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We follow the notation from [3] to introduce basic definitions of the Henstock-Kurzweil inte- gral. Let R = R and [a, b] be a non-degenerative interval in R. A partition P of [a, b] is a finite collection∪{±∞} of non-overlapping intervals such that
[a, b]= I I ... In. 1 ∪ 2 ∪ ∪
Specifically, the partition itself is the set of endpoints of each sub-interval Ii.
a = x x x ... xn = b, 0 ≤ 1 ≤ 2 ≤ ≤ where Ii := [xi−1, xi] for i =1, . . . , n, Observe that [a, b] can be unbounded.
Definition 2.1. A tagged partition of I
n P˙ = (Ii, ti) . { }i=1 n is a finite set of ordered pairs (Ii, ti) , where the collection of subintervals Ii forms a { }i=1 { } partition of I and the point ti Ii is called a tag of Ii. ∈ Definition 2.2. A map δ :[a, b] (0, ) is called gauge function on [a, b]. Given a gauge → ∞ n function δ on [a, b] it is said that a tagged partition P˙ = ([xi , xi]; ti) of [a, b] is δ-fine { −1 }i=1 according to the following cases: For a R and b = : ∈ ∞
1. a = x , b = xn = tn = . 0 ∞
2. [xi , xi] [ti δ(ti), ti + δ(ti)], for all i =1, 2,...,n 1. −1 ⊂ − − 1 3. [xn−1, ] [ , ]. ∞ ⊂ δ(tn ) ∞ For a = and b R: −∞ ∈
1. a = x = t = , b = xn. 0 1 −∞
2. [xi , xi] [ti δ(ti), ti + δ(ti)], for all i =2,...,n. −1 ⊂ − 3. [ , x ] [ , 1 ]. −∞ 1 ⊂ −∞ − δ(t1 ) For a = and b = −∞ ∞
1. a = x = t = , b = xn = tn = . 0 1 −∞ ∞
2. [xi , xi] [ti δ(ti), ti + δ(ti)], for all i =2,...,n 1. −1 ⊂ − − 1 1 3. [xn−1, ] [ , ] and [ , x1] [ , ]. ∞ ⊂ δ(tn ) ∞ −∞ ⊂ −∞ − δ(t1 ) For a, b R: ∈
1. [xi , xi] [ti δ(ti), ti + δ(ti)], for all i =1, 2, . . . , n. −1 ⊂ − 2 According to the convention concerning to the “ arithmetic ” in R, 0 ( ) = 0, a · ±∞ real-valued function f defined over R can be extended by setting f( ) = 0. Thus, it is ±∞ introduced the definition of the Henstock-Kurzweil integral over intervals in R. Definition 2.3. Let [a, b] be an interval in R. The real-valued function f defined over I is said to be Henstock-Kurzweil integrable on [a, b] iff there exists A R such that for every ˙ ∈ n ǫ> 0 there exists a gauge function δǫ over [a, b], such that if P = ([xi−1, xi]; ti) i=1 is a δǫ-fine partition of [a, b], then { } n f(ti)(xi xi−1) A < ǫ. − − Xi=1 b The number A is the integral of f over [a, b] and it is denoted by a f = A. R The set of all Henstock-Kurzweil integrable functions on the interval I is denoted by HK(I), and the set of Henstock-Kurzweil integrable functions over each compact interval is denoted by HKloc(R). The integrals will be in the Henstock-Kurzweil sense, if not specified. The Multiplier Theorem in [3] states that the bounded variation functions are the mul- tipliers of the Henstock-Kurzweil integrable functions. Moreover, this concept is related to the Riemann-Stieltjes integral, which generalizes the Riemann integral and also it is useful to calculate the Fourier transform, see Theorem 4.1 below. There exist several versions of the Riemann-Stieltjes integral. In this work it is considered the Riemann-Stieljes (δ)-integral, also called norm Riemann-Stieltjes integral [19, 22]. Here for simplicity, it is called as Riemann- Stieltjes integral. The set of bounded variation functions over I R is denoted by BV (I), and BV (R) ⊆ 0 denotes the functions in BV (R) vanishing at infinity, [15, 20].
2.1 The Fourier transform operator in the classical sense We will enunciate basic results about the Fourier Analysis in the classical sense, i.e. conside- ring Lebesgue’s integral. Definition 2.4. Let 1 p< . For any Lebesgue measurable function f : R R we define ≤ ∞ → 1 p p f p = f dµ . k k ZR | |
For each p 1, the set of functions f such that f p < (called p-integrable functions) ≥ k k ∞ p is a normed space (considering equivalence classes respect to p) and is denoted by (R). The Fourier transform has been developed in the context ofk·k the Lebesgue theory andL over the spaces p(R), with important implications in different areas, e.g. optics, signal theory, statistics, probabilityL theory [4]. The set 1(R) is well known as the space of Lebesgue integrable functions on R or ab- L solutely integrable functions. 2(R) is a normed space with an inner product that provides algebraic and geometric techniquesL applicable to spaces of arbitrary dimension. These spaces are usually considered to define the Fourier transform [9, 25, 26]. Definition 2.5. Let f 1(R). The Fourier transform of f at the point s is defined as ∈L ∞ 1 −isx 1(f)(s)= e f(x)dx. F √2π Z−∞
3 It is well known that (f) is pointwise defined and by the Riemann-Lebesgue Lemma, F1 (f) belongs to C (R), the set of complex-valued continuous functions on R vanishing at F1 ∞ [23, 25]. On the other hand, the space 2(R) is not contained in 1(R). Thus, the ±∞ L L operator is not well defined over 2(R). Nevertheless, the Fourier transform on 2(R) is F1 L L given as an extension of the Fourier transform (f) initially defined on 1(R) 2(R). It F1 L ∩L means that the Fourier transform over 2(R) is defined as a limit [6], and is denoted as . L F2 Definition 2.6 ([9, 24, 25]). The Fourier transform operator in p(R), 1
At the beginning of this century, the Fourier theory was developed using the Henstock- Kurzweil theory. In [29], E. Talvila showed some existence theorems and continuity of the Fourier transform over Henstock-Kurzweil integrable functions. Moreover, the Fourier trans- form has been studied as a Henstock-Kurzweil integral over non-classical spaces of functions [21, 23]. It is well known that if I is a compact interval, then BV (I) 1(I) HK(I). ⊂L ⊂ However, when I is an unbounded interval, BV (I) * 1(I) L and 1(I) * HK(I) BV (I). L ∩ Thereby, when I is unbounded, there is no inclusion relation between 1(I) and HK(I) L ∩ BV (I). On the other hand, BV (I) HK(I) 2(I) (by the Multiplier Theorem [3]). In ∩ ⊂ L [28, Lemma 4.1] it is proved that BV (I) HK(I) BV (R). Accordingly, it is possible to ∩ ⊂ 0 analyze the Fourier transform via the Henstock-Kurzweil integral over BV0(R). Thus, in [21] it was shown a generalized Riemann-Lebesgue lemma on unbounded intervals, giving rise to the definition of the HK-Fourier transform [23]. Definition 3.1. 1(R)+ BV (R) denotes the vector space of functions f = f + f , where L 0 1 2 f 1(R) and f BV (R). 1 ∈L 2 ∈ 0 Definition 3.2. The HK-Fourier transform is defined as 1 HK : (R)+ BV (R) C (R 0 ), F L 0 → ∞ \{ }
1 −isx HK(f)(s) := e f(x)dx F √2π ZR 1 1 = cos(sx)f(x)dx i sin(sx)f(x)dx (3.1) √2π ZR − √2π ZR C (f)(s) i S (f)(s) ≡ FHK − FHK C S where the integrals are in Henstock-Kurzweil sense. HK (f) and HK(f) are called the HK-Cosine Fourier and the HK-Sine Fourier transformsF of f, respectively.F
4 Proposition 1. The HK-Fourier transform is well defined.
1 Proof. Suppose f = u +v = u +v with ui (R) and vi BV (R) for i =1, 2. Therefore, 1 1 2 2 ∈L ∈ 0 u u = v v 1(R) BV (R). 1 − 2 2 − 1 ∈L ∩ 0 This yields the result since the Henstock-Kurzweil integral coincides with the Lebesgue 1 integral on the intersection (R) BV (R), see [23]. Therefore, HK(f) does not depend on L ∩ 0 F the representation of the function f.
Note that some integrals in (3.1) might not converge at s = 0. Recently, in [1] was shown that the HK-Fourier Cosine transform is a bounded linear operator from BV0(R) into HK(R). This is related to the question about whether the HK-Fourier transform is continuous at s = 0. By [21, Theorem 2.5.] the HK-Fourier Cosine and Sine transforms of C S any function f in BV0(R), HK(f) and HK(f) are continuous functions (except at s = 0) and vanish at infinity as o( Fs −1). F | | Theorem 3.3 ([23]). If f p(R) ( 1(R)+ BV (R)), for 1 p 2, then ∈L ∩ L 0 ≤ ≤ q HK(f) (R) C (R 0 ), F ∈L ∩ ∞ \{ } where p−1 + q−1 =1. Moreover,
p(f)(s)= HK(f)(s), F F almost everywhere. In particular, if f BV (R), then ∈ 0
HK(f) C (R 0 ). F ∈ ∞ \{ } 4 An approach of via Fp FHK According to the classical theory, it is not always possible to achieve a pointwise expression of the Fourier transform operator p in Lebesgue’s theory of integration, for 1
p Theorem 4.1. If φ (R) BV (R) ACloc(R), for 1 p 2, then ∈L ∩ 0 ∩ ≤ ≤
(i) HK(φ) C (R 0 ). F ∈ ∞ \{ }
(ii) HK(φ)(s)= p(φ)(s) a.e. F F (iii) For every s R 0 , ∈ \{ } i ′ HK(φ)(s)= (φ )(s). (4.1) F −sF1
5 (iv) Moreover, 1 1 ′ HK(φ)(s) φ 1 . |F | ≤ √2π · s k k | | 2 Proof. Let φ (R) BV (R) ACloc(R). By Theorem 3.3 we get (i) and (ii). Note that ∈ L ∩ 0 ∩ HK(φ)(s) is defined for any s = 0, whereas p(φ)(s) is defined almost everywhere. Applying F 6 F the Hake Theorem we get, 1 T T HK(φ)(s)= lim cos(st)φ(t)dt i sin(st)φ(t)dt . (4.2) F √2π T →∞ Z−T − Z−T From the Multiplier Theorem [3] and the hypothesis for φ we have T sin(st) + sin(sT ) cos(st)φ(t)dt = lim dφ. T ZR − →∞ Z−T s Similarly for the Sine Fourier transform we get T cos(st) + cos(sT ) sin(st)φ(t)dt = lim − dφ. T ZR − →∞ Z−T s Where we have used that φ BV (R) vanishes at infinity. This yields, from (4.2), ∈ 0 1 T sin(st) + sin(sT ) HK(φ)(s) = lim dφ F − T →∞ √2π Z−T s T cos(st) + cos(sT ) + i lim − dφ. T →∞ Z−T s Note that the Stieltjes-type integrals below exist as Riemann-Stieltjes and Lebesgue-Stieltjes integrals [2, 10, 22]. Since φ ACloc(R), by [22, Theorem 6.2.12] and [19, Exercise 2, pag. 186] it follows that ∈ T T (sin(st) + sin(sT ))dφ = (sin(st) + sin(sT )) φ′(t)dt Z−T Z−T and T T ( cos(st) + cos(sT ))dφ = ( cos(st) + cos(sT )) φ′(t)dt. Z−T − Z−T −
Since φ BV (R) ACloc(R), one gets ∈ 0 ∩ T lim (cos(sT ) sin(sT )) φ′(t)dt =0. (4.3) T →∞ − Z−T and [13, Corollary 2.23] implies that φ′ 1(R). Therefore, we get ∈L T 1 1 ′ HK(φ)(s) = lim [sin(st)+ i cos(st)]φ (t)dt F √2π s T →∞ − Z−T 1 = (φ′)(s). isF1 Furthermore, 1 1 ′ HK(φ)(s) φ 1 . |F | ≤ √2π · s k k | | For 1 p< 2 the same formulas and argumentation are valid. ≤ 6 Remark 4.2. Since f BV (R) ACloc(R), by [3, Theorem 7.5, pag. 281] and [13, Theorem ∈ 0 ∩ 3.39] we have that φ′ = V ar(φ, R) and φ AC(R). Thus, BV (R) AC(R)= BV (R) || ||1 ∈ 0 ∩ 0 ∩ ACloc(R). From Theorem 4.1 we have the following result.
p Corollary 1. Let φ (R) BV (R) ACloc(R), for 1 p 2. ∈L ∩ 0 ∩ ≤ ≤ (i) If φ is an even function, then
∞ 2 1 ′ HK(φ)(s)= sin(st)φ (t)dt. (4.4) F −rπ · s Z0 (ii) If φ is an odd function, then
∞ 2 1 ′ HK(φ)(s)= i [cos(st) 1] φ (t)dt. (4.5) F − rπ · s Z0 −
In either case, HK(φ)(s)= p(φ)(s) a.e., where HK(φ) C (R 0 ). F F F ∈ ∞ \{ } E. Liflyand in [14]-[17] worked on a subspace of BV (R) ACloc(R) to obtain integra- 0 ∩ bility and asymptotic formulas for the Fourier transform. We restrict to the domain of the Fourier transform operator in order to provide new integral expressions of the classical Fourier transform p. F The implications from these results are that the classical Fourier transform p(f)(s) for f F in a dense subspace of p(R) is represented by a Lebesgue integral, is a continuous function, except at s = 0 and vanishesL at infinity as o( s −1). | | The algorithms of numerical integration are very important in applications, for example, approximation of the Fourier transform have implications in digital image processing, eco- nomic estimates, acoustic phonetics, among others [4], [27]. There exist integrable functions whose primitives cannot be calculated explicitly; thus numerical integration is fundamental to achieve explicit results. Also note that the Lebesgue integral is not suitable for numerical approximations. Alter- natively, (4.4) and (4.5) provide expressions that might be used to approximate numerically p(f) at specific values. Actually, as a consequence of the Hake Theorem, it is possible to F approximate HK(φ)(s) via the relation F
1 −ist HK(φ)(s) e φ(t)dt (M ), (4.6) F ≈ √2π Z|t|≤M →∞ for any s = 0, φ p(R) BV (R) 1(R) (1
5 Differentiability of the Fourier transform
A classical theorem in Lebesgue’s theory is about differentiability under the integral sign [26] and [7]. The following result is a generalization.
7 Theorem 5.1. Let f 1(R)+ BV (R) such that g(t) := tf(t) belongs to 1(R)+ BV (R). ∈L 0 L 0 Then HK(f) is continuously differentiable away from zero and F d HK(f)(s)= i HK(g)(s), (s = 0). (5.1) dsF − F 6 Proof. For the case f,g BV (R), ∈ 0 let us define G(s, t) := cos(st)f(t) in 1 (R) with respect to s for all t R, where [α, β] is Lloc ∈ any compact interval such that 0 [α, β] and let us consider the sequence (Φn) where 6∈ n d Φn(s) := G(s, t)dt (5.2) Z−n ds with n N. Since ∈ 2 Φn(s) V ar(g, R), | | ≤ s | | 1 then Φn L [α, β], where [α, β] is any compact interval such that 0 [α, β]. Applying the ⊂ 6∈ Dominated Convergence Theorem, Fubini’s Theorem and Hake’s Theorem [3], we get
1 β ∞ d 1 β ∞ G(s, t) dt ds = sin(st)g(t) dt ds √2π Zα Z−∞ ds √2π Zα Z−∞ − 1 β = lim Φn(s) ds (5.3) √2π Zα n→∞ 1 n = lim f(t) cos(βt) cos(αt) dt n √2π →∞ Z−n h − i = C (f)(β) C (f)(α). FHK −FHK On the other hand, 1 ∞ β d 1 ∞ G(s, t) ds dt = cos(αt)f(t) cos(βt)f(t)dt √2π Z−∞ Zα ds √2π Z−∞ − = C (f)(β) C (f)(α). (5.4) FHK −FHK From (5.3), (5.4) and [30, Theorem 4], we get that the HK-Cosine Fourier transform is C ′ differentiable under the integral sing. Since g BV0(R), by Theorem 3.3, HK(f) is a con- ∈ F S ′ tinuous function (except at s = 0) vanishing at infinity. By similar arguments, HK(f) (s)= C F HK(g)(s) for any s = 0. F 6 1 For the general case, we suppose tf = g1+g2 (R)+BV0(R). Then (5.2) with g = g1+g2 1 ∈L obeys also Φn L [α, β], so that (5.3) remains valid. Therefore, the HK-Fourier transform { } ⊂ is differentiable and (5.1) is obtained. Corollary 2. Under the assumptions of Theorem 5.1. Then
S (f) ACG∗ (R). FHK ∈ loc Proof. Theorem 5.1 implies that S (f) is a continuously differentiable function away from FHK zero, and [1, Corollary 1] yields that its derivate is actually a function in HKloc(R). Therefore [30, Theorem 2] gives the result.
8 We will extend this theorem to study differentiability of the Fourier transform p(f) for F 1 p 2. ≤ ≤ p Lemma 5.2. Suppose 1 p 2 and f (R). Then there exists a subsequence (nk) N such that ≤ ≤ ∈ L ∈ nk 1 −isx lim e f(x)dx = p(f)(s), (5.5) k→∞ √2π Z−nk F almost everywhere on R. Proof. The cases p =1 or p = 2 follow from [25, 26]. For 1
p(f)= (f )+ (f ). F F1 1 F2 2 Applying once again [25, 26], we obtain a sequence (nk) N such that ⊂ nk 1 −isx lim e f(x)dx = 2(f2)(s) k→∞ √2π Z−nk F almost everywhere in R. This yields,
nk nk 1 −isx 1 −isx lim e f(x)dx = lim e f1(x)dx k→∞ k→∞ √2π Z−nk √2π Z−nk nk 1 −isx + lim e f2(x)dx k→∞ √2π Z−nk = (f )(s)+ (f )(s) F1 1 F2 2 = p(f)(s), F almost everywhere. This proves the statement. Below we use the notation p−1 + q−1 = 1. Proposition 2. Let 1 p 2 be fixed. If f p(R) and g(t) := tf(t) belongs to 1(R)+ ≤ ≤ ∈L L BV (R), then by redefining p(f)(s) on a set of measure zero, it yields 0 F d p(f)(s)= i HK (g)(s), (s = 0). dsF − F 6 Proof. Take values s = α, and s = β such that (5.5) is valid. Suppose that 0 <α<β. Proceeding similarly as in Theorem 5.1, we have
β β nk i −ist i HK(g)(s)ds = lim − e tf(t)dtds k→∞ − Zα F Zα √2π Z−nk i nk β = − lim e−isttf(t)dsdt k→∞ √2π Z−nk Zα i nk = − lim (e−iβt e−iαt)f(t))dt k→∞ √2π Z−nk − = p(f)(β) p(f)(α), (5.6) F −F where (5.6) holds almost everywhere by Lemma 5.2. This implies the statement of the propo- sition.
9 Corollary 3. Assume f p(R) and tf p(R) BV (R). Then, by redefining S(f) on ∈ L ∈ L ∩ 0 Fp a set of measure zero yields
S(f) ACG∗(R). Fp ∈ Proof. This follows from Proposition 2, [1, Corollary 1] and [30, Theorem 2].
p p Proposition 3. Let 1 < p 2 be fixed, f (R) and tf = hp + h (R)+ BV (R). ≤ ∈ L 0 ∈ L 0 Then, by redefining p(f)(s) on a set of measure zero, it yields F q p(f) ACloc(R 0 ) (R) F ∈ \{ } ∩L and d p(f)(s)= i[ p(hp)(s)+ HK (h )(s)], a.e. dsF − F F 0
Proof. Due to n 1 −ist e hp(t)dt p(hp) (n ), √2π Z−n →F →∞ there exists M > 0 such that n 1 −ist e hp(t)dt M < √2π Z−n q ≤ ∞
uniformly on n N. As argued in equation (5.3), ∈
β nk β nk β i −ist −ist i p(hp)(s)+ HK(h0)(s)ds = − lim e hp(t)dsdt + lim e h0(t)dsdt k→∞ k→∞ − Zα F F √2π Z−nk Zα Z−nk Zα nk β i −ist = − lim e (hp + h0)(s)dsdt k→∞ √2π Z−nk Zα i nk β = − lim e−isttf(t)dsdt k→∞ √2π Z−nk Zα 1 nk = lim (e−iβt e−iαt)f(t)dt k→∞ √2π Z−nk − = p(f)(β) p(f)(α). F −F
Where we take a subsequence of (nk), if neccessary. Here α, β are values such that (5.5) is valid.
S Corollary 4. Assume the hypothesis of Proposition 3. Then, by redefining p (f) on a set of measure zero F S(f) ACG∗ (R). Fp ∈ loc Proof. Similar arguments as above give the result.
Now we show some examples.
10 Example 5.3. Let φ(t) := (1 + t2)−1 on [0, ) and zero otherwise. It is easy to see that φ 1 2 ∞ 1 belongs to (R) (R). Moreover, g(t)= tφ(t) belongs to BV0(R) (R). By Proposition 2 we have L ∩L \L d d (φ)(s)= (φ)(s)= i HK (g)(s) (s = 0). dsF2 dsF1 − F 6
Example 5.4. Let φ(t) = arctan t π . Note that φ BV (R) 2(R) 1(R). However, | | − 2 ∈ 0 ∩L \L tφ(t) does not belong to 1(R)+ BV (R). By Corollary 1 we have that L 0 2 1 (φ)(s)= S(τ ′)(s), F2 −rπ · sF1 ′ 1 2 S ′ S ′ where τ(t) = φ χ(0,∞)(t). Note that τ (R) (R), hence 1 (τ ) = 2 (τ ). Applying Proposition 2 to· S(τ ′) we have that ∈ L ∩L F F F2 d S(τ ′)(s)= C (g)(s), dsF2 FHK here g(t)= tτ ′(t) BV (R). Thus, (φ) is a continuously differentiable function away from ∈ 0 F2 zero and d 2 1 1 (φ)(s)= S(τ ′)(s) C (g)(s) , a.e. dsF2 rπ s2 F1 − sFHK
3 Example 5.5. Let h1(t) := 1 C(2/π arctan(t)) and h2(t) := √t sin(1/t), where C is the Cantor function [5] . We take − t−1 h (t) if t> 2, · 1 f(t)= h2(t) if 0 An integral representation of the Fourier transform is obtained on the subspace p(R) 1 L ∩ BV (R) ACloc(R) (R), for 1 < p 2. This is possible by switching to the Henstock- 0 ∩ \L ≤ Kurzweil integral. Furthermore, expressions (4.4) and (4.5) give explicit formulas of p over that subspace. Using our results, specific values of the Fourier transform of particularF func- tions might be approximated with arbitrary accuracy. Moreover, it was shown differentiability of p(φ)(s) by extending a classical theorem in Lebesgue’s theory. This illustrates the appli- F cability of the results obtained, which are original in Fourier Analysis over p(R). L 11 Acknowledgements The first author is supported by GA 20-11846S of the Czech Science Foundation. J.H.A. and M.B. acknowledge partial support from CONACyT–SNI. References [1] Arredondo, J. 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Henstock-Kurzweil integral transforms. Int. J. Math. Math. Sci. 2012 Art. ID 209462, 11 pp. https://doi.org/10.5772/59766 13 [29] Talvila, E. Henstock-Kurzweil Fourier transforms. Illinois J. Math. 46 (2016), no. 4, 1207–1226. http://dx.doi.org/10.1155/2012/209462 [30] Talvila, E. Necessary and sufficient conditions for differentiating under the integral sign. Amer. Math. Monthly 108 (2001), no. 6, 544–548. https://www.jstor.org/stable/2695709 [31] Wiener, N. and Wintner, A. Fourier-Stieltjes transforms and singular infinite convolu- tions. Amer. J. Math. 60 (1938), 513522. https://www.jstor.org/stable/2371591 J. H. Arredondo Departamento de Matem´aticas, Universidad Aut´onoma Metropolitana - Iztapalapa Av. San Rafael Atlixco 186, M´exico City, 09340, M´exico. e-mail: [email protected] M. G. Morales Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotl´aˇrsk´a2, 611 37 Brno, Czech Republic. e-mail: [email protected] M. Bernal Departamento de Matem´aticas, Universidad Aut´onoma Metropolitana - Iztapalapa Av. San Rafael Atlixco 186, M´exico City, 09340, M´exico. e-mail: [email protected] 14