# Convolutions with the Continuous Primitive Integral

Authors: Erik Talvila
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

IfF is a continuous function on the real line and f=F′ is its distributional derivative, then the continuous primitive integral of distribution f is ∫abf=F(b)−F(a). This integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals. Under
the Alexiewicz norm, the space of integrable distributions is a Banach space. We define the
convolution f∗g(x)=∫−∞∞f(x−y)g(y)dy for f an integrable distribution and g a function of bounded variation or an L1 function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. For g of bounded variation,
f∗g is uniformly continuous and we have the estimate ∥f∗g∥∞≤∥f∥∥g∥ℬ𝒱, where ∥f∥=supI|∫If| is the Alexiewicz norm. This supremum is taken over all intervals
I⊂ℝ. When g∈L1, the estimate is ∥f∗g∥≤∥f∥∥g∥1. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.


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## Body

## 1. Introduction and Notation

The convolution of two functionsf and g on the real line is f*g(x)=∫-∞∞f(x-y)g(y)dy. Convolutions play an important role in pure and applied mathematics in Fourier analysis, approximation theory, differential equations, integral equations, and many other areas. In this paper, we consider convolutions for the continuous primitive integral. This integral extends the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals on the real line and has a very simple definition in terms of distributional derivatives.Some of the main results for Lebesgue integral convolutions are that the convolution defines a Banach algebra onL1 and *:L1×L1→L1 such that ∥f*g∥1≤∥f∥1∥g∥1. The convolution is commutative, associative, and commutes with translations. If f∈L1 and g∈Cn, then f*g∈Cn and (f*g)(n)(x)=f*g(n)(x). Convolutions also have the approximation property that if f∈Lp (1≤p<∞) and g∈L1, then ∥f*gt-af∥p→0 as t→0, where gt(x)=g(x/t)/t and a=∫-∞∞g. When f is bounded and continuous, there is a similar result for p=∞. For these results see, for example, [1]; see [2] for related results with the Henstock-Kurzweil integral. Using the Alexiewicz norm, all of these results have generalizations to continuous primitive integrals that are proven in what follows.We now define the continuous primitive integral. For this, we need some notation for distributions. The space oftest functions is 𝒟=Cc∞(ℝ)={ϕ:ℝ→ℝ∣ϕ∈C∞(ℝ)andsupp(ϕ)iscompact}. The support of function ϕ is the closure of the set on which ϕ does not vanish and is denoted supp(ϕ). Under usual pointwise operations, 𝒟 is a linear space over field ℝ. In 𝒟, we have a notion of convergence. If {ϕn}⊂𝒟, then ϕn→0 as n→∞ if there is a compact set K⊂ℝ such that for each n, supp(ϕn)⊂K, and for each m≥0, we have ϕn(m)→0 uniformly on K as n→∞. The  distributions  are denoted 𝒟′ and are the continuous linear functionals on 𝒟. For T∈𝒟′ and ϕ∈𝒟, we write 〈T,ϕ〉∈ℝ. For ϕ,ψ∈𝒟 and a,b∈ℝ, we have 〈T,aϕ+bψ〉=a〈T,ϕ〉+b〈T,ψ〉. Moreover, if ϕn→0 in 𝒟, then 〈T,ϕn〉→0 in ℝ. Linear operations are defined in 𝒟′ by 〈aS+bT,ϕ〉=a〈S,ϕ〉+b〈T,ϕ〉 for S,T∈𝒟′; a,b∈ℝ and ϕ∈𝒟. If f∈Lloc1, then 〈Tf,ϕ〉=∫-∞∞f(x)ϕ(x)dx defines a distribution Tf∈𝒟′. The integral exists as a Lebesgue integral. All distributions have derivatives of all orders that are themselves distributions. For T∈𝒟′ and ϕ∈𝒟, the  distributional derivative of T is T′ where 〈T′,ϕ〉=-〈T,ϕ′〉. This is also called the  weak derivative. If p:ℝ→ℝ is a function that is differentiable in the pointwise sense at x∈ℝ, then we write its derivative as p′(x). If p is a C∞ bijection such that p′(x)≠0 for  any x∈ℝ, then the composition  with  distribution T is  defined  by 〈T∘p,ϕ〉=〈T,(ϕ∘p-1)/(p′∘p-1)〉 for all ϕ∈𝒟. Translations are a special case. For x∈ℝ, define the  translationτx on  distribution T∈𝒟′ by 〈τxT,ϕ〉=〈T,τ-xϕ〉 for test function ϕ∈𝒟, where τxϕ(y)=ϕ(y-x). All  of  the  results  on distributions we use can be found in [3].The following Banach space will be of importance:ℬC={F:ℝ→ℝ∣F∈C0(ℝ),F(-∞)=0,F(∞)∈ℝ}. We use the notation F(-∞)=limx→-∞F(x) and F(∞)=limx→∞F(x). The extended real line is denoted by ℝ¯=[-∞,∞]. The space ℬC then consists of functions continuous on ℝ¯ with a limit of 0 at -∞. We denote the functions that are continuous on ℝ that have real limits at ±∞ by C0(ℝ¯). Hence, ℬC is properly contained in C0(ℝ¯), which is itself properly contained in the space of uniformly continuous functions on ℝ. The space ℬC is a Banach space under the uniform norm; ∥F∥∞=supx∈ℝ|F(x)|=maxx∈ℝ¯|F(x)| for F∈ℬC. The continuous primitive integral is defined by taking ℬC as the space of primitives. The space of integrable distributions is 𝒜C={f∈𝒟′∣f=F′forF∈ℬC}. If f∈𝒜C, then ∫abf=F(b)-F(a) for a,b∈ℝ¯. The distributional differential equation T′=0 has only constant solutions so the primitive F∈ℬC satisfying F′=f is unique. Integrable distributions are then tempered and of order one. This integral, including a discussion of extensions to ℝn, is described in [4]. A more general integral is obtained by taking the primitives to be regulated functions, that is, functions with a left and right limit at each point, see [5].Examples of distributions in𝒜C are Tf for functions f that have a finite Lebesgue, Henstock-Kurzweil, or wide Denjoy integral. We identify function f with the distribution Tf. Pointwise function values can be recovered from Tf at points of continuity of f by evaluating the limit 〈Tf,ϕn〉 for a delta sequence converging to x∈ℝ. This is a sequence of test functions {ϕn}⊂𝒟 such that for each n, ϕn≥0, ∫-∞∞ϕn=1, and the support of ϕn tends to {x} as n→∞. Note that if F∈C0(ℝ¯) is an increasing function with F′(x)=0 for almost all x∈ℝ, then the Lebesgue integral ∫abF′(x)dx=0 but F′∈𝒜C and ∫abF′=F(b)-F(a). For another example of a distribution in 𝒜C, let F∈C0(ℝ¯) be continuous and nowhere differentiable in the pointwise sense. Then F′∈𝒜C and ∫abF′=F(b)-F(a) for all a,b∈ℝ¯.The space𝒜C is a Banach space under the Alexiewicz norm; ∥f∥=supI⊂ℝ|∫If|, where the supremum is taken over all intervals I⊂ℝ. An equivalent norm is ∥f∥′=supx∈ℝ|∫-∞xf|. The continuous primitive integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals since their primitives are continuous functions. These three spaces of functions are not complete under the Alexiewicz norm and in fact 𝒜C is their completion. The lack of a Banach space has hampered application of the Henstock-Kurzweil integral to problems outside of real analysis. As we will see in what follows, the Banach space 𝒜C is a suitable setting for applications of nonabsolute integration.We will also need to use functions of bounded variation. Letg:ℝ→ℝ. The variation of g is Vg=sup∑|g(xi)-g(yi)| where the supremum is taken over all disjoint intervals {(xi,yi)}. The functions of bounded variation are denoted ℬ𝒱={g:ℝ→ℝ∣Vg<∞}. This is a Banach space under the norm ∥g∥ℬ𝒱=|g(-∞)|+Vg. Equivalent norms are ∥g∥∞+Vg and |g(a)|+Vg for each a∈ℝ¯. Functions of bounded variation have a left and right limit at each point in ℝ and limits at ±∞, so, as above, we will define g(±∞)=limx→±∞g(x).Ifg∈Lloc1, then the essential variation of g is  essvarg=sup∫-∞∞gϕ′, where the supremum is taken over all ϕ∈𝒟 with ∥ϕ∥∞≤1. Then ℰℬ𝒱={g∈Lloc1∣essvarg<∞}. This is a Banach space under the norm ∥g∥ℰℬ𝒱=esssup|g|+essvarg. Let 0≤γ≤1. For g:ℝ→ℝ, define gγ(x)=(1-γ)g(x-)+γg(x+). For left continuity, γ=0 and for right continuity γ=1. The functions  of  normalized bounded variation are 𝒩ℬ𝒱γ={gγ∣g∈ℬ𝒱}. If g∈ℰℬ𝒱, then essvarg=infVh such that h=g almost everywhere. For each 0≤γ≤1, there is exactly one function h∈𝒩ℬ𝒱γ such that g=h almost everywhere. In this case, essvarg=Vh. Changing g on a set of measure zero does not affect its essential variation. Each function of essential bounded variation has a distributional derivative that is a signed Radon measure. This will be denoted μg where 〈g′,ϕ〉=-〈g,ϕ′〉=-∫-∞∞gϕ′=∫-∞∞ϕdμg for all ϕ∈𝒟.We will see that*:𝒜C×ℬ𝒱→C0(ℝ¯) and that ∥f*g∥∞≤∥f∥∥g∥ℬ𝒱. Similarly for g∈ℰℬ𝒱. Convolutions for f∈𝒜C and g∈L1 will be defined using sequences in ℬ𝒱∩L1 that converge to g in the L1 norm. It will be shown that *:𝒜C×L1→𝒜C and that ∥f*g∥≤∥f∥∥g∥1.Convolutions can be defined for distributions in several different ways.Definition 1.1.
LetS,T∈𝒟′ and ϕ,ψ∈𝒟. Define ϕ̃(x)=ϕ(-x): (i) 〈T*ψ,ϕ〉=〈T,ϕ*ψ̃〉, (ii) for each x∈ℝ, let T*ψ(x)=〈T,τxψ̃〉; (iii) 〈S*T,ϕ〉=〈S(x),〈T(y),ϕ(x+y)〉〉.In (i),*:𝒟′×𝒟→𝒟′. This definition also applies to other spaces of test functions and their duals, such as the Schwartz space of rapidly decreasing functions or the compactly supported distributions. In (ii), *:𝒟′×𝒟→C∞. In [1], it is shown that definitions (i) and (ii) are equivalent. In (iii), *:𝒟′×𝒟′→𝒟′. However, this definition requires restrictions on the supports of S and T. It suffices that one of these distributions has compact support. Other conditions on the supports can be imposed  (see [3, 6]). This definition is an instance of the tensor product, 〈S⊗T,Φ〉=〈S(x),〈T(y),Φ(x,y)〉〉, where now Φ∈𝒟(ℝ2).Under (i),T*ψ is in C∞. It satisfies (T*ψ)*ϕ=T*(ψ*ϕ), τx(T*ψ)=(τxT)*ψ=T*(τxψ), and (T*ψ)(n)=T*ψ(n)=T(n)*ψ. Under (iii), with appropriate support restrictions, S*T is in 𝒟′. It is commutative and associative, commutes with translations, and satisfies (S*T)(n)=S(n)*T=S*T(n). It is weakly continuous in 𝒟′, that is, if Tn→T in 𝒟′, then Tn*ψ→T*ψ in 𝒟′ see [1, 3, 6, 7] for additional properties of convolutions of distributions.Although elements of𝒜C are distributions, we show in this paper that their behavior as convolutions is more like that of integrable functions.An appendix contains the proof of a type of Fubini theorem.


## 2. Convolution in𝒜C×ℬ𝒱

In this section, we prove basic results for the convolution whenf∈𝒜C and g∈ℬ𝒱. Under these conditions, f*g is commutative, continuous on ℝ¯, and commutes with translations. It can be estimated in the uniform norm in terms of the Alexiewicz and ℬ𝒱 norms. There is also an associative property. We first need the result that ℬ𝒱 forms the space of multipliers for 𝒜C, that is, if f∈𝒜C, then fg∈𝒜C for all g∈ℬ𝒱. The integral ∫Ifg is defined using the integration by parts formula in the appendix. The Hölder inequality (A.5) shows that ℬ𝒱 is the dual space of 𝒜C.We define the convolution off∈𝒜C and g∈ℬ𝒱 as f*g(x)=∫-∞∞(f∘rx)g, where rx(t)=x-t. We write this as f*g(x)=∫-∞∞f(x-y)g(y)dy.Theorem 2.1.
Letf∈𝒜C and let g∈ℬ𝒱. Then (a)  f*g exists on ℝ.  (b) Let f*g=g*f.  (c)  Let  ∥f*g∥∞≤|∫-∞∞f|infℝ|g|+∥f∥Vg≤∥f∥∥g∥ℬ𝒱.   (d)   Assume f*g∈C0(ℝ¯), limx→±∞f*g(x)=g(±∞)∫-∞∞f.   (e) If h∈L1, then f*(g*h)=(f*g)*h∈C0(ℝ¯).    (f) Let x,z∈ℝ,   then τz(f*g)(x)=(τzf)*g(x)=(f*τzg)(x).   (g)   For each  f∈𝒜C,   define Φf:ℬ𝒱→C0(ℝ¯) by Φf[g]=f*g.   Then Φf is a bounded linear operator and ∥Φf∥≤∥f∥.   There exists a nonzero distribution  f∈𝒜C such that ∥Φf∥=∥f∥. For each g∈ℬ𝒱, define Ψg:𝒜C→C0(ℝ¯) by Ψg[f]=f*g. Then Ψg is a bounded linear operator and ∥Ψg∥≤∥g∥ℬ𝒱. There exists a nonzero function g∈ℬ𝒱 such that ∥Ψg∥=∥g∥ℬ𝒱.   (h)  supp(f*g)⊂supp(f)+supp(g).Proof.
(a) Existence is given via the integration by parts formula (A.1) in the appendix. (b) See [4, Theorem  11] for a change of variables theorem that can be used with y↦x-y. (c) This inequality follows from the Hölder inequality (A.5). (d) Let x,t∈ℝ. From (c), we have
(2.1)|f*g(t)-f*g(x)|≤∥f(t-·)-f(x-·)∥∥g∥ℬ𝒱=∥f(t-x-·)-f(·)∥∥g∥ℬ𝒱→0ast→x.
The last line follows from continuity in the Alexiewicz norm [4, Theorem  22]. Hence, f*g is uniformly continuous on ℝ. Also, it follows that limx→∞∫-∞∞f(y)g(x-y)dy=∫-∞∞f(y)limx→∞g(x-y)dy=g(∞)∫-∞∞f. The limit x→∞ can be taken under the integral sign since g(x-y) is of uniform bounded variation, that is, Vy∈ℝg(x-y)=Vg. Theorem  22 in [4] then applies. Similarly, as x→-∞. (e) First show g*h∈ℬ𝒱. Let {(si,ti)} be disjoint intervals in ℝ. Then
(2.2)∑|g*h(si)-g*h(ti)|≤∑∫-∞∞|g(si-y)-g(ti-y)||h(y)|dy=∫-∞∞∑|g(si-y)-g(ti-y)||h(y)|dy.
Hence, V(g*h)≤Vg∥h∥1. The interchange of sum and integral follows from the Fubini-Tonelli theorem. Now (d) shows f*(g*h)∈C0(ℝ¯). Write
(2.3)f*(g*h)(x)=∫-∞∞f(y)∫-∞∞g(x-y-z)h(z)dzdy=∫-∞∞h(z)∫-∞∞f(y)g(x-y-z)dydz=(f*g)*h(x).
We can interchange orders of integration using Proposition A.3. For (ii) in Proposition A.3, the function z↦Vy∈ℝg(x-y-z)h(z)=Vgh(z) is in L1 for each fixed x∈ℝ. Since g is of bounded variation, it is bounded so |g(x-y-z)h(z)|≤∥g∥∞|h(z)| and condition (iii) is satisfied. (f) This follows from a linear change of variables as in (a). (g) From (c), we have ∥Φf∥=sup∥g∥ℬ𝒱=1∥f*g∥∞≤sup∥g∥ℬ𝒱=1∥f∥∥g∥ℬ𝒱=∥f∥. Let f>0 be in L1. If g=1, then ∥g∥ℬ𝒱=1 and f*g(x)=∫-∞∞f so ∥Φf∥=∥f∥=∥f∥1. To prove ∥Ψg∥≤∥g∥ℬ𝒱, note that ∥Ψg∥=sup∥f∥=1∥f*g∥∞≤sup∥f∥=1∥f∥∥g∥ℬ𝒱=∥g∥ℬ𝒱. Let g=χ(0,∞). Then ∥Ψg∥=sup∥f∥=1∥f*g∥∞=sup∥f∥=1supx∈ℝ|∫-∞xf|=1=∥g∥ℬ𝒱. (h) Suppose x∉supp(f)+supp(g). Note that we can write f*g(x)=∫-∞∞g(x-y)dF(y) in terms of a Henstock-Stieltjes integral, see [4] for details. This integral is approximated by Riemann sums ∑n=1Ng(x-zn)[F(tn)-F(tn-1)] where zn∈[tn-1,tn], -∞=t0<t1<⋯<tN=∞ and there is a gauge function γ mapping ℝ¯ to the open intervals in ℝ¯ such that [tn-1,tn]⊂γ(zn). If zn∉supp(f), then since ℝ∖supp(f) is open, there is an open interval zn⊂I⊂ℝ∖supp(f). We can take γ such that [tn-1,tn]⊂I for all 1≤n≤N. Also, F is constant on each interval in ℝ∖supp(f). Therefore, g(x-zn)[F(tn)-F(tn-1)]=0 and only tags zn∈supp(f) can contribute to the Riemann sum. However, for all zn∈supp(f), we have x-zn∉supp(g) so g(x-zn)[F(tn)-F(tn-1)]=0. It follows that f*g(x)=0.Similar results are proven forf∈Lp in [1, Section  8.2].If we use the equivalent norm∥f∥′=supx∈ℝ|∫-∞xf|, then ∥Φf∥=∥f∥′. Also, integration by parts gives ∥Φf∥≤∥f∥′. Now, given f∈𝒜C, let g=χ(0,∞). Then ∥g∥ℬ𝒱=1,  and f*g(x)=∫-∞xf. Hence, ∥f*g∥∞=∥f∥′ and ∥Φf∥=∥f∥′. We can have strict inequality in ∥Ψg∥≤∥g∥ℬ𝒱. For example, let g=χ{0},  then ∥g∥ℬ𝒱=2 but integration by parts shows f*g=0 for each f∈𝒜C.Remark 2.2.
Iff∈𝒜C and g∈ℰℬ𝒱, one can use Definition A.2 to define f*g(x)=f*gγ(x) where gγ=g almost everywhere and gγ∈𝒩ℬ𝒱γ. All of the results in Theorem 2.1 and the rest of this paper have analogues. Note that f*g(x)=F(∞)gγ(-∞)+F*μg.Proposition 2.3.
The three definitions of convolution for distributions in Definition1.1 are compatible with f*g for f∈𝒜C and g∈ℬ𝒱.Proof.
Letf∈𝒜C, g∈ℬ𝒱, and ϕ,ψ∈𝒟. Definition 1.1(i) gives
(2.4)〈f,ψ̃*ϕ〉=∫-∞∞f(x)∫-∞∞ψ(y-x)ϕ(y)dydx=∫-∞∞∫-∞∞f(x)ψ(y-x)ϕ(y)dxdy=〈f*ψ,ϕ〉.
Since ψ∈ℬ𝒱 and ϕ∈L1, Proposition A.3 justifies the interchange of integrals. Definition 1.1(ii) gives
(2.5)〈f,τxψ̃〉=∫-∞∞f(y)ψ(x-y)dy=f*ψ(x).
Definition 1.1(iii) gives
(2.6)〈f(y),〈g(x),ϕ(x+y)〉〉=∫-∞∞f(y)∫-∞∞g(x)ϕ(x+y)dxdy=∫-∞∞f(y)∫-∞∞g(x-y)ϕ(x)dxdy=∫-∞∞ϕ(x)∫-∞∞f(y)g(x-y)dxdx=〈f*g,ϕ〉.
The interchange of integrals is accomplished using Proposition A.3 since g∈ℬ𝒱 and ϕ∈L1.The locally integrable distributions are defined as𝒜C(loc)={f∈𝒟′∣f=F′forsomeF∈C0(ℝ)}. Let f∈𝒜C(loc) and let g∈ℬ𝒱 with support in the compact interval [a,b]. By the Hake theorem [4, Theorem  25], f*g(x) exists if and only if the limits of ∫αβf(x-y)g(y)dy exist as α→-∞ and β→∞. This gives(2.7)f*g(x)=∫abf(x-y)g(y)dy=∫x-bx-af(y)g(x-y)dy.There are analogues of the results in Theorem2.1. For example, |f*g(x)|≤|∫x-bx-af|inf[a,b]|g|+∥fχ[x-b,x-a]∥V[a,b]g. There are also versions where the supports are taken to be semi-infinite intervals.We can also define the distributions with bounded primitive as𝒜C(bd)={f∈𝒟′∣f=F′forsomeboundedF∈C0(ℝ)withF(0)=0}. Let f∈𝒜C(bd) and let F be its unique primitive. If g∈ℬ𝒱 such that g(±∞)=0, then(2.8)f*g(x)=limα→-∞β→∞∫αβf(x-y)g(y)dy=limα→-∞β→∞[F(x-α)g(α)-F(x-β)g(β)+∫αβF(x-y)dg(y)]=∫-∞∞F(x-y)dg(y)=∫-∞∞F(y)dg(x-y).It follows that∥f*g∥∞≤∥F∥∞Vg.It is possible to formulate other existence criteria. For example, iff(x)=log|x|sin(x) and g(x)=|x|-α for some 0<α<1, then f and g are not in 𝒜C, ℬ𝒱 or Lp for any 1≤p≤∞ but f*g exists on ℝ because f,g∈Lloc1 and if F(x)=∫0xf, then lim|x|→∞F(x)g(x)=0.The following example shows thatf*g needs not to be of bounded variation and hence not absolutely continuous. Let g=χ(0,∞). For f∈𝒜C, we have f*g(x)=∫-∞xf=F(x), where F∈ℬC is the primitive of f. However, F needs not to be of bounded variation or even of local bounded variation. For example, let f(x)=sin(x-2)-2x-2cos(x-2) and let F be its primitive in ℬC. Finally, although f*g is continuous, it needs not to be integrable over ℝ. For example, let g=1, then f*g(x)=∫-∞∞f and ∫-∞∞f*g only exists if ∫-∞∞f=0.


## 3. Convolution in𝒜C×L1

We now extend the convolutionf*g to f∈𝒜C and g∈L1. Since there are functions in L1 that are not of bounded variation, there are distributions f∈𝒜C and functions g∈L1 such that the integral ∫-∞∞f(x-y)g(y)dy does not exist. The convolution is then defined as the limit in ∥·∥ of a sequence f*gn for gn∈ℬ𝒱∩L1 such that gn→g in the L1 norm. This is possible since ℬ𝒱∩L1 is dense in L1. We also give an equivalent definition using the fact that L1 is dense in 𝒜C. Take a sequence {fn}⊂L1 such that ∥fn-f∥→0. Then f*g is the limit in ∥·∥ of fn*g. In this more general setting of convolution defined in 𝒜C×L1, we now have an Alexiewicz norm estimate for f*g in terms of estimates of f in the Alexiewicz norm and g in the L1 norm. There is associativity with L1 functions and commutativity with translations.Definition 3.1.
Letf∈𝒜C and let g∈L1. Let {gn}⊂ℬ𝒱∩L1 such that ∥gn-g∥1→0. Define f*g as the unique element in 𝒜C such that ∥f*gn-f*g∥→0.To see that the definition makes sense, first note thatℬ𝒱∩L1 is dense in L1 since step functions are dense in L1. Hence, the required sequence {gn} exists. Let [α,β]⊂ℝ be a compact interval. Let F∈ℬC be the primitive of f. Then(3.1)∫αβf*gn(x)dx=∫αβ∫-∞∞f(y)gn(x-y)dydx=∫-∞∞f(y)∫αβgn(x-y)dxdy(3.2)=∫-∞∞f(y)∫α-yβ-ygn(x)dxdy=-∫-∞∞F(y)d[∫α-yβ-ygn](3.3)=∫-∞∞F(y)[gn(β-y)-gn(α-y)]dy=∫-∞∞(∫α-yβ-yf)gn(y)dy.
The interchange of orders of integration in (3.1) is accomplished with Proposition A.3 using g(x,y)=gn(x-y)χ[α,β](x). Integration by parts gives (3.2) since limy→∞∫α-yβ-ygn=0. As F is continuous and the function y↦∫α-yβ-ygn is absolutely continuous, we get (3.3). Taking the supremum over α,β∈ℝ gives(3.4)∥f*gn∥≤∥f∥∥gn∥1.
We now have(3.5)∥f*gm-f*gn∥=∥f*(gm-gn)∥≤∥f∥∥gm-gn∥1and{f*gn} is a Cauchy sequence in 𝒜C. Since 𝒜C is complete, this sequence has a limit in 𝒜C which we denote f*g. The definition does not depend on the choice of sequence {gn}, thus if {hn}⊂ℬ𝒱∩L1 such that ∥hn-g∥1→0, then ∥f*gn-f*hn∥≤∥f∥(∥gn-g∥1+∥hn-g∥1)→0 as n→∞. The previous calculation also shows that if g∈ℬ𝒱∩L1, then the integral definition f*g(x)=∫-∞∞f(x-y)g(y)dy and the limit definition agree.Definition 3.2.
Letf∈𝒜C and let g∈L1. Let {fn}⊂L1 such that ∥fn-f∥→0. Define f*g as the unique element in 𝒜C such that ∥fn*g-f*g∥→0.To show this definition makes sense, first showL1 is dense in 𝒜C.Proposition 3.3.
L1 is dense in 𝒜C.Proof.
LetAC(ℝ¯) be the functions that are absolutely continuous on each compact interval and which are of bounded variation on the real line. Then, f∈L1 if and only if there exists F∈AC(ℝ¯) such that F′(x)=f(x) for almost all x∈ℝ. Let f∈𝒜C be given. Let F∈ℬC be its primitive. For ϵ>0, take M>0 such that |F(x)|<ϵ for x<-M and |F(x)-F(∞)|<ϵ for x>M. Due to the Weierstrass approximation theorem, there is a continuous function P:ℝ→ℝ such that P(x)=F(-M) for x≤-M, P(x)=F(M) for x≥M, |P(x)-F(x)|<ϵ for |x|≤M and P is a polynomial on [-M,M]. Hence, P∈AC(ℝ¯) and ∥P′-f∥<3ϵ.In Definition3.2, the required sequence {fn}⊂L1 exists. Let [α,β]⊂ℝ be a compact interval. Then, by the usual Fubini-Tonelli theorem in L1,(3.6)∫αβfn*g(x)dx=∫αβ∫-∞∞fn(x-y)g(y)dydx=∫-∞∞g(y)∫αβfn(x-y)dxdy.Take the supremum overα,β∈ℝ and use the L1-L∞ Hölder inequality to get(3.7)∥fn*g∥≤∥fn∥∥g∥1.
It now follows that {fn*g} is a Cauchy sequence. It then converges to an element of 𝒜C. However, (3.7) also shows that this limit is independent of the choice of {fn}. To see that Definitions 3.1 and 3.2 agree, take {fn}⊂L1 with ∥fn-f∥→0 and {gn}⊂ℬ𝒱∩L1 with ∥gn-g∥1→0. Then(3.8)∥fn*g-f*gn∥=∥(fn-f)*g-f*(gn-g)∥≤∥(fn-f)*g∥+∥f*(gn-g)∥≤∥fn-f∥∥g∥1+∥f∥∥gn-g∥1Lettingn→∞ shows that the limits of fn*g in Definition 3.2 and f*gn in Definition 3.1 are the same.Theorem 3.4.
Letf∈𝒜C and g∈L1. Define f*g as in Definition 3.1. Then (a)∥f*g∥≤∥f∥∥g∥1.  (b)  Let h∈L1.  Then (f*g)*h=f*(g*h)∈𝒜C.  (c) For each z∈ℝ, τz(f*g)=(τzf)*g=(f*τzg).  (d) For each f∈𝒜C, define Φf:L1→𝒜C by Φf[g]=f*g. Then Φf is a bounded linear operator and ∥Φf∥≤∥f∥. There exists a nonzero distribution f∈𝒜C such that ∥Φf∥=∥f∥. For each g∈L1, define Ψg:𝒜C→𝒜C by Ψg[f]=f*g. Then Ψg is a bounded linear operator and ∥Ψg∥≤∥g∥1. There exists a nonzero function g∈L1 such that ∥Ψg∥=∥g∥ℬ𝒱.  (e) Define gt(x)=g(x/t)/t for t>0. We have a=∫-∞∞gt(x)dx=∫-∞∞g. Then ∥f*gt-af∥→0 as t→0.  (f)  Let supp(f*g)⊂supp(f)+supp(g).Proof.
Let{gn} be as in Definition 3.1. (a) Since ∥f*gn∥→∥f*g∥, (3.4) shows ∥f*g∥≤∥f∥∥g∥1. (b) Let {hn}⊂ℬ𝒱∩L1 such that ∥hn-h∥1→0. Then (f*g)*h:=ξ∈𝒜C such that ∥(f*g)*hn-ξ∥→0. Since g*h∈L1, there is {pn}⊂ℬ𝒱∩L1 such that ∥pn-g*h∥1→0. Then f*(g*h):=η∈𝒜C such that ∥f*pn-η∥→0. Now,
(3.9)∥ξ-η∥≤∥(f*g)*hn-ξ∥+∥f*pn-η∥+∥(f*g)*hn-(f*gn)*hn∥+∥(f*gn)*hn-f*pn∥.
Using (3.4),
(3.10)∥(f*g)*hn-(f*gn)*hn∥=∥[f*(g-gn)]*hn∥≤∥f∥∥gn-g∥1∥hn∥1→0asn→∞.
Finally, use Theorem 2.1(e) and (3.4) to write
(3.11)∥(f*gn)*hn-f*pn∥=∥f*(gn*hn-pn)∥≤∥f∥(∥gn-g∥1∥hn∥1+∥g∥1∥hn-h∥1+∥pn-g*h∥1)→0asn→∞.(c) The Alexiewicz norm is invariant under translation [4, Theorem  28] so τz(f*g)∈𝒜C. Use Theorem 2.1(f) to write ∥τz(f*g)-τz(f*gn)∥=∥f*g-f*gn∥=∥τz(f*g)-(τzf)*gn)∥=∥τz(f*g)-f*(τzgn)∥. Translation invariance of the L1 norm completes the proof. (d) From (a), we have ∥Φf∥=sup∥g∥1=1∥f*g∥≤sup∥g∥1=1∥f∥∥g∥1=∥f∥. We get equality by considering f and g to be positive functions in L1. To prove ∥Ψg∥≤∥g∥1, note that ∥Ψg∥=sup∥f∥=1∥f*g∥≤sup∥f∥=1∥f∥∥g∥1=∥g∥1. We get equality by considering f and g to be positive functions in L1. (e) First consider g∈ℬ𝒱∩L1. We have
(3.12)f*gt(x)=∫-∞∞f(x-y)g(yt)dyt=∫-∞∞f(x-ty)g(y)dy.
For -∞<α<β<∞,
(3.13)|∫αβ[f*gt(x)-af(x)]dx|=|∫αβ∫-∞∞[f(x-ty)-f(x)]g(y)dydx|=|∫-∞∞∫αβ[f(x-ty)-f(x)]g(y)dxdy|(3.14)≤∫-∞∞∥τtyf-f∥|g(y)|dy≤2∥f∥∥g∥1.
By dominated convergence, we can take the limit t→0 inside the integral (3.14). Continuity of f in the Alexiewicz norm then shows ∥f*gt-af∥→0 as t→0.
Now take a sequence{g(n)}⊂ℬ𝒱∩L1 such that ∥g(n)-g∥1→0. Define gt(n)(x)=g(n)(x/t)/t and a(n)=∫-∞∞g(n)(x)dx. We have
(3.15)∥f*gt-af∥≤∥f*gt(n)-a(n)f∥+∥f*gt(n)-f*gt∥+∥a(n)f-af∥.
By the inequality in (a), ∥f*gt(n)-f*gt∥≤∥f∥∥gt(n)-gt∥1. Whereas,
(3.16)∥gt(n)-gt∥1=∫-∞∞|g(n)(xt)-g(xt)|dxt=∥g(n)-g∥1→0asn→∞,
and ∥a(n)f-af∥=|a(n)-a|∥f∥=∥g(n)-g∥1∥f∥. Given ϵ>0 fix n large enough so that ∥f*gt(n)-f*gt∥+∥a(n)f-af∥<ϵ. Now let t→0 in (3.15).
The interchange of order of integration in (3.13) is justified as follows. A change of variables and Proposition A.3 give
(3.17)∫αβ∫-∞∞f(x-ty)g(y)dydx=∫αβ∫-∞∞f(y)g(x-yt)dytdx=∫-∞∞∫αβf(y)g(x-yt)dxdyt,∫-∞∞∫αβf(x-ty)g(y)dxdy=∬-∞∞f(x)g(yt)χ(α-y,β-y)(x)dxdyt=∬-∞∞f(x)g(yt)χ(α-y,β-y)(x)dytdx=∫-∞∞∫αβf(x)g(y-xt)dytdx.
Note that ∫αβ∫-∞∞f(x)g(y)dydx=∫-∞∞∫αβf(x)g(y)dxdy by Corollary A.4. (f) This follows from the equivalence of Definitions 1.1 and 3.1, proved in Proposition 3.5, see [6, Theorems  5.4-2 and 5.3-1].Young's inequality states that∥f*g∥p≤∥f∥p∥g∥1 when f∈Lp for some 1≤p≤∞ and g∈L1. Part (a) of Theorem 3.4 extends this to f∈𝒜C. see [1] for other results when f∈Lp.The fact that convolution is linear in both arguments, together with (b), shows that𝒜C is an L1-module over the L1 convolution algebra, see [8] for the definition. It does not appear that 𝒜C is a Banach algebra under convolution.We now show that Definition1.1(iii) and the aforementioned definitions agree.Proposition 3.5.
Letf∈𝒜C, g∈L1, and ϕ∈𝒟. Define F(y)=∫-∞yf and G(x)=∫-∞xg. Definitions 1.1 and 3.1 both give
(3.18)〈f*g,ϕ〉=∫-∞∞f(y)∫-∞∞g(x)ϕ(x+y)dxdy=∬-∞∞F(y)G(x)ϕ′(x+y)dxdy.Proof.
LetΦ(y)=∫-∞∞g(x)ϕ(x+y)dx. Then Φ∈C∞(ℝ) and Φ′(y)=∫-∞∞g(x)ϕ′(x+y)dx. Also, ∫-∞∞|Φ′(y)|dy≤∫-∞∞|g(x)|∫-∞∞|ϕ′(x+y)|dydx≤∥g∥1∥ϕ′∥1, so Φ∈AC(ℝ¯). Dominated convergence then shows lim|y|→∞Φ(y)=0. Integration by parts now gives (3.18).
Let{gn}⊂ℬ𝒱∩L1 such that ∥gn-g∥1→0. Since convergence in ∥·∥ implies convergence in 𝒟′, we have
(3.19)〈f*g,ϕ〉=limn→∞〈f*gn,ϕ〉=limn→∞∬-∞∞f(y)gn(x-y)ϕ(x)dydx=limn→∞∫-∞∞f(y)∫-∞∞gn(x-y)ϕ(x)dxdy.
Proposition A.3 allows interchange of the iterated integrals. Define Φn(y)=∫-∞∞gn(x)ϕ(x+y)dx. Then, VΦn≤∥gn∥1∥ϕ′∥1≤(∥g∥1+1)∥ϕ′∥1 for large enough n. Hence, Φn is of uniform bounded variation. Theorem  22 in [4], then gives 〈f*g,ϕ〉=∫-∞∞f(y)limn→∞Φn(y)dy=∫-∞∞f(y)∫-∞∞g(x-y)ϕ(x)dxdy. The last step follows since ∥gn-g∥1→0.Ifg∈L1∖ℬ𝒱, then f*g needs not to be continuous or bounded. For example, take 1/2≤α<1 and let f(x)=g(x)=x-αχ(0,1)(x). Then, f∈L1⊂𝒜C and g∈L1∖ℬ𝒱. We have f*g(x)=0 for x≤0. For 0<x≤1, we have f*g(x)=∫0xy-α(x-y)-αdy=x1-2α∫01y-α(1-y)-αdy=x1-2αΓ2(1-α)/Γ(2-2α). Hence, f*g is not continuous at 0. If 1/2<α<1, then f*g is unbounded at 0.As another example, considerf(x)=sin(πx)/log|x| and g(x)=χ(0,1)(x). Then f∈𝒜C and for each 1≤p≤∞, we have g∈ℬ𝒱∩Lp. And,(3.20)f*g(x)=∫x-1xsin(πy)log(y)dyforx≥2=cos(π(x-1))πlog(x-1)-cos(πx)πlog(x)-1π∫x-1xcos(πy)ylog2(y)dy~-2cos(πx)πlog(x)asx→∞.Therefore, by Theorem2.1(d), f*g∈C0(ℝ¯), and lim|x|→∞f*g(x)=0 but for each 1≤p<∞, we have f*g∉Lp.


## 4. Differentiation and Integration

Ifg is sufficiently smooth, then the pointwise derivative is (f*g)′(x)=f*g′(x). Recall the definition AC(ℝ¯) of primitives of L1 functions given in the proof of Proposition 3.3. In the following theorem, we require pointwise derivatives of g to exist at each point in ℝ.Theorem 4.1.
Letf∈𝒜C, n∈ℕ, and g(k)∈AC(ℝ¯) for each 0≤k≤n. Then f*g∈Cn(ℝ) and (f*g)(n)(x)=f*g(n)(x) for each x∈ℝ.Proof.
First considern=1. Let x∈ℝ. Then
(4.1)(f*g)′(x)=limh→0∫-∞∞f(y)[g(x+h-y)-g(x-y)h]dy.
To take the limit inside the integral we can show that the bracketed term in the integrand is of uniform bounded variation for 0<|h|≤1. Let h≠0. Since g∈AC(ℝ¯) it follows that the variation is given by the Lebesgue integrals
(4.2)Vy∈ℝ[g(x+h-y)-g(x-y)h]=∫-∞∞|g′(x+h-y)-g′(x-y)h|dy≤∫-∞∞|g′′(y)|dy+∫-∞∞|g′(x+h-y)-g′(x-y)h-g′′(x-y)|dy.
Since g′∈AC(ℝ¯), we have g′′∈L1. The second integral on the right of (4.2) gives the L1 derivative of g′ in the limit h→0; see [1, page 246]. Hence, in (4.1), we can use [4, Theorem  22] to take the limit under the integral sign. This then gives (f*g)′(x)=f*g′(x). Theorem 2.1(d) now shows (f*g)′∈C0(ℝ¯). Induction on n completes the proof.For similar results whenf∈L1, see [1, Proposition  8.10].Note thatg′∈AC(ℝ¯) does not imply g∈AC(ℝ¯). For example, g(x)=x. The conditions g(k)∈ℬ𝒱 for 0≤k≤n+1 imply those in Theorem 4.1. To see this, it suffices to consider n=1. If g′,g′′∈ℬ𝒱, then g′′ exists at each point and is bounded. Hence, the Lebesgue integral g′(x)=g′(0)+∫0xg′′(y)dy exists for each x∈ℝ and g′ is absolutely continuous. Since g′∈ℬ𝒱, we then have g′∈AC(ℝ¯). Similarly, for n>1. The example g(x)=|x|1.5sin(1/[1+x2]) shows that the AC(ℝ¯) condition in the theorem is weaker than the aforementioned ℬ𝒱 condition since g,g′∈AC(ℝ¯) but g′′(0) does not exist so g′′∉ℬ𝒱.We found that wheng∈ℬ𝒱∩L1, then f*g∈𝒜C. We can compute the distributional derivative (F*g)′=f*g, where F is a primitive of f.Proposition 4.2.
LetF∈C0(ℝ¯) and write f=F′∈𝒜C. Let g∈ℬ𝒱∩L1. Then F*g∈C0(ℝ¯) and (F*g)′=f*g∈𝒜C.Proof.
Letx,t∈ℝ. Then by the usual Hölder inequality,
(4.3)|F*g(x)-F*g(t)|=|∫-∞∞[F(x-y)-F(t-y)]g(y)dy|≤∥F(x-·)-F(t-·)∥∞∥g∥1→0ast→xsinceFisuniformlycontinuousonℝ.
Hence, F*g is continuous on ℝ. Dominated convergence shows that limx→±∞F*g(x)=F(±∞)∫-∞∞g. Therefore, F*g∈C0(ℝ¯).
Letϕ∈𝒟. Then
(4.4)〈(F*g)′,ϕ〉=-〈F*g,ϕ′〉=-∬-∞∞F(x-y)g(y)ϕ′(x)dydx=-∫-∞∞g(y)∫-∞∞F(x-y)ϕ′(x)dxdy(Fubini-Tonellitheorem).
Integrate by parts and use the change of variables x↦x+y to get
(4.5)〈(F*g)′,ϕ〉=∫-∞∞g(y)∫-∞∞f(x)ϕ(x+y)dxdy=∫-∞∞f(x)∫-∞∞g(y)ϕ(x+y)dydx(byPropositionA.3)=∫-∞∞f(x)∫-∞∞g(y-x)ϕ(y)dydx=∫-∞∞ϕ(y)∫-∞∞f(x)g(y-x)dxdy(byPropositionA.3)=〈f*g,ϕ〉.This gives an alternate definition off*g for f∈𝒜C and g∈L1.Theorem 4.3.
Letf∈𝒜C, let F∈ℬC be the primitive of f and let g∈L1. Define f*g as in Definition 3.1. Then (F*g)′=f*g∈𝒜C.Proof.
Let-∞<α<β<∞. Let {gn}⊂ℬ𝒱∩L1 such that ∥gn-g∥1→0. By Proposition 4.2, we have
(4.6)∫αβ(F*g)′=F*g(β)-F*g(α)=∫-∞∞F(y)[g(β-y)-g(α-y)]dy.
As in (3.3), ∫αβf*gn=∫-∞∞F(y)[gn(β-y)-gn(α-y)]dy. Hence,
(4.7)|∫αβ[(F*g)′-f*gn]|=|∫-∞∞F(y)[(g(β-y)-gn(β-y))-(g(α-y)-gn(α-y))]dy|≤∥F∥∞(∥g(β-·)-gn(β-·)∥1+∥g(α-·)-gn(α-·)∥1)=2∥f∥∥gn-g∥1.
Therefore, ∥(F*g)′-f*gn∥≤2∥f∥∥gn-g∥1→0 as n→∞.The following theorem and its corollary give results on integrating convolutions.Theorem 4.4.
Letf∈𝒜C and let g∈L1. Define F(x)=∫-∞xf and G(x)=∫-∞xg. Then, f*G∈C0(ℝ¯) and f*G(x)=F*g(x) for all x∈ℝ.Proof.
SinceG∈AC(ℝ¯), Theorem 2.1(d) shows f*G∈C0(ℝ¯). We have
(4.8)f*G(x)=∫-∞∞f(y)∫-∞x-yg(z)dzdy=∬-∞∞f(y)χ(-∞,x-y)(z)g(z)dzdy=∬-∞∞f(y)χ(-∞,x-y)(z)g(z)dydz=∫-∞∞g(z)∫-∞x-zf(y)dydz=F*g(x).
Proposition A.3 justifies the interchange of orders of integration.Corollary 4.5.
Note that the following hold: (a)f*g=(F*g)′=(f*G)′,(b) for all -∞≤α<β≤∞, one has ∫αβf*g=F*g(β)-F*g(α)=f*G(β)-f*G(β).Hence, the convolutionf*g can be evaluated by taking the distributional derivative of the Lebesgue integral F*g. Since f*G∈C0(ℝ¯), when f∈𝒜C and G∈ℬ𝒱, we can use the equation f*g=(f*G)′ to define f*g for f∈𝒜C and g=G′ for G∈ℬ𝒱. In this case, g will be a signed Radon measure. As G(x)=∫-∞xg and this integral is a regulated primitive integral [5], we will save this case for discussion elsewhere.



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