𝑠+ as 𝑛→∞. That is, [𝐸⁻¹_𝑛,𝑘/𝑠_𝑘]→[𝐸_𝑘⁻¹/𝑠_𝑘] as 𝑛→∞. Using (42) we get 𝐊^−1𝑝_𝑣[𝐸_𝑛,𝑘/𝑠_𝑘]→𝐊^−1𝑝_𝑣[𝐸_𝑘/𝑠_𝑘] as 𝑛→∞.
Now, we establish continuity of 𝐊^𝑝_𝑣 and 𝐊^−1𝑝_𝑣 with respect to Δ₊ convergence. Let 𝛽−→_𝑛^Δ𝛽 in 𝛽_𝑠+as 𝑛→∞. Then, we find 𝑓_𝑛∈𝒮^′₊(ℒ^𝑟,𝛼,(𝑎_𝑖),𝑎), and (𝑠_𝑛)∈Δ₊ such that (𝛽_𝑛−𝛽)⋅𝑠_𝑛=[(𝑓_𝑛⋅𝑠_𝑘)/𝑠_𝑘] and 𝑓_𝑛→0 as 𝑛→∞. Employing (41) we get 𝐊^𝑝_𝑣𝛽_𝑛−𝛽⋅𝑠_𝑛=𝐾^𝑝_𝑣𝑓_𝑛⋅𝑠_𝑘𝑠_𝑘. (47) Hence, from (41) and Lemma 19 we have 𝐊^𝑝_𝑣((𝛽_𝑛−𝛽)⋅𝑠_𝑛)=[(𝐸_𝑛⊚𝑠_𝑘)/𝑠_𝑘]=𝐸_𝑘→0 as 𝑛→∞ in 𝚉₊. Therefore 𝐊^𝑝_𝑣𝛽_𝑛−𝛽⋅𝑠_𝑛=𝐊^𝑝_𝑣𝛽_𝑛−𝐊^𝑝_𝑣𝛽⊚𝑠_𝑛⟶0as𝑛⟶∞. (48) Hence,𝐊^𝑝_𝑣𝛽−→_𝑛^Δ𝐊^𝑝_𝑣𝛽 as 𝑛→∞. Finally, let 𝑔−→_𝑛^Δ𝑔 in 𝛽_𝚉+ as 𝑛→∞ and then we find 𝐸_𝑘∈𝚉₊ such that (𝑔_𝑛−𝑔)⊚𝑠_𝑛=[(𝐸_𝑘⊚𝑠_𝑘)/𝑠_𝑘] and 𝐸_𝑘→0 as 𝑛→∞ for some (𝑠_𝑛)∈Δ₊ and 𝐸_𝑘=𝐾^𝑝_𝑣𝑓_𝑛.
Next, using (42), we obtain 𝐊^−1𝑝_𝑣((𝑔_𝑛−𝑔)⊚𝑠_𝑛)=[𝐾^−1𝑝_𝑣(𝐸_𝑘⊚𝑠_𝑘)/𝑠_𝑘]. Lemma 19 implies that 𝐊^−1𝑝_𝑣𝑔_𝑛−𝑔⊚𝑠_𝑛=𝑓_𝑛⋅𝑠_𝑘𝑠_𝑘=𝑓_𝑛⟶0as𝑛⟶∞in𝒮^′₊ℒ^𝑟,𝛼,𝑎_𝑖,𝑎. (49) Thus 𝐊^−1𝑝_𝑣((𝑔_𝑛−𝑔)⊚𝑠_𝑛)=(𝐊^−1𝑝_𝑣𝑔_𝑛−𝐊^−1𝑝_𝑣𝑔)⋅𝑠_𝑛→0 as 𝑛→∞. Hence, we have 𝐊^−1𝑝_𝑣𝑔−→_𝑛^Δ𝐊^−1𝑝_𝑣𝑔 as 𝑛→∞ in 𝛽_𝑠+.
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.