Henstock-Kurzweil Integral Transforms

Abstract

We show conditions for the existence, continuity, and differentiability of functions defined by Γ(𝑠)=∫^∞_−∞𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑡, where 𝑓 is a function of bounded variation on ℝ with lim_|𝑡|→∞𝑓(𝑡)=0.

1. Introduction

Let 𝑔 be a complex function defined on a certain subset of ℝ². Many functions on functional analysis are integrals of the following form:Γ(𝑠)=^∞_−∞𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑡. (1.1)

We discuss the above function Γ, where the integral that we use is that of Henstock-Kurzweil. This integral introduced independently by Kurzweil and Henstock in 1957-58 encompasses the Riemann and Lebesgue integrals, as well as the Riemann and Lebesgue improper integrals.

In Lebesgue theory, there are well-known results about the existence, continuity, and differentiability of Γ. For Henstock-Kurzweil integrals also there are results about this, for example, Theorems 12.12 and 12.13 of [1]. However, they all need the stronger condition: 𝑓(𝑡)𝑔(𝑡,𝑠) is bounded by a Henstock-Kurzweil integrable function 𝑟(𝑡). We provide other conditions for the existence, continuity, and differentiability of Γ.

2. Preliminaries

Let us begin by recalling the definition of Henstock-Kurzweil integral. For finite intervals in ℝ it is defined in the following way.

Definition 2.1. Let 𝑓∶[𝑎,𝑏]→ℝ be a function. One can say that 𝑓 is Henstock-Kurzweil (shortly, HK-) integrable, if there exists 𝐴∈ℝ such that, for each 𝜖>0, there is a function 𝛾_𝜖∶[𝑎,𝑏]→(0,∞) (named a gauge) with the property that for any 𝛿_𝜖-fine partition 𝑃={([𝑥_𝑖−1,𝑥_𝑖],𝑡_𝑖)}^𝑛_𝑖=1 of [𝑎,𝑏] (i.e., for each 𝑖, [𝑥_𝑖−1,𝑥_𝑖]⊂[𝑡_𝑖−𝛾_𝜖(𝑡_𝑖),𝑡_𝑖+𝛾_𝜖(𝑡_𝑖)]), one has ^𝑛_𝑖=1𝑓𝑡_𝑖𝑥_𝑖−𝑥_𝑖−1−𝐴<𝜖. (2.1)
The number 𝐴 is the integral of 𝑓 over [𝑎,𝑏] and it is denoted as 𝐴=∫^𝑏_𝑎𝑓.
In the unbounded case, the Henstock-Kurzweil integral is defined as follows.

Definition 2.2. Given a gauge function 𝛾∶[𝑎,∞]→(0,∞), one can say that a tagged partition 𝑃={([𝑥_𝑖−1,𝑥_𝑖],𝑡_𝑖)}^𝑛+1_𝑖=1 of [𝑎,∞] is 𝛾-fine, if (a)a = 𝑥₀,𝑥_𝑛+1=𝑡_𝑛+1=∞,(b)[𝑥_𝑖−1,𝑥_𝑖]⊂[𝑡_𝑖−𝛾(𝑡_𝑖),𝑡_𝑖+𝛾(𝑡_𝑖)] for all𝑖=1,2,…,𝑛,(c) [𝑥_𝑛,∞]⊆[1/𝛾(𝑡_𝑛+1),∞].

Definition 2.3. A function 𝑓∶[𝑎,∞]→ℝ is Henstock-Kurzweil integrable on [𝑎,∞], if there exists 𝐴∈ℝ such that, for each 𝜖>0, there is a gauge 𝛾_𝜖∶[𝑎,∞]→(0,∞) for which (2.1) is satisfied for all tagged partition 𝑃 which is 𝛿_𝜖-fine according to Definition 2.2.
Let 𝑓 be a function defined on an infinite interval [𝑎,∞), One can suppose that 𝑓 is defined on [𝑎,∞] assuming that 𝑓(∞)=0. Thus, 𝑓 is Henstock-Kurzweil integrable on [𝑎,∞) if 𝑓 extended on [𝑎,∞] is HK-integrable. For functions defined over intervals (−∞,𝑎] and (−∞,∞)One can makes similar considerations.
Let 𝐼 be a finite or infinite interval. The space of all Henstock-Kurzweil integrable functions over 𝐼 is denoted by ℋ𝒦(𝐼). This space will be considered with the Alexiewicz seminorm, which it is defined as follows: 𝑓_𝐼=sup_{𝐽⊆𝐼𝐽}𝑓, (2.2) where the supremum is being taken over all intervals 𝐽 contained in 𝐼.

Definition 2.4. Let 𝜑∶𝐼→ℝ be a function, where 𝐼⊆ℝ is a finite interval. The variation of 𝜑 over the interval 𝐼 is defined as follows: 𝑉_𝐼𝜑=sup^𝑛_𝑖=1𝜑𝑥_𝑖−𝜑𝑥_𝑖−1∶𝑃ispartitionof𝐼. (2.3)
We say that the function 𝜑 is of bounded variation on 𝐼 if 𝑉_𝐼𝜑<∞. Now if 𝜑 is a function defined on an infinite interval 𝐼, then 𝜑 is of bounded variation on 𝐼, if 𝜑 is of bounded variation on each finite subinterval of 𝐼 and there is 𝑀>0 such that 𝑉_[𝑎,𝑏]𝜑≤𝑀 for all [𝑎,𝑏]⊆𝐼. The variation of 𝜑 on 𝐼 is 𝑉_𝐼𝜑=sup{𝑉_[𝑎,𝑏]𝜑|[𝑎,𝑏]⊆𝐼}.
Given an interval 𝐼, the space of all bounded variation functions on 𝐼 is denoted by ℬ𝒱(𝐼). We set ℬ𝒱₀(ℝ)={𝑓∈ℬ𝒱(ℝ)∣lim_|𝑡|→∞𝑓(𝑡)=0}. The following are some classical theorems that are used throughout this paper. The first is given in [2, Lemma 24] and is an immediate consequence of [1, Theorem 10.12,and Corollary H.4].

Theorem 2.5. If 𝑔 is a HK-integrable function on [𝑎,𝑏]⊆ℝ and 𝑓 is a function of bounded variation on [𝑎,𝑏], then 𝑓𝑔 is HK-integrable on [𝑎,𝑏] and ^𝑏_𝑎𝑓𝑔≤inf_{[]𝑡∈𝑎,𝑏}𝑓(𝑡)^𝑏_𝑎𝑔(𝑡)𝑑𝑡+𝑔_[𝑎,𝑏]𝑉_[𝑎,𝑏]𝑓. (2.4)

Theorem 2.6 ([1] Chartier-Dirichlet's test). Let 𝑓 and 𝑔 be functions defined on [𝑎,∞). Suppose that(i)𝑔∈ℋ𝒦([𝑎,𝑐]) for every 𝑐≥𝑎, and 𝐺 defined by 𝐺(𝑥)=∫^𝑥_𝑎𝑔 is bounded on [𝑎,∞); (ii)𝑓 is of bounded variation on [𝑎,∞) and lim_𝑥→∞𝑓(𝑥)=0.Then 𝑓𝑔∈ℋ𝒦([𝑎,∞)).

Definition 2.7 (see [3]). Let 𝐸⊆[𝑎,𝑏]. A function 𝑓∶[𝑎,𝑏]→ℝ is 𝐴𝐶_𝛿 on 𝐸, if for every 𝜖>0, there exist 𝜂_𝜖>0 and a gauge 𝛿_𝜖 on 𝐸 such that ^𝑠_𝑖=1𝑓𝑣_𝑖−𝑓𝑢_𝑖<𝜖, (2.5) whenever 𝑃={([𝑢_𝑖,𝑣_𝑖],𝑡_𝑖)}^𝑠_𝑖=1 is a (𝛿_𝜖,𝐸)-fine subpartition of [𝑎,𝑏] (i.e., 𝑃 is 𝛿_𝜖-fine and the tags 𝑡_𝑖 belong to 𝐸) and ∑^𝑠_𝑖=1|𝑣_𝑖−𝑢_𝑖|<𝜂_𝜖.
We say that 𝑓 is 𝐴𝐶𝐺_𝛿 on [𝑎,𝑏], if [𝑎,𝑏] can be written as a countable union of sets on each of which the function 𝑓 is 𝐴𝐶_𝛿.
If ℎ(𝑡,𝑠) is a function on ℝ×ℝ, then we use the notation 𝐷₂ℎ for the partial derivative of ℎ with respect to the second component 𝑠.

Theorem 2.8 (4, Theorem4). Let 𝑎,𝑏∈ℝ. If ℎ∶ℝ×[𝑎,𝑏]→ℂ is such that (i)ℎ(𝑡,⋅) is 𝐴𝐶𝐺_𝛿 on [𝑎,𝑏] for almost all 𝑡∈ℝ; (ii)ℎ(⋅,𝑠) is HK-integrable on ℝ for all 𝑠∈[𝑎,𝑏].
Then 𝐻∶=∫^∞_−∞ℎ(𝑡,⋅)𝑑𝑡 is 𝐴𝐶𝐺_𝛿 on [𝑎,𝑏] and 𝐻^′(𝑠)=∫^∞_−∞𝐷₂ℎ(𝑡,𝑠)𝑑𝑡 for almost all 𝑠∈(𝑎,𝑏), if and only if, ^𝑡_𝑠^∞_−∞𝐷₂ℎ(𝑡,𝑠)𝑑𝑡𝑑𝑠=^∞_−∞^𝑡_𝑠𝐷₂ℎ(𝑡,𝑠)𝑑𝑠𝑑𝑡, (2.6) for all [𝑠,𝑡]⊆[𝑎,𝑏]. In particular, 𝐻^′𝑠₀=^∞_−∞𝐷₂ℎ𝑡,𝑠₀𝑑𝑡, (2.7) when 𝐻₂∶=∫^∞_−∞𝐷₂ℎ(𝑡,⋅)𝑑𝑡 is continuous at 𝑠₀.

3. Main Results

All results in this paper are based on functions in the vector space ℬ𝒱₀(ℝ). Note that ℬ𝒱₀(ℝ)⊈𝐿(ℝ), where 𝐿(ℝ) is the space of Lebesgue integrable functions. Indeed, the function 𝑓∶ℝ→ℝ defined by 𝑓(𝑥)=(),10if𝑥∈−∞,1𝑥if𝑥∈[1,∞), (3.1) is in ℬ𝒱₀(ℝ)⧵𝐿(ℝ). However, for bounded intervals 𝐼, functions in ℬ𝒱(𝐼) are Lebesgue integrables on 𝐼.

To facilitate the statement of these results, it seems appropriate to introduce some additional terminology. If 𝑔∶ℝ×ℝ→ℂ is a function and 𝑠₀∈ℝ, we say that 𝑠₀ satisfies Hypothesis (H) relative to 𝑔 if (𝐇) there exist 𝛿=𝛿(𝑠₀)>0 and 𝑀=𝑀(𝑠₀)>0, such that, if |𝑠−𝑠₀|<𝛿 then ^𝑣_𝑢𝑔(𝑡,𝑠)𝑑𝑡≤𝑀, (3.2)for all [𝑢,𝑣]⊆ℝ.

This type of condition plays a major role in the results of the present work.

Theorem 3.1. Let 𝑓∶ℝ→ℝ and 𝑔∶ℝ×ℝ→ℂ   be functions. If 𝑓∈ℬ𝒱₀(ℝ), and 𝑠₀∈ℝ satisfies Hypothesis (H) relative to 𝑔, then Γ(𝑠)=^∞_−∞𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑡 (3.3) exists for all 𝑠 in a neighborhood of 𝑠₀.

Proof. It follows by Theorem 2.6.

Theorem 3.2. Let 𝑓∶ℝ→ℝ and 𝑔∶ℝ×ℝ→ℂ be functions such that (i)𝑓∈ℬ𝒱₀(ℝ), 𝑔 is bounded, and (ii)𝑔(𝑡,⋅) is continuous for all 𝑡∈ℝ. If 𝑠₀∈ℝ satisfies Hypothesis (H) relative to 𝑔, then the function Γ defined in Theorem 3.1 is continuous at 𝑠₀.

Proof. There exist 𝛿₁>0 and 𝑀>0, such that, if |𝑠−𝑠₀|<𝛿₁ then ^𝑣_𝑢𝑔(𝑡,𝑠)𝑑𝑡≤𝑀, (3.4) for all [𝑢,𝑣]⊆ℝ. From Theorem 3.1, Γ(𝑠) exists for all 𝑠∈𝐵_𝛿1(𝑠₀).
Let 𝜖>0 be given. By Hake's Theorem, there exists 𝐾₁>0 such that _{|𝑡|≥𝑢}𝑓(𝑡)𝑔𝑡,𝑠₀<𝜖𝑑𝑡3, (3.5) for all 𝑢≥𝐾₁. On the other hand, as lim_{𝑡→−∞}𝑉_{(−∞,𝑡]}𝑓=0,lim_𝑡→∞𝑉_[𝑡,∞)𝑓=0, (3.6) there is 𝐾₂>0 such that for each 𝑡>𝐾₂, 𝑉_{(−∞,−𝑡]}𝑓+𝑉_[𝑡,∞)𝜖𝑓<.3𝑀 (3.7)
Let 𝐾=max{𝐾₁,𝐾₂}. From Theorem 2.5, it follows that for every 𝑣≥𝐾 and every 𝑠∈𝐵_𝛿1(𝑠₀), ^𝑣_𝐾𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑡≤𝑔(⋅,𝑠)_[𝐾,𝑣]inf_{[]𝑡∈𝐾,𝑣}𝑓(𝑡)+𝑉_[𝐾,𝑣]𝑓≤𝑀𝑓(𝑣)+𝑉_[𝐾,∞)𝑓, (3.8) where the second inequality is true due to (3.4). This implies, since lim_𝑡→∞|𝑓(𝑡)|=0, that ^∞_𝐾𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑡≤𝑀⋅𝑉_[𝐾,∞)𝑓. (3.9)
Analogously we have that ^−𝐾_−∞𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑡≤𝑀⋅𝑉_{(−∞,−𝐾]}𝑓. (3.10)
Therefore, for each 𝑠∈𝐵_𝛿1(𝑠₀), _{|𝑡|≥𝐾}𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑡≤𝑀𝑉_{(−∞,−𝐾]}𝑓+𝑉_[𝐾,∞)𝑓𝜖<𝑀=𝜖3𝑀3. (3.11)
By hypothesis, 𝑓 is Lebesgue integrable on [−𝐾,𝐾], 𝑔 is bounded, and 𝑔(𝑡,⋅) is continuous for all 𝑡∈ℝ. From this it is easy to see, for example using [1, Theorem 12.12], that Γ_𝐾∶ℝ→ℝ defined as Γ_𝐾(𝑠)=^𝐾_−𝐾𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑡,𝑠∈ℝ, (3.12) is continuous at 𝑠₀. This implies that there is 𝛿₂>0 such that for every 𝑠∈𝐵_𝛿2(𝑠₀), ^𝐾_−𝐾𝑓(𝑡)𝑔(𝑡,𝑠)−𝑔𝑡,𝑠₀<𝜖𝑑𝑡3. (3.13)
Let 𝛿=min{𝛿₁,𝛿₂}. Then for all 𝑠∈𝐵_𝛿(𝑠₀), Γ(𝑠)−Γ𝑠₀≤^𝐾_−𝐾𝑓(𝑡)𝑔(𝑡,𝑠)−𝑔𝑡,𝑠₀+𝑑𝑡_{|𝑡|≥𝐾}𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑡+_{|𝑡|≥𝐾}𝑓(𝑡)𝑔𝑡,𝑠₀𝑑𝑡. (3.14)
Thus, from (3.5), (3.11), and (3.13), |Γ(𝑠)−Γ(𝑠₀)|<𝜖/3+𝜖/3+𝜖/3=𝜖, for all 𝑠∈𝐵_𝛿(𝑠₀).

Theorem 3.3. Let 𝑎,𝑏∈ℝ. If 𝑓∶ℝ→ℝ and 𝑔∶ℝ×[𝑎,𝑏]→ℂ are functions such that (i)𝑓∈ℬ𝒱₀(ℝ), 𝑔 is measurable, bounded, and (ii) for all 𝑠∈[𝑎,𝑏], 𝑠 satisfies hypothesis (H) relative to 𝑔, then ^𝑏_𝑎^∞_−∞𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑡𝑑𝑠=^∞_−∞^𝑏_𝑎𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑠𝑑𝑡. (3.15)

Proof. From condition (ii) and by the compactness of [𝑎,𝑏], we claim that there exists 𝑀>0 such that, for each 𝑠∈[𝑎,𝑏], |∫^𝑣_𝑢𝑔(𝑡,𝑠)𝑑𝑡|≤𝑀, for all [𝑢,𝑣]⊆ℝ.
For each 𝑟>0 and 𝑠∈[𝑎,𝑏], let Γ_𝑟(𝑠)=∫^𝑟_−𝑟𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑡. Observe, by Theorem 2.5, Γ_𝑟(𝑠)=^𝑟_−𝑟𝑓(𝑡)𝑔()≤𝑔(𝑡,𝑠𝑑𝑡⋅,𝑠)_{[−𝑟,𝑟]}inf_{[]𝑡∈−𝑟,𝑟}𝑓(𝑡)+𝑉_{[−𝑟,𝑟]}𝑓≤𝑀𝑓(0)+𝑉𝑓, (3.16) for all 𝑠∈[𝑎,𝑏].
So, for each 𝑟>0, Γ_𝑟 is HK-integrable on [𝑎,𝑏] and is bounded for a fixed constant. Moreover, by Theorem 3.1 and Hake's Theorem, lim_𝑟→∞Γ_𝑟(𝑠)=Γ(𝑠), (3.17) for all 𝑠∈[𝑎,𝑏].
Therefore, by dominated convergence theorem, Γ is HK-integrable on [𝑎,𝑏] and ^𝑏_𝑎Γ(𝑠)𝑑𝑠=lim_𝑟→∞^𝑏_𝑎Γ_𝑟(𝑠)𝑑𝑠. (3.18)
Now, since 𝑓 is Lebesgue integrable on [−𝑟,𝑟], 𝑔 is measurable and bounded; it follows by Fubini's Theorem that ^𝑏_𝑎^𝑟_−𝑟𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑡𝑑𝑠=^𝑟_−𝑟^𝑏_𝑎𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑠𝑑𝑡. (3.19)
Consequently, lim_𝑟→∞^𝑟_−𝑟^𝑏_𝑎𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑠𝑑𝑡=lim_𝑟→∞^𝑏_𝑎Γ_𝑟(𝑠)𝑑𝑠=^𝑏_𝑎Γ(𝑠)𝑑𝑠. (3.20)
So by Hake's Theorem, ^∞_−∞^𝑏_𝑎𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑠𝑑𝑡=^𝑏_𝑎Γ(𝑠)𝑑𝑠=^𝑏_𝑎^∞_−∞𝑓(𝑡)𝑔(𝑡,𝑠)𝑑𝑡𝑑𝑠. (3.21)

Theorem 3.4. Consider 𝑓∶ℝ→ℝ and 𝑔∶ℝ×ℝ→ℂ functions, where 𝑓∈ℬ𝒱₀(ℝ) and the partial derivative 𝐷₂𝑔 exists on ℝ×ℝ and is bounded and continuous. If 𝑠₀∈ℝ such that(i)there is 𝐾>0 for which ‖𝑔(⋅,𝑠₀)‖_[𝑢,𝑣]≤𝐾 for all [𝑢,𝑣]⊆ℝ, and (ii)𝑠₀ satisfies Hypothesis (H) relative to 𝐷₂𝑔;then Γ is differentiable at 𝑠₀, and Γ^′𝑠₀=^∞_−∞𝑓(𝑡)𝐷₂𝑔𝑡,𝑠₀𝑑𝑡. (3.22)

Proof. It is not difficult to prove, using conditions (i), (ii), and the Mean Value Theorem, that there exist 𝛿>0 and 𝑀>0 such that, for each 𝑠∈(𝑠₀−𝛿,𝑠₀+𝛿), ^𝑣_𝑢𝐷₂𝑔(𝑡,𝑠)𝑑𝑡<𝑀,^𝑣_𝑢𝑔(𝑡,𝑠)𝑑𝑡<𝑀, (3.23) for all [𝑢,𝑣]⊆ℝ.
Consider 𝑎,𝑏∈ℝ with 𝑠₀−𝛿<𝑎<𝑠₀<𝑏<𝑠₀+𝛿. In order to show (3.22), we use Theorem 2.8. The function 𝑓(𝑡)𝑔(𝑡,⋅) is differentiable on [𝑎,𝑏] for each 𝑡∈ℝ, so 𝑓(𝑡)𝑔(𝑡,⋅) is 𝐴𝐶𝐺_𝛿 on [𝑎,𝑏] for all 𝑡∈ℝ. Also, by (3.23) and Theorem 2.6, 𝑓(⋅)𝑔(⋅,𝑠) is HK-integrable on ℝ for all 𝑠∈[𝑎,𝑏]. Then Γ^′𝑠₀=^∞_−∞𝑓(𝑡)𝐷₂𝑔𝑡,𝑠₀𝑑𝑡, (3.24) if Γ₂∶=^∞_−∞𝑓(𝑡)𝐷₂𝑔(𝑡,⋅)𝑑𝑡 (3.25) is continuous at 𝑠₀, and ^𝑡_𝑠^∞_−∞𝑓(𝑡)𝐷₂𝑔(𝑡,𝑠)𝑑𝑡𝑑𝑠=^∞_−∞^𝑡_𝑠𝑓(𝑡)𝐷₂𝑔(𝑡,𝑠)𝑑𝑠𝑑𝑡, (3.26) for all [𝑠,𝑡]⊆[𝑎,𝑏]. The first affirmation is true by (3.23) and Theorem 3.2. The second affirmation is true due to (3.23) and Theorem 3.3.

Remark 3.5. In the previous theorems the kernel 𝑔(𝑡,𝑠) satisfies |∫^𝑣_𝑢𝑔(𝑡,𝑠)𝑑𝑡|≤𝑀, for all [𝑢,𝑣]⊆ℝ. Moreover, if 𝑔 will satisfy ^𝑣_𝑢𝑔()≤𝑀𝑡,𝑠𝑑𝑡⁰|𝑠|, (3.27) for all [𝑢,𝑣]⊆ℝ, thenlim_|𝑠|→∞Γ(𝑠)=0, when 𝑓∈ℬ𝒱₀(ℝ) (a version of Riemann-Lebesgue Lemma).

4. Applications

If 𝑓∶ℝ→ℝ, then its Fourier transform at 𝑠∈ℝ is defined as follows: 𝑓(𝑠)=^∞_−∞𝑓(𝑡)𝑒^{−𝑖𝑡𝑠}𝑑𝑡. (4.1)

Talvila in [2] has done an extensive work about the Fourier transform using the Henstock-Kurzweil integral: existence, continuity, inversion theorems and so forth. Nevertheless, there are some omissions in those results that use [2, Lemma 25(a)]. Also Mendoza Torres et al. in [5] have studied existence, continuity, and Riemann-Lebesgue Lemma about the Fourier transform of functions belonging to ℋ𝒦(𝑅)∩ℬ𝒱(ℝ). Following the line of [5], in Theorem 4.2, we include some results from them as consequences of theorems in the above section.

Let 𝑓 and 𝑔 be real-valued functions on ℝ. The convolution of 𝑓 and 𝑔 is the function 𝑓∗𝑔 defined by 𝑓∗𝑔(𝑥)=^∞_−∞𝑓𝑥−𝑦𝑔𝑦𝑑𝑦, (4.2) for all 𝑥 such that the integral exists. Various conditions can be imposed on 𝑓 and 𝑔 to guarantee that 𝑓∗𝑔 is defined on ℝ, for example, if 𝑓 is HK-integrable and 𝑔 is of bounded variation.

Lemma 4.1. If 𝑓∈ℋ𝒦(ℝ)∩ℬ𝒱(ℝ), then lim_|𝑥|→∞𝑓(𝑥)=0.

Proof. Since 𝑓 is of bounded variation on ℝ, then lim_{𝑥→−∞}𝑓(𝑥) and lim_𝑥→∞𝑓(𝑥) exist. Suppose that lim_𝑥→∞𝑓(𝑥)=𝛼≠0. If 𝛼>0, there exists 𝐴>0 such that 𝛼/2<𝑓(𝑥), for all 𝑥>𝐴. If 𝛼<0, there is 𝐵>0 such that −𝛼/2<−𝑓(𝑥), for all 𝑥>𝐵. This shows that 𝑓∉ℋ𝒦([𝐴,∞)) or −𝑓∉ℋ𝒦([𝐵,∞)), which contradicts 𝑓∈ℋ𝒦(ℝ), so lim_𝑥→∞𝑓(𝑥)=0. Using a similar argument, we show that lim_{𝑥→−∞}𝑓(𝑥)=0.

As consequence of Lemma 4.1, the vector space ℋ𝒦(ℝ)∩ℬ𝒱(ℝ) is contained in ℬ𝒱₀(ℝ). So the next theorem is an immediate consequence of the above section.

Theorem 4.2. If 𝑓∈ℋ𝒦(ℝ)∩ℬ𝒱(ℝ), then (a)𝑓exists on ℝ. (b)𝑓 is continuous on ℝ⧵{0}.(c)lim_|𝑠|→∞𝑓(𝑠)=0. (d) Define 𝑔(𝑡)=𝑡𝑓(𝑡) and suppose that 𝑔∈ℋ𝒦(ℝ)∩ℬ𝒱(ℝ), then 𝑓 is differentiable on ℝ⧵{0}, and 𝑓^′(𝑠)=−𝑖𝑔(𝑠),∀𝑠∈ℝ⧵{0}. (4.3)(e) If ℎ∈𝐿(ℝ)∩ℬ𝒱(ℝ), then 𝑓∗ℎ(𝑠)=𝑓(𝑠)ℎ(𝑠) for all 𝑠∈ℝ.

Proof. First observe that ^𝑣_𝑢𝑒^{−𝑖𝑡𝑠}≤2𝑑𝑡|𝑠|, (4.4) for all [𝑢,𝑣]⊆ℝ. Then, each 𝑠₀≠0 satisfies Hypothesis (H) relative to 𝑒^{−𝑖𝑡𝑠}.
(a) Theorem 3.1 implies that 𝑓(𝑠₀) exists for all 𝑠₀≠0 and, since 𝑓∈ℋ𝒦(ℝ), 𝑓(0) exists. Thus, 𝑓 exists on ℝ.
(b) From Theorem 3.2, 𝑓 is continuous at 𝑠₀, for all 𝑠₀≠0.
(c) It follows by Remark 3.5 and (4.4).
(d) It follows by Theorem 2.8 in a similar way to the proof of Theorem 3.4.
(e) Take 𝑠∈ℝ and let 𝑘(𝑥,𝑦)=𝑓(𝑦−𝑥)𝑒^{−𝑖𝑦𝑠}. Then, for each 𝑦∈ℝ and all [𝑢,𝑣]⊆ℝ, ^𝑣_𝑢𝑘𝑥,𝑦𝑑𝑥=^𝑣_𝑢𝑓𝑦−𝑥𝑑𝑥=^𝑦−𝑣_𝑦−𝑢𝑓(𝑧)𝑑𝑧≤𝑓. (4.5)
Thus, for every 𝑦∈ℝ, 𝑦 satisfies Hypothesis (H) relative to 𝑘. Now, observe that ℎ∈ℬ𝒱₀(ℝ) and 𝑘 is measurable and bounded. So by Theorem 3.3, ^𝑎_−𝑎^∞_−∞ℎ(𝑥)𝑘𝑥,𝑦𝑑𝑥𝑑𝑦=^∞_−∞^𝑎_−𝑎ℎ(𝑥)𝑘𝑥,𝑦𝑑𝑦𝑑𝑥, (4.6) for all 𝑎>0.
On the other hand, ℎ(𝑥)^𝑎_−𝑎𝑓𝑦−𝑥𝑒^{−𝑖𝑦𝑠}𝑑𝑦≤|ℎ(𝑥)|^𝑎−𝑥_{−𝑎−𝑥}𝑓(𝑧)𝑒^{−𝑖𝑧𝑠}≤|ℎ(𝑥𝑓(⋅)𝑒𝑑𝑧)|^{−𝑖(⋅)𝑠}. (4.7)
Thus, since ℎ∈𝐿(ℝ), dominated convergence theorem implies that 𝑓(𝑠)ℎ(𝑠)=^∞_−∞ℎ(𝑥)^∞_−∞𝑓𝑦−𝑥𝑒^{−𝑖𝑦𝑠}𝑑𝑦𝑑𝑥=lim_𝑎→∞^∞_−∞ℎ(𝑥)^𝑎_−𝑎𝑓𝑦−𝑥𝑒^{−𝑖𝑦𝑠}𝑑𝑦𝑑𝑥, (4.8) but from (4.6), we have that 𝑓(𝑠)ℎ(𝑠)=lim_𝑎→∞^𝑎_−𝑎^∞_−∞ℎ(𝑥)𝑓𝑦−𝑥𝑒^{−𝑖𝑦𝑠}𝑑𝑥𝑑𝑦=lim_𝑎→∞^𝑎_−𝑎𝑓∗ℎ𝑦𝑒^{−𝑖𝑦𝑠}𝑑𝑦. (4.9) Therefore, by Hake's Theorem, 𝑓∗ℎ(𝑠)=𝑓(𝑠)ℎ(𝑠). (4.10)

If 𝑓∶[0,∞)→ℝ, then its Laplace transform at 𝑧∈ℂ is defined as follows: 𝐿𝑓(𝑧)=^∞₀𝑓(𝑡)𝑒^−𝑧𝑡𝑑𝑡. (4.11) Here, also the Laplace transform is considered as Henstock-Kurzweil integral.

Theorem 4.3. If 𝑓∈ℋ𝒦([0,∞))∩ℬ𝒱([0,∞)), then (a)𝐿(𝑓)(𝑧) exists for all 𝑧∈ℂ with 𝑅𝑒𝑧≥0. (b) If 𝐹(𝑥,𝑦)=𝐿(𝑓)(𝑥+𝑖𝑦), then 𝐹(⋅,𝑦) is continuous on ℝ⁺∪{0} for all 𝑦≠0, and 𝐹(𝑥,⋅) is continuous on ℝ for all 𝑥>0.

Proof. It is an easy consequence from Theorems 3.1 and 3.2, since |∫^𝑣_𝑢𝑒^{−(𝑥+𝑖𝑦)𝑡}𝑑𝑡|≤2/|𝑥+𝑖𝑦| for all 𝑢,𝑣,𝑥∈ℝ⁺∪{0}, 𝑦∈ℝ with 𝑥+𝑖𝑦≠0.

Moreover, the Riemann-Lebesgue Lemma holds the following.

Theorem 4.4. If 𝑓∈ℋ𝒦([0,∞))∩ℬ𝒱([0,∞)) and 𝑧=𝑥+𝑖𝑦, with 𝑥≥0, then lim_𝑦→∞𝐿(𝑓)(𝑧)=0.

Proof. It results by Remark 3.5 and (4.4), because 𝑓(⋅)𝑒^−𝑥(⋅) is in ℋ𝒦([0,∞))∩ℬ𝒱([0,∞)).