Henstock-Kurzweil Integral Transforms and the Riemann-Lebesgue Lemma Henstock-Kurzweil Integral Transforms and the Riemann-Lebesgue Lemma

• The classical Riemann-Lebesgue Lemma is an important tool used when proving several results related with convergence of Fourier Series and Fourier transform. In turn, these theorems have applications in the Harmonic Analysis which has many applications in the physics, biology, engineering and others sciences. For example, it is directly applied to the study of periodic perturbations of a class of resonant problems. • An important problem is to consider an orthogonal basis, different to the trigonometric basis, and study the Fourier expansion of a function with respect to this basis. In this case, it is obtained an expression in this way


Introduction
Let f be a function defined on a closed interval [a, b] in the extended real line R, its Fourier transform at s ∈ R is defined asf The classical Riemann-Lebesgue Lemma states that lim |s|→∞ b a e −ixs f (x)dx = 0, whenever f ∈ L 1 ([a, b]).
We consider important to study analogous results about this lemma due to the following reasons: • The classical Riemann-Lebesgue Lemma is an important tool used when proving several results related with convergence of Fourier Series and Fourier transform. In turn, these theorems have applications in the Harmonic Analysis which has many applications in the physics, biology, engineering and others sciences. For example, it is directly applied to the study of periodic perturbations of a class of resonant problems.
• An important problem is to consider an orthogonal basis, different to the trigonometric basis, and study the Fourier expansion of a function with respect to this basis. In this case, it is obtained an expression in this way b a h(xs) f (x)dx.
• In some cases the expression (1) exists and the expression (2) is true even if the function f is not Lebesgue integrable.
Thus, a variant of the Riemann-Lebesgue Lemma is to get conditions for the functions f and h which ensure that (3) Some results of this type and related results are found in [1], [2], [3], and [4].
In the space of Henstock-Kurzweil integrable functions over R, HK(R), the Fourier transform does not always exist. In [5] was proven that e −i(·)s f is Henstock-Kurzweil integrable under certain conditions and that, in general, does not satisfy the Riemann-Lebesgue Lemma. Subsequently, it was shown in [6], [3] and [4] that the Fourier transform exists and the equation (2) is true when −∞ = a, b = ∞ and f belongs to BV 0 (R), the space of bounded variation functions that vanish at infinity. A special case arises when f is in the intersection of functions of bounded variation and Henstock-Kurzweil integrable functions.
There exist Henstock-Kurzweil (HK) integrable functions f which f ∈ HK(I) \ L 1 (I) such that (2) is not fulfilled, when I is a bounded interval. In [7], Zygmund exhibited Henstock-Kurzweil integrable functions such that their Fourier coefficients do not tend to zero. In [8] are given necessary and sufficient conditions in order to b a f (x)g n (x)dx −→ b a f (x)g(x)dx, for all f ∈ HK([a, b]). Thus, we will prove that the Fourier transform has the asymptotic behavior:f (s) = o(s), as |s| → ∞, where f ∈ HK(I) \ L 1 (I) Moreover in [9], Titchmarsh proved that it is the best possible approximation for functions with improper Riemann integral.
This chapter is divided into 5 sections; we present the main results we have obtained in recent years: [3], [4], [6] and [10]. In this section we introduce basic concepts and important theorems about the Henstock-Kurzweil integral and bounded variation functions. In the second part of this study we prove some generalizations about the convergence of integrals of products in the completion of the space HK([a, b]), HK([a, b]), where [a, b] can be a bounded or unbounded interval. As a consequence, some results related to the Riemann-Lebesgue Lemma in the context of the Henstock-Kurzweil integral are proved over bounded intervals. Besides, for elements in the completion of the space of Henstock-Kurzweil integrable functions, we get a similar result to the Riemann-Lebesgue property for the Dirichlet Kernel, as well as the asymptotic behavior of the n-th partial sum of Fourier series.
In the third section, we consider a complex function g defined on certain subset of R 2 . Many functions on functional analysis are integrals of the form Γ(s) = ∞ −∞ f (t)g(t, s)dt. We study the function Γ when f belongs to BV 0 (R) and g(t, ·) is continuous for all t. The integral we use is Henstock-Kurzweil integral. There are well known results about existence, continuity and differentiability of Γ, considering the Lebesgue theory. In the HK integral context there are results about this too, for example, Theorems 12.12 and 12.13 from [11]. But they all need the stronger condition that the function f (t)g(t, s) is bounded by a HK integrable function. We give more conditions for existence, continuity and differentiability of Γ. Finally we give some applications such as some properties about the convolution of the Fourier and Laplace transforms.
In section 4, we exhibit a family of functions in HK(R) included in BV 0 (R) \ L 1 (R). At the last section we get a version of Riemann-Lebesgue Lemma for bounded variation functions that vanish at infinity. With this result we get properties for the Fourier transform of functions in BV 0 (R): it is well defined, is continuous on R − {0}, and vanishes at ±∞, as classical results. Moreover, we obtain a result on pointwise inversion of the Fourier transform.

Basic concepts and nomenclature
We will refer to a finite or infinite interval if its Lebesgue measure is finite or infinite. Let I ⊂ R be a closed interval, finite or infinite. A partition P of I is a increasing finite collection of points {t 0 , t 1 , ..., Let us consider I ⊂ R as a closed interval finite. A tagged partition of I is a set of ordered pairs then It is said that a tagged partition is called γ ε −fine if satisfies (5).
This definition can be extended on infinite intervals as follows.
for all t ≥ a. The total variation of f on I is equal to For I = (−∞, b] the considerations are analogous.  (R). We will refer to BV 0 (R) as the subspace of functions f belong to BV(R) such that vanishing at ±∞.
At the Lemma 25 we prove that: HK(R) ∩ BV(R) ⊂ BV 0 (R). It is not hard prove that BV 0 (R) L 1 (R) and BV 0 (R) HK(R). Furthermore, there are functions in HK(R) or L 1 (R) but they are not in BV 0 (R). For example, the function f (t) defined by 0 for t ∈ (−∞, 1) and 1/t for t ∈ [1, ∞) belongs to BV 0 (R) but does not belong to L 1 (R), neither in HK(R). In addition, other examples are the characteristic function of Q on a compact interval and g(t) = t 2 sin(exp(t 2 )) are in HK(R) \ BV 0 (R).
We consider the completion of HK([a, b]) as where the convergence is respect Alexiewicz norm, and will be denoted by HK ([a, b]). It is possible to prove that HK ([a, b]) is isometrically isomorphic to the subspace of distributions each of which is the distributional derivative of a continuous function, see [12]. The indefinite Thus, HK ([a, b]) is a Banach space with the Alexiewicz norm (6). The completion is also defined in [13]. Besides, basic results of the integral continue being true on the completion. More details see [12].
To facilitate reading, we recall the following results. The first one is a well known result, and it can be found for example in [14] and [15].
where Z is a dense subset on X implies that for each x ∈ X, the sequence (T n (x)) is convergent in Y and the linear operator T : X → Y defined by

Theorem 7. [17] If g is a HK integrable function on [a, b] ⊆ R and f is a bounded variation function on [a, b], then f g is HK integrable on
a+ε ϕ(t)dt exists. In this case, this limit is ∞ a ϕ(t)dt. Theorem 9. [11, Chartier-Dirichlet's Test] Let f and g be functions defined on [a, ∞). Suppose that 1. g ∈ HK([a, c]) for every c ≥ a, and G defined by Then f g ∈ HK([a, ∞)).
Moreover, by Multiplier Theorem, Hake's Theorem and Chartier-Dirichlet Test, we have the following lemma. Let I = [a, b] and E ⊂ I. We say that the function F : On the other hand, F belongs to the class ACG δ (I) if there exists a sequence {E n } ∞ 1 of sets in I such that I = ∪ ∞ n=1 E n and F ∈ AC δ (E n ) for each n ∈ N. A characterization of this type of functions is the following.

Theorem 11. A function f ∈ HK(I) if and only if there exists a function F
2. and h(·, s) is a HK integrable function on R for all s ∈ [a, b].

Fourier coefficients for functions in the Henstock-Kurzweil completion.
For finite intervals, the Theorem 12.11 of [19] tells us that: , it is necessary and sufficient that: The Theorem 3 of [8] proves that above theorem is valid for infinite intervals. In this section we show that [19,Theorem 12.11] and [8,Theorem 3] are true for functions belonging to the completion of the Henstock-Kurzweil space. First, we need to prove the next lemma. The class of step functions on [a, b] will be denoted as K([a, b]).

The convergence of integrals of products in the completion
The following result appears in [3]. Here, we present a detailed proof.
Proof. The necessity follows from [19,Theorem 12.11]. Now we will prove the sufficiency condition. Define the linear functionals T, T n : Supposing i) and ii), we have, by Multiplier Theorem, that the sequence {T n } is bounded by sup{||g n || ∞ + ||g n || BV }. Owing to Lemma 13, the space of step functions is dense in by the hypothesis iii), we have that {T n (χ (c,d) )} converges to T(χ (c,d) ), as n → ∞. Now, let f be a step function. Being that each T n is a linear functional, then {T n ( f )} converges to T( f ), as n → ∞. Thus, the result holds.

Remark 15. On HK([a, b]
). The hypothesis iii) can be replaced by: g n converges pointwise to g, then the result follows from Corollary 3.2 of [21]. The result on the completion holds by Theorem 6. Note that the conditions i), ii) and iii) do not imply converges pointwise from {g n } to g, see example 2 of [8].
For the case of functions defined on a finite interval we get Theorem 16, and a lemma of Riemann-Lebesgue type for functions in the Henstock-Kurzweil space completion. by the integral definition on the completion. We will prove that lim k→∞ f (k, n) = b a f g n converges uniformly on n. Let ǫ > 0 be given, there Therefore, by Theorem 5, The following result is a "generalization" of Riemann-Lebesgue Lemma on the completion of the space HK([a, b]), over finite intervals, it also appears in [3].
Considering an similar argument from above proof, it follows that π r f (t) For n ∈ N ∪ {0}, we define the function Φ n (t) = sin(n+1/2)t t/2 for t = 0 and Φ n (0) = 2n + 1, it is called the discrete Fourier Kernel of order n. This kernel provides a very good approximation to the Dirichlet Kernel D n for |t| < 2, but Φ n decreases more rapidly than D n , see [1].
Theorem 19. Let f ∈ HK([0, π]) and r ∈ (0, π]. Then, assuming that any of next limits exist, Proof. Define g : [0, π] → R by Since g ∈ BV([0, π]), f g ∈ HK([0, π]). By Corollary 17, we have Now, by (14), we have By Theorem 18 and (13), Therefore, assuming that any of the limits exist, we have The following result is a characterization of the asymptotic behavior of n − th partial sum of the Fourier series, it can be found in [3].
Corollary 20. Let f ∈ HK([−π, π]) be 2π− periodic. The n − th partial sum of the Fourier series at t has the following asymptotic behavior S n ( Proof. Since S n ( f , t) = π −π f (t + u)D n (u)du, realizing a change of variable (see section 6 of [13]), then by Theorem 19 we get the result.

Henstock-Kurzweil integral transform
The results in this section are based for functions in the vector space BV 0 (R), and they have to [10] as principal reference.
We will introduce some additional terminology in order to facilitate the following results.
If g : R × R → C is a function and s 0 ∈ R, we say that s 0 fulfills hypothesis (H) relative to g if: This condition plays a significant role in the following results. Also, the next theorems can be found in [10].
Theorem 21. Let f : R → R and g : R × R → C be functions. Assume that f ∈ BV 0 (R), and s 0 ∈ R fulfills Hypothesis (H) relative to g, then is well defined for all s in a neighborhood of s 0 .
Proof. Applying Theorem 9 the result holds.
Theorem 22. Let f : R → R and g : R × R → C be functions assume that 1. f belongs to BV 0 (R), g is bounded, and 2. g(t, ·) is continuous for all t ∈ R.
If s 0 ∈ R fulfills Hypothesis (H) relative to g, then the function Γ is continuous at s 0 . there is K 2 > 0 such that for each t > K 2 , Let K = max{K 1 , K 2 }. From Theorem 7, it follows that for every v ≥ K and every s where the second inequality is true due to (15). This implies, since lim t→∞ | f (t)| = 0, that Analogously we have that Therefore, for each s ∈ B δ 1 (s 0 ), Since f is L 1 [−K, K], g is bounded and g(t, ·) is continuous for all t ∈ R. For example, using Theorem 12.12 of [11], it is easy to show that the function is continuous at s 0 . This implies that there is δ 2 > 0 such that for every s ∈ B δ 2 (s 0 ), Let δ = min{δ 1 , δ 2 }. Then for all s ∈ B δ (s 0 ), Thus, from (16), (17) and (18),   So by Hake's theorem, Theorem 24. Let f ∈ BV 0 (R) and g : R × R → C be a function such that its partial derivative D 2 g is bounded and continuous on R × R. If s 0 ∈ R is such that 1. there is K > 0 for which g(·, s 0 ) [u,v] ≤ K for all [u, v] ⊆ R, and 2. s 0 satisfies Hypothesis (H) relative to D 2 g.
Then Γ is derivable at s 0 , and Proof. Using conditions (1) and (2) and the Mean Value theorem, there exist δ > 0 and M > 0 such that, for each s ∈ (s 0 − δ, for all [u, v] ⊆ R. Let a, b be real numbers with s 0 − δ < a < s 0 < b < s 0 + δ. We use Theorem 12 to prove (19). The function for all t ∈ R. By (20) and Theorem 9, f (·)g(·, s) is HK-integrable on R for all s ∈ [a, b]. Then is continuous at s 0 , and The first affirmation is true by (20) and Theorem 22, and the second affirmation is true due to (20) and Theorem 23

Some applications
An important work about the Fourier transform using the Henstock-Kurzweil integral: existence, continuity, inversion theorems etc. was published in [5]. Nevertheless, there are some omissions in that results that use the Lemma 25 (a) of [5]. Also the authors of this book chapter in [6], [3] and [4] have studied existence, continuity and Riemann-Lebesgue lemma about the Fourier transform of functions belong to HK(R) ∩ BV(R) and BV 0 (R). Following the line of [6], in Theorem 26 we include some results from them as consequences of theorems 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 for all x such that the integral exists. Several 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.
Proof. Since f is a bounded variation function on R then the limit of f (x), as |x| → ∞, exists.
Suppose that lim ∞)). Therefore the constant function α − ǫ is Lebesgue integrable on [A, ∞), which is a contradiction.
Observe, as consequence of above Lemma, we have that the vector space HK(R) ∩ BV(R) is contained in BV 0 (R). So the next theorem is an immediately consequence of above section.

f is continuous on
for all [u, v] ⊆ R. Thus, each s 0 = 0 satisfies Hypothesis (H) relative to e −its .
(a) Theorem 21 implies that f (s 0 ) exists for all s 0 = 0 and, since f ∈ HK(R), f (0) exists. Therefore f exists on R.
(b) By Theorem 22, f is continuous at s 0 , for all s 0 = 0.
(c) It follows by Theorem 12 in similar way to the proof of Theorem 24.
where s is a fixed real number. We get, for each y ∈ R and all So, every real number y satisfies Hypothesis (H) relative to k. Now, observe that h ∈ BV 0 (R) and k is measurable and bounded. Thus, by Theorem 23, for all a > 0.
2. If F(x, y) = L( f )(x + iy), then F(·, y) is continuous on R for all y = 0, and F(x, ·) is continuous on R for all x = 0.

A set of functions in HK(R) ∩ BV 0 (R) \ L 1 (R)
Taking into account Lemma 25, the set HK(R) ∩ BV(R) is included in BV 0 (R) and does not have inclusion relations with L 1 (R). Since the step functions belong to HK(R) ∩ BV(R), then by Lemma 13, we have that HK(R) ∩ BV(R) is dense in HK(R). In this section we exhibit a set of functions in HK(R) ∩ BV 0 (R) \ L 1 (R). be bounded on [a, ∞). Moreover, assume that ϕ : [a, ∞) → R is a nonnegative and monotone decreasing function which satisfies the next conditions: HK([a, ∞)).
Corollary 29 and Proposition 30 provide us Henstock-Kurzweil integrable functions defined on unbounded intervals which are not Lebesgue integrable.
Taking into account the above functions we have the following corollary.
From the above example belongs to HK(R) ∩ BV(R) \ L( R). By the Multiplier theorem it follows that HK(R) ∩ BV(R) ⊂ L 2 (R), so the above function is in BV 0 (R) ∩ L 2 (R) \ L( R). Therefore, there exist functions in L 2 (R) \ L( R) such that their Fourier transforms exist as in (1), as an integral in HK sense.

The Riemann-Lebesgue Lemma and the Dirichlet-Jordan Theorem for BV 0 functions
The Riemann-Lebesgue lemma is a fundamental result of the Harmonic Analysis. An novel aspect is the validity of this lemma for functions which are not Lebesgue integrable, since this fact could help to expand the space of functions where the inversion of the Fourier transform is possible. In this section we prove a generalization of the Riemann-Lebesgue Lemma for functions of bounded variation which vanish at infinity. As consequence, it is obtained a proof of the Dirichlet-Jordan theorem for this kind of functions. This theorem provides a pointwise inversion of the Fourier transform.
We observe that the implications 1 and 2 of Theorem 26 are particularizations of the next result.
Theorem 34 (Generalization of Riemann-Lebesgue Lemma). Let ϕ ∈ HK loc (R) be a function such that where d f i (t) is the Lebesgue-Sieltjes measure generated by f i , i = 1, 2.
Let β a positive number and let M the upper bound of |Φ|. For w ∈ [β, ∞) we have that Since Φ(wt)/w is continuous over [β, ∞) and the measures d f i (t) are finite, then by the Dominated Convergence Theorem applied to right side integrals in (28), it follows that for each w 0 ∈ [β, ∞). Since β is arbitrary, we obtain the continuity of H on (0, ∞).
Moreover, by (28), we have for w ∈ (0, ∞) that Thus, we conclude that To complete the proof, we use similar arguments for the interval (−∞, 0]. The above theorem confirms that H ∈ C 0 (R \ {0}), for each f ∈ BV 0 (R). As corollary we have the Riemann-Lebesgue Lemma.
We know that if g, h ∈ BV([a, ∞]) then gh ∈ BV([a, ∞]). Employing this fact and Theorem 34 we get the following corollary.
Corollary 36. Suppose that δ, α > 0 and f ∈ BV(R), then The Sine Integral is defined as which has the properties: 1. Si(0) = 0, lim x→∞ Si(x) = 1 and We use the Sine Integral function in the proof of the following lemma.
Proof. By Lemma 10 we have Since for each t ∈ [a, ∞): lim ε→0 tε δε sin u u du = 0 and tε δε sin u u du ≤ πSi(π) for all ε > 0. Then, we obtain the result applying the Lebesgue Dominated Convergence theorem to the integral on the right in (30).
Proof. We will do the proof for 0 < α < β. Let f 0b (s) = To conclude the proof, we follow a similar process over the interval [a, 0] leading a to minus infinity.
To obtain the Dirichlet-Jordan theorem we state the following lemma [22,Theorem 11.8]. We conclude the proof combining (33), (34) and the above expression.
We observe that the classical theorem of Dirichlet-Jordan on L(R) is a particular case of Theorem 40. Taking into account that HK(R) ∩ BV(R) ⊂ BV 0 (R), then from Theorem 34 and Theorem 40 we get that if f ∈ HK(R) ∩ BV(R), then its Fourier transform f (s) exists in each s ∈ R; f ∈ C 0 (R\ {0}), and the expression (32) holds for each x ∈ R.
Corollary 41. There exist functions in L 2 (R) \ L( R) such that their Fourier transforms exist as in (1) and, for each x ∈ R, the expression (32) is true.

Conclusions
We present theorems about convergence of integrals of products in the completion of HK(I), which those we have a version of Riemann-Lebesgue Lemma (over compact intervals) and analogous results at Riemann-Lebesgue property, a characterization of behavior of n-th partial sum of the Fourier series. Moreover, we have gotten basic properties (existence as integral, continuity, asymptotic behavior) about Fourier transform using Henstock-Kurzweil Integral, for this was necessary to get a generalization of Riemann-Lebesgue Lemma over BV 0 (R), in particular those characteristics are valid over HK(R) ∩ BV(R). This intersection does not have relation inclusion with Lebesgue integrable functions space, we give a set of functions such that it belongs to HK(R) ∩ BV(R) \ L(R). Finally we have a generalization of Dirichlet-Jordan over BV 0 (R).