Convolutions with the Continuous Primitive Integral

Abstract

If 𝐹 is a continuous function on the real line and 𝑓=𝐹^ is its distributional derivative, then the continuous primitive integral of distribution 𝑓 is ∫^𝑏𝑎𝑓=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‖=sup_𝐼|∫_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.

1. Introduction and Notation

The convolution of two functions 𝑓 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 on 𝐿¹ 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∈𝐶^𝑛, then 𝑓∗𝑔∈𝐶^𝑛 and (𝑓∗𝑔)^(𝑛)(𝑥)=𝑓∗𝑔^(𝑛)(𝑥). Convolutions also have the approximation property that if 𝑓∈𝐿^𝑝 (1≤𝑝<∞) and 𝑔∈𝐿¹, then ‖𝑓∗𝑔_𝑡−𝑎𝑓‖_𝑝→0 as 𝑡→0, where 𝑔_𝑡(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 of test functions is 𝒟=𝐶^∞𝑐(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 {𝜙_𝑛}⊂𝒟, then 𝜙_𝑛→0 as 𝑛→∞ if there is a compact set 𝐾⊂ℝ such that for each 𝑛, supp(𝜙_𝑛)⊂𝐾, and for each 𝑚≥0, we have 𝜙^𝑛(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 𝜙_𝑛→0 in 𝒟, then ⟨𝑇,𝜙_𝑛⟩→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∈𝐿^1loc, then ⟨𝑇_𝑓,𝜙⟩=∫^∞−∞𝑓(𝑥)𝜙(𝑥)𝑑𝑥 defines a distribution 𝑇_𝑓∈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  translation 𝜏_𝑥 on  distribution 𝑇∈𝒟^ by ⟨𝜏_𝑥𝑇,𝜙⟩=⟨𝑇,𝜏_−𝑥𝜙⟩ for test function 𝜙∈𝒟, where 𝜏_𝑥𝜙(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: ℬ_𝐶={𝐹∶ℝ→ℝ∣𝐹∈𝐶⁰(ℝ),𝐹(−∞)=0,𝐹(∞)∈ℝ}. We use the notation 𝐹(−∞)=lim_{𝑥→−∞}𝐹(𝑥) and 𝐹(∞)=lim_𝑥→∞𝐹(𝑥). The extended real line is denoted by ℝ=[−∞,∞]. The space ℬ_𝐶 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, ℬ_𝐶 is properly contained in 𝐶⁰(ℝ), which is itself properly contained in the space of uniformly continuous functions on ℝ. The space ℬ_𝐶 is a Banach space under the uniform norm; ‖𝐹‖_∞=sup_𝑥∈ℝ|𝐹(𝑥)|=max_𝑥∈ℝ|𝐹(𝑥)| for 𝐹∈ℬ_𝐶. The continuous primitive integral is defined by taking ℬ_𝐶 as the space of primitives. The space of integrable distributions is 𝒜_𝐶={𝑓∈𝒟^∣𝑓=𝐹^for𝐹∈ℬ_𝐶}. If 𝑓∈𝒜_𝐶, then ∫^𝑏𝑎𝑓=𝐹(𝑏)−𝐹(𝑎) for 𝑎,𝑏∈ℝ. The distributional differential equation 𝑇^=0 has only constant solutions so the primitive 𝐹∈ℬ_𝐶 satisfying 𝐹^=𝑓 is unique. Integrable distributions are then tempered and of order one. This integral, including a discussion of extensions to ℝ^𝑛, 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 in 𝒜_𝐶 are 𝑇_𝑓 for functions 𝑓 that have a finite Lebesgue, Henstock-Kurzweil, or wide Denjoy integral. We identify function 𝑓 with the distribution 𝑇_𝑓. Pointwise function values can be recovered from 𝑇_𝑓 at points of continuity of 𝑓 by evaluating the limit ⟨𝑇_𝑓,𝜙_𝑛⟩ for a delta sequence converging to 𝑥∈ℝ. This is a sequence of test functions {𝜙_𝑛}⊂𝒟 such that for each 𝑛, 𝜙_𝑛≥0, ∫^∞−∞𝜙_𝑛=1, and the support of 𝜙_𝑛 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 ∫^𝑏𝑎𝐹^(𝑥)𝑑𝑥=0 but 𝐹^∈𝒜_𝐶 and ∫^𝑏𝑎𝐹^=𝐹(𝑏)−𝐹(𝑎). For another example of a distribution in 𝒜_𝐶, let 𝐹∈𝐶⁰(ℝ) be continuous and nowhere differentiable in the pointwise sense. Then 𝐹^∈𝒜_𝐶 and ∫^𝑏𝑎𝐹^=𝐹(𝑏)−𝐹(𝑎) for all 𝑎,𝑏∈ℝ.

The space 𝒜_𝐶 is a Banach space under the Alexiewicz norm; ‖𝑓‖=sup_𝐼⊂ℝ|∫_𝐼𝑓|, where the supremum is taken over all intervals 𝐼⊂ℝ. An equivalent norm is ‖𝑓‖^=sup_𝑥∈ℝ|∫^𝑥−∞𝑓|. 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 𝒜_𝐶 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 𝒜_𝐶 is a suitable setting for applications of nonabsolute integration.

We will also need to use functions of bounded variation. Let 𝑔∶ℝ→ℝ. The variation of 𝑔 is 𝑉𝑔=sup∑|𝑔(𝑥_𝑖)−𝑔(𝑦_𝑖)| where the supremum is taken over all disjoint intervals {(𝑥_𝑖,𝑦_𝑖)}. 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(±∞)=lim_𝑥→±∞𝑔(𝑥).

If 𝑔∈𝐿^1loc, 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∈𝐿^1loc∣essvar𝑔<∞}. This is a Banach space under the norm ‖𝑔‖_ℰℬ𝒱=esssup|𝑔|+essvar𝑔. Let 0≤𝛾≤1. For 𝑔∶ℝ→ℝ, define 𝑔_𝛾(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 𝒩ℬ𝒱_𝛾={𝑔_𝛾∣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∈𝒩ℬ𝒱_𝛾 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 𝜇_𝑔 where ⟨𝑔^,𝜙⟩=−⟨𝑔,𝜙^⟩=−∫^∞−∞𝑔𝜙^=∫^∞−∞𝜙𝑑𝜇_𝑔 for all 𝜙∈𝒟.

We will see that ∗∶𝒜_𝐶×ℬ𝒱→𝐶⁰(ℝ) and that ‖𝑓∗𝑔‖_∞≤‖𝑓‖‖𝑔‖_ℬ𝒱. Similarly for 𝑔∈ℰℬ𝒱. Convolutions for 𝑓∈𝒜_𝐶 and 𝑔∈𝐿¹ will be defined using sequences in ℬ𝒱∩𝐿¹ that converge to 𝑔 in the 𝐿¹ norm. It will be shown that ∗∶𝒜_𝐶×𝐿¹→𝒜_𝐶 and that ‖𝑓∗𝑔‖≤‖𝑓‖‖𝑔‖₁.

Convolutions can be defined for distributions in several different ways.

Definition 1.1. Let 𝑆,𝑇∈𝒟^ and 𝜙,𝜓∈𝒟. Define 𝜙(𝑥)=𝜙(−𝑥): (i) ⟨𝑇∗𝜓,𝜙⟩=⟨𝑇,𝜙∗𝜓⟩, (ii) for each 𝑥∈ℝ, let 𝑇∗𝜓(𝑥)=⟨𝑇,𝜏_𝑥𝜓⟩; (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), 𝜏_𝑥(𝑇∗𝜓)=(𝜏_𝑥𝑇)∗𝜓=𝑇∗(𝜏_𝑥𝜓), 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 𝑇_𝑛→𝑇 in 𝒟^, then 𝑇_𝑛∗𝜓→𝑇∗𝜓 in 𝒟^ see [1, 3, 6, 7] for additional properties of convolutions of distributions.

Although elements of 𝒜_𝐶 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 in 𝒜_𝐶×ℬ𝒱

In this section, we prove basic results for the convolution when 𝑓∈𝒜_𝐶 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 𝒜_𝐶, that is, if 𝑓∈𝒜_𝐶, then 𝑓𝑔∈𝒜_𝐶 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 𝒜_𝐶.

We define the convolution of 𝑓∈𝒜_𝐶 and 𝑔∈ℬ𝒱 as 𝑓∗𝑔(𝑥)=∫^∞−∞(𝑓∘𝑟_𝑥)𝑔, where 𝑟_𝑥(𝑡)=𝑥−𝑡. We write this as 𝑓∗𝑔(𝑥)=∫^∞−∞𝑓(𝑥−𝑦)𝑔(𝑦)𝑑𝑦.

Theorem 2.1. Let 𝑓∈𝒜_𝐶 and let 𝑔∈ℬ𝒱. Then (a)  𝑓∗𝑔 exists on ℝ.  (b) Let 𝑓∗𝑔=𝑔∗𝑓.  (c)  Let  ‖𝑓∗𝑔‖_∞≤|∫^∞−∞𝑓|inf_ℝ|𝑔|+‖𝑓‖𝑉𝑔≤‖𝑓‖‖𝑔‖_ℬ𝒱.   (d)   Assume 𝑓∗𝑔∈𝐶⁰(ℝ), lim_𝑥→±∞𝑓∗𝑔(𝑥)=𝑔(±∞)∫^∞−∞𝑓.   (e) If ℎ∈𝐿¹, then 𝑓∗(𝑔∗ℎ)=(𝑓∗𝑔)∗ℎ∈𝐶⁰(ℝ).    (f) Let 𝑥,𝑧∈ℝ,   then 𝜏_𝑧(𝑓∗𝑔)(𝑥)=(𝜏_𝑧𝑓)∗𝑔(𝑥)=(𝑓∗𝜏_𝑧𝑔)(𝑥).   (g)   For each  𝑓∈𝒜_𝐶,   define Φ_𝑓∶ℬ𝒱→𝐶⁰(ℝ) by Φ_𝑓[𝑔]=𝑓∗𝑔.   Then Φ_𝑓 is a bounded linear operator and ‖Φ_𝑓‖≤‖𝑓‖.   There exists a nonzero distribution  𝑓∈𝒜_𝐶 such that ‖Φ_𝑓‖=‖𝑓‖. For each 𝑔∈ℬ𝒱, define Ψ_𝑔∶𝒜_𝐶→𝐶⁰(ℝ) by Ψ_𝑔[𝑓]=𝑓∗𝑔. Then Ψ_𝑔 is a bounded linear operator and ‖Ψ_𝑔‖≤‖𝑔‖_ℬ𝒱. There exists a nonzero function 𝑔∈ℬ𝒱 such that ‖Ψ_𝑔‖=‖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 ||𝑓∗𝑔(𝑡)−𝑓∗𝑔(𝑥)||≤‖𝑓(𝑡−⋅)−𝑓(𝑥−⋅)‖‖𝑔‖_ℬ𝒱=‖𝑓(𝑡−𝑥−⋅)−𝑓(⋅)‖‖𝑔‖_ℬ𝒱⟶0as𝑡⟶𝑥.(2.1) 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 lim_𝑥→∞∫^∞−∞𝑓(𝑦)𝑔(𝑥−𝑦)𝑑𝑦=∫^∞−∞𝑓(𝑦)lim_𝑥→∞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, 𝑉_𝑦∈ℝ𝑔(𝑥−𝑦)=𝑉𝑔. Theorem  22 in [4] then applies. Similarly, as 𝑥→−∞. (e) First show 𝑔∗ℎ∈ℬ𝒱. Let {(𝑠_𝑖,𝑡_𝑖)} be disjoint intervals in ℝ. Then ||𝑔∗ℎ𝑠^𝑖−𝑔∗ℎ𝑡_𝑖||≤^∞−∞=||𝑔𝑠_𝑖−𝑦−𝑔𝑡_𝑖−𝑦||||ℎ(𝑦)||𝑑𝑦_∞−∞||𝑔𝑠^𝑖−𝑦−𝑔𝑡_𝑖−𝑦||||ℎ(𝑦)||𝑑𝑦.(2.2) 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 𝑓∗(𝑔∗ℎ)(𝑥)==^∞−∞𝑓(𝑦)^∞−∞𝑔(𝑥−𝑦−𝑧)ℎ(𝑧)𝑑𝑧𝑑𝑦_∞−∞=(𝑓∗ℎ𝑔)(𝑧∗)ℎ(𝑥^∞−∞)𝑓.(𝑦)𝑔(𝑥−𝑦−𝑧)𝑑𝑦𝑑𝑧(2.3) We can interchange orders of integration using Proposition A.3. For (ii) in Proposition A.3, the function 𝑧↦𝑉_𝑦∈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 ‖Φ_𝑓‖=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 ‖Φ_𝑓‖=‖𝑓‖=‖𝑓‖₁. To prove ‖Ψ_𝑔‖≤‖𝑔‖_ℬ𝒱, note that ‖Ψ_𝑔‖=sup_‖𝑓‖=1‖𝑓∗𝑔‖_∞≤sup_‖𝑓‖=1‖𝑓‖‖𝑔‖_ℬ𝒱=‖𝑔‖_ℬ𝒱. Let 𝑔=𝜒_(0,∞). Then ‖Ψ_𝑔‖=sup_‖𝑓‖=1‖𝑓∗𝑔‖_∞=sup_‖𝑓‖=1sup_𝑥∈ℝ|∫^𝑥−∞𝑓|=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 ∑^𝑁𝑛=1𝑔(𝑥−𝑧_𝑛)[𝐹(𝑡_𝑛)−𝐹(𝑡_𝑛−1)] where 𝑧_𝑛∈[𝑡_𝑛−1,𝑡_𝑛], −∞=𝑡₀<𝑡₁<⋯<𝑡_𝑁=∞ and there is a gauge function 𝛾 mapping ℝ to the open intervals in ℝ such that [𝑡_𝑛−1,𝑡_𝑛]⊂𝛾(𝑧_𝑛). If 𝑧_𝑛∉supp(𝑓), then since ℝ⧵supp(𝑓) is open, there is an open interval 𝑧_𝑛⊂𝐼⊂ℝ⧵supp(𝑓). We can take 𝛾 such that [𝑡_𝑛−1,𝑡_𝑛]⊂𝐼 for all 1≤𝑛≤𝑁. Also, 𝐹 is constant on each interval in ℝ⧵supp(𝑓). Therefore, 𝑔(𝑥−𝑧_𝑛)[𝐹(𝑡_𝑛)−𝐹(𝑡_𝑛−1)]=0 and only tags 𝑧_𝑛∈supp(𝑓) can contribute to the Riemann sum. However, for all 𝑧_𝑛∈supp(𝑓), we have 𝑥−𝑧_𝑛∉supp(𝑔) so 𝑔(𝑥−𝑧_𝑛)[𝐹(𝑡_𝑛)−𝐹(𝑡_𝑛−1)]=0. It follows that 𝑓∗𝑔(𝑥)=0.

Similar results are proven for 𝑓∈𝐿^𝑝 in [1, Section  8.2].

If we use the equivalent norm ‖𝑓‖^=sup_𝑥∈ℝ|∫^𝑥−∞𝑓|, then ‖Φ_𝑓‖=‖𝑓‖^. Also, integration by parts gives ‖Φ_𝑓‖≤‖𝑓‖^. Now, given 𝑓∈𝒜_𝐶, let 𝑔=𝜒_(0,∞). Then ‖𝑔‖_ℬ𝒱=1,  and 𝑓∗𝑔(𝑥)=∫^𝑥−∞𝑓. Hence, ‖𝑓∗𝑔‖_∞=‖𝑓‖^ and ‖Φ_𝑓‖=‖𝑓‖^. We can have strict inequality in ‖Ψ_𝑔‖≤‖𝑔‖_ℬ𝒱. For example, let 𝑔=𝜒_{0},  then ‖𝑔‖_ℬ𝒱=2 but integration by parts shows 𝑓∗𝑔=0 for each 𝑓∈𝒜_𝐶.

Remark 2.2. If 𝑓∈𝒜_𝐶 and 𝑔∈ℰℬ𝒱, one can use Definition A.2 to define 𝑓∗𝑔(𝑥)=𝑓∗𝑔_𝛾(𝑥) where 𝑔_𝛾=𝑔 almost everywhere and 𝑔_𝛾∈𝒩ℬ𝒱_𝛾. All of the results in Theorem 2.1 and the rest of this paper have analogues. Note that 𝑓∗𝑔(𝑥)=𝐹(∞)𝑔_𝛾(−∞)+𝐹∗𝜇_𝑔.

Proposition 2.3. The three definitions of convolution for distributions in Definition 1.1 are compatible with 𝑓∗𝑔 for 𝑓∈𝒜_𝐶 and 𝑔∈ℬ𝒱.

Proof. Let 𝑓∈𝒜_𝐶, 𝑔∈ℬ𝒱, and 𝜙,𝜓∈𝒟. Definition 1.1(i) gives ⟨𝑓,𝜓∗𝜙⟩=^∞−∞𝑓(𝑥)^∞−∞𝜓(𝑦−𝑥)𝜙(𝑦)𝑑𝑦𝑑𝑥=^∞−∞^∞−∞𝑓(𝑥)𝜓(𝑦−𝑥)𝜙(𝑦)𝑑𝑥𝑑𝑦=⟨𝑓∗𝜓,𝜙⟩.(2.4) Since 𝜓∈ℬ𝒱 and 𝜙∈𝐿¹, Proposition A.3 justifies the interchange of integrals. Definition 1.1(ii) gives ⟨𝑓,𝜏_𝑥𝜓⟩=^∞−∞𝑓(𝑦)𝜓(𝑥−𝑦)𝑑𝑦=𝑓∗𝜓(𝑥).(2.5) Definition 1.1(iii) gives ⟨𝑓(𝑦),⟨𝑔(𝑥),𝜙(𝑥+𝑦)⟩⟩==^∞−∞𝑓(𝑦)^∞−∞𝑔(𝑥)𝜙(𝑥+𝑦)𝑑𝑥𝑑𝑦_∞−∞=𝑓(𝑦)^∞−∞𝑔(𝑥−𝑦)𝜙(𝑥)𝑑𝑥𝑑𝑦_∞−∞=⟨𝑓∗𝜙𝑔,(𝜙𝑥⟩).^∞−∞𝑓(𝑦)𝑔(𝑥−𝑦)𝑑𝑥𝑑𝑥(2.6) The interchange of integrals is accomplished using Proposition A.3 since 𝑔∈ℬ𝒱 and 𝜙∈𝐿¹.

The locally integrable distributions are defined as 𝒜_𝐶(loc)={𝑓∈𝒟^∣𝑓=𝐹^forsome𝐹∈𝐶⁰(ℝ)}. Let 𝑓∈𝒜_𝐶(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 ∫^𝛽𝛼𝑓(𝑥−𝑦)𝑔(𝑦)𝑑𝑦 exist as 𝛼→−∞ and 𝛽→∞. This gives

𝑓∗𝑔(𝑥)=^𝑏𝑎𝑓(𝑥−𝑦)𝑔(𝑦)𝑑𝑦=^{𝑥−𝑎𝑥−𝑏}𝑓(𝑦)𝑔(𝑥−𝑦)𝑑𝑦.(2.7)

There are analogues of the results in Theorem 2.1. For example, |𝑓∗𝑔(𝑥)|≤|∫^{𝑥−𝑎𝑥−𝑏}𝑓|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 as 𝒜_𝐶(𝑏𝑑)={𝑓∈𝒟^∣𝑓=𝐹^forsomebounded𝐹∈𝐶⁰(ℝ)with𝐹(0)=0}. Let 𝑓∈𝒜_𝐶(𝑏𝑑) and let 𝐹 be its unique primitive. If 𝑔∈ℬ𝒱 such that 𝑔(±∞)=0, then

𝑓∗𝑔(𝑥)=lim_{𝛼→−∞𝛽→∞}^𝛽𝛼=lim_{𝛼→−∞𝛽→∞}𝑓(𝑥−𝑦)𝑔(𝑦)𝑑𝑦𝐹(𝑥−𝛼)𝑔(𝛼)−𝐹(𝑥−𝛽)𝑔(𝛽)+^𝛽𝛼=𝐹(𝑥−𝑦)𝑑𝑔(𝑦)_∞−∞𝐹(𝑥−𝑦)𝑑𝑔(𝑦)=^∞−∞𝐹(𝑦)𝑑𝑔(𝑥−𝑦).(2.8)

It follows that ‖𝑓∗𝑔‖_∞≤‖𝐹‖_∞𝑉𝑔.

It is possible to formulate other existence criteria. For example, if 𝑓(𝑥)=log|𝑥|sin(𝑥) and 𝑔(𝑥)=|𝑥|^−𝛼 for some 0<𝛼<1, then 𝑓 and 𝑔 are not in 𝒜_𝐶, ℬ𝒱 or 𝐿^𝑝 for any 1≤𝑝≤∞ but 𝑓∗𝑔 exists on ℝ because 𝑓,𝑔∈𝐿^1loc and if 𝐹(𝑥)=∫^𝑥0𝑓, then lim_|𝑥|→∞𝐹(𝑥)𝑔(𝑥)=0.

The following example shows that 𝑓∗𝑔 needs not to be of bounded variation and hence not absolutely continuous. Let 𝑔=𝜒_(0,∞). For 𝑓∈𝒜_𝐶, we have 𝑓∗𝑔(𝑥)=∫^𝑥−∞𝑓=𝐹(𝑥), where 𝐹∈ℬ_𝐶 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 ℬ_𝐶. 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 in 𝒜_𝐶×𝐿¹

We now extend the convolution 𝑓∗𝑔 to 𝑓∈𝒜_𝐶 and 𝑔∈𝐿¹. Since there are functions in 𝐿¹ that are not of bounded variation, there are distributions 𝑓∈𝒜_𝐶 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∗𝑔_𝑛 for 𝑔_𝑛∈ℬ𝒱∩𝐿¹ such that 𝑔_𝑛→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 𝒜_𝐶. Take a sequence {𝑓_𝑛}⊂𝐿¹ such that ‖𝑓_𝑛−𝑓‖→0. Then 𝑓∗𝑔 is the limit in ‖⋅‖ of 𝑓_𝑛∗𝑔. In this more general setting of convolution defined in 𝒜_𝐶×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. Let 𝑓∈𝒜_𝐶 and let 𝑔∈𝐿¹. Let {𝑔_𝑛}⊂ℬ𝒱∩𝐿¹ such that ‖𝑔_𝑛−𝑔‖₁→0. Define 𝑓∗𝑔 as the unique element in 𝒜_𝐶 such that ‖𝑓∗𝑔_𝑛−𝑓∗𝑔‖→0.

To see that the definition makes sense, first note that ℬ𝒱∩𝐿¹ is dense in 𝐿¹ since step functions are dense in 𝐿¹. Hence, the required sequence {𝑔_𝑛} exists. Let [𝛼,𝛽]⊂ℝ be a compact interval. Let 𝐹∈ℬ_𝐶 be the primitive of 𝑓. Then

^𝛽𝛼𝑓∗𝑔_𝑛(𝑥)𝑑𝑥=^𝛽𝛼^∞−∞=𝑓(𝑦)𝑔_𝑛(𝑥−𝑦)𝑑𝑦𝑑𝑥_∞−∞=𝑓(𝑦)^𝛽𝛼𝑔_𝑛(𝑥−𝑦)𝑑𝑥𝑑𝑦(3.1)_∞−∞=−_∞−∞𝑓(𝑦)^{𝛽−𝑦𝛼−𝑦}𝑔_𝑛(𝑥)𝑑𝑥𝑑𝑦𝐹(𝑦)𝑑^{𝛽−𝑦𝛼−𝑦}𝑔_𝑛=(3.2)_∞−∞=𝐹(𝑦)𝑔_𝑛(𝛽−𝑦)−𝑔_𝑛(𝛼−𝑦)𝑑𝑦_∞−∞^{𝛽−𝑦𝛼−𝑦}𝑓𝑔_𝑛(𝑦)𝑑𝑦.(3.3) The interchange of orders of integration in (3.1) is accomplished with Proposition A.3 using 𝑔(𝑥,𝑦)=𝑔_𝑛(𝑥−𝑦)𝜒_[𝛼,𝛽](𝑥). Integration by parts gives (3.2) since lim_𝑦→∞∫^{𝛽−𝑦𝛼−𝑦}𝑔_𝑛=0. As 𝐹 is continuous and the function 𝑦↦∫^{𝛽−𝑦𝛼−𝑦}𝑔_𝑛 is absolutely continuous, we get (3.3). Taking the supremum over 𝛼,𝛽∈ℝ gives

‖‖𝑓∗𝑔_𝑛‖‖≤‖𝑓‖‖‖𝑔_𝑛‖‖₁.(3.4) We now have

‖‖𝑓∗𝑔_𝑚−𝑓∗𝑔_𝑛‖‖=‖‖𝑓∗𝑔_𝑚−𝑔_𝑛‖‖≤‖𝑓‖‖‖𝑔_𝑚−𝑔_𝑛‖‖₁(3.5)

and {𝑓∗𝑔_𝑛} is a Cauchy sequence in 𝒜_𝐶. Since 𝒜_𝐶 is complete, this sequence has a limit in 𝒜_𝐶 which we denote 𝑓∗𝑔. The definition does not depend on the choice of sequence {𝑔_𝑛}, thus if {ℎ_𝑛}⊂ℬ𝒱∩𝐿¹ such that ‖ℎ_𝑛−𝑔‖₁→0, then ‖𝑓∗𝑔_𝑛−𝑓∗ℎ_𝑛‖≤‖𝑓‖(‖𝑔_𝑛−𝑔‖₁+‖ℎ_𝑛−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. Let 𝑓∈𝒜_𝐶 and let 𝑔∈𝐿¹. Let {𝑓_𝑛}⊂𝐿¹ such that ‖𝑓_𝑛−𝑓‖→0. Define 𝑓∗𝑔 as the unique element in 𝒜_𝐶 such that ‖𝑓_𝑛∗𝑔−𝑓∗𝑔‖→0.

To show this definition makes sense, first show 𝐿¹ is dense in 𝒜_𝐶.

Proposition 3.3. 𝐿¹ is dense in 𝒜_𝐶.

Proof. Let 𝐴𝐶(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∈𝒜_𝐶 be given. Let 𝐹∈ℬ_𝐶 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 Definition 3.2, the required sequence {𝑓_𝑛}⊂𝐿¹ exists. Let [𝛼,𝛽]⊂ℝ be a compact interval. Then, by the usual Fubini-Tonelli theorem in 𝐿¹,

^𝛽𝛼𝑓_𝑛∗𝑔(𝑥)𝑑𝑥=^𝛽𝛼^∞−∞=𝑓_𝑛(𝑥−𝑦)𝑔(𝑦)𝑑𝑦𝑑𝑥_∞−∞𝑔(𝑦)^𝛽𝛼𝑓_𝑛(𝑥−𝑦)𝑑𝑥𝑑𝑦.(3.6)

Take the supremum over 𝛼,𝛽∈ℝ and use the 𝐿¹−𝐿^∞ Hölder inequality to get

‖‖𝑓_𝑛∗𝑔‖‖≤‖‖𝑓_𝑛‖‖‖𝑔‖₁.(3.7) It now follows that {𝑓_𝑛∗𝑔} is a Cauchy sequence. It then converges to an element of 𝒜_𝐶. However, (3.7) also shows that this limit is independent of the choice of {𝑓_𝑛}. To see that Definitions 3.1 and 3.2 agree, take {𝑓_𝑛}⊂𝐿¹ with ‖𝑓_𝑛−𝑓‖→0 and {𝑔_𝑛}⊂ℬ𝒱∩𝐿¹ with ‖𝑔_𝑛−𝑔‖₁→0. Then

‖‖𝑓_𝑛∗𝑔−𝑓∗𝑔_𝑛‖‖≤=‖‖‖‖𝑓𝑓_𝑛−𝑓∗𝑔−𝑓∗𝑔_𝑛−𝑔_𝑛‖‖≤‖‖𝑓−𝑓∗𝑔‖‖+‖‖𝑓∗𝑔_𝑛−𝑔_𝑛‖‖−𝑓‖‖‖𝑔‖₁+‖𝑓‖‖‖𝑔_𝑛−𝑔‖‖₁(3.8)

Letting 𝑛→∞ shows that the limits of 𝑓_𝑛∗𝑔 in Definition 3.2 and 𝑓∗𝑔_𝑛 in Definition 3.1 are the same.

Theorem 3.4. Let 𝑓∈𝒜_𝐶 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)∈𝒜_𝐶.  (c) For each 𝑧∈ℝ, 𝜏_𝑧(𝑓∗𝑔)=(𝜏_𝑧𝑓)∗𝑔=(𝑓∗𝜏_𝑧𝑔).  (d) For each 𝑓∈𝒜_𝐶, define Φ_𝑓∶𝐿¹→𝒜_𝐶 by Φ_𝑓[𝑔]=𝑓∗𝑔. Then Φ_𝑓 is a bounded linear operator and ‖Φ_𝑓‖≤‖𝑓‖. There exists a nonzero distribution 𝑓∈𝒜_𝐶 such that ‖Φ_𝑓‖=‖𝑓‖. For each 𝑔∈𝐿¹, define Ψ_𝑔∶𝒜_𝐶→𝒜_𝐶 by Ψ_𝑔[𝑓]=𝑓∗𝑔. Then Ψ_𝑔 is a bounded linear operator and ‖Ψ_𝑔‖≤‖𝑔‖₁. There exists a nonzero function 𝑔∈𝐿¹ such that ‖Ψ_𝑔‖=‖𝑔‖_ℬ𝒱.  (e) Define 𝑔_𝑡(𝑥)=𝑔(𝑥/𝑡)/𝑡 for 𝑡>0. We have 𝑎=∫^∞−∞𝑔_𝑡(𝑥)𝑑𝑥=∫^∞−∞𝑔. Then ‖𝑓∗𝑔_𝑡−𝑎𝑓‖→0 as 𝑡→0.  (f)  Let supp(𝑓∗𝑔)⊂supp(𝑓)+supp(𝑔).

Proof. Let {𝑔_𝑛} be as in Definition 3.1. (a) Since ‖𝑓∗𝑔_𝑛‖→‖𝑓∗𝑔‖, (3.4) shows ‖𝑓∗𝑔‖≤‖𝑓‖‖𝑔‖₁. (b) Let {ℎ_𝑛}⊂ℬ𝒱∩𝐿¹ such that ‖ℎ_𝑛−ℎ‖₁→0. Then (𝑓∗𝑔)∗ℎ∶=𝜉∈𝒜_𝐶 such that ‖(𝑓∗𝑔)∗ℎ_𝑛−𝜉‖→0. Since 𝑔∗ℎ∈𝐿¹, there is {𝑝_𝑛}⊂ℬ𝒱∩𝐿¹ such that ‖𝑝_𝑛−𝑔∗ℎ‖₁→0. Then 𝑓∗(𝑔∗ℎ)∶=𝜂∈𝒜_𝐶 such that ‖𝑓∗𝑝_𝑛−𝜂‖→0. Now, ‖𝜉−𝜂‖≤+‖‖‖‖(𝑓(𝑓∗∗𝑔𝑔))∗∗ℎℎ_𝑛𝑛−𝜉‖‖+‖‖𝑓∗𝑝_𝑛−𝜂‖‖−𝑓∗𝑔_𝑛∗ℎ_𝑛‖‖+‖‖𝑓∗𝑔_𝑛∗ℎ_𝑛−𝑓∗𝑝_𝑛‖‖.(3.9) Using (3.4), ‖‖(𝑓=∗𝑔‖‖)∗𝑓ℎ∗_𝑛−𝑔−𝑓𝑔∗_𝑛𝑔_𝑛∗ℎ_𝑛‖‖∗ℎ_𝑛‖‖≤‖𝑓‖‖‖𝑔_𝑛−𝑔‖‖₁‖‖ℎ_𝑛‖‖₁⟶0as𝑛⟶∞.(3.10) Finally, use Theorem 2.1(e) and (3.4) to write ‖‖𝑓=∗‖‖𝑔_𝑛𝑓∗∗ℎ𝑔_𝑛𝑛−𝑓∗𝑝_𝑛‖‖≤‖𝑓‖‖‖𝑔∗_𝑛ℎ_𝑛−𝑝_𝑛‖‖−𝑔‖‖₁‖‖ℎ_𝑛‖‖₁+‖𝑔‖₁‖‖ℎ_𝑛−ℎ‖‖₁+‖‖𝑝_𝑛−𝑔∗ℎ‖‖₁⟶0as𝑛⟶∞.(3.11)(c) The Alexiewicz norm is invariant under translation [4, Theorem  28] so 𝜏_𝑧(𝑓∗𝑔)∈𝒜_𝐶. Use Theorem 2.1(f) to write ‖𝜏_𝑧(𝑓∗𝑔)−𝜏_𝑧(𝑓∗𝑔_𝑛)‖=‖𝑓∗𝑔−𝑓∗𝑔_𝑛‖=‖𝜏_𝑧(𝑓∗𝑔)−(𝜏_𝑧𝑓)∗𝑔_𝑛)‖=‖𝜏_𝑧(𝑓∗𝑔)−𝑓∗(𝜏_𝑧𝑔_𝑛)‖. Translation invariance of the 𝐿¹ norm completes the proof. (d) From (a), we have ‖Φ_𝑓‖=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 ‖Ψ_𝑔‖≤‖𝑔‖₁, note that ‖Ψ_𝑔‖=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 𝑓∗𝑔_𝑡(𝑥)=^∞−∞𝑓(𝑥−𝑦)𝑔𝑦𝑡𝑑𝑡𝑦=^∞−∞𝑓(𝑥−𝑡𝑦)𝑔(𝑦)𝑑𝑦.(3.12) For −∞<𝛼<𝛽<∞, ||||^𝛽𝛼𝑓∗𝑔_𝑡(𝑥)−𝑎𝑓(𝑥)𝑑𝑥||||==||||||||^𝛽𝛼^∞−∞[𝑓(𝑥−𝑡𝑦)−𝑓(𝑥)]𝑔(𝑦)𝑑𝑦𝑑𝑥_∞−∞||||^𝛽𝛼≤[𝑓(𝑥−𝑡𝑦)−𝑓(𝑥)]𝑔(𝑦)𝑑𝑥𝑑𝑦||||(3.13)_∞−∞≤2‖𝑓‖‖‖𝜏‖𝑔_𝑡𝑦𝑓‖₁−𝑓‖‖||𝑔(𝑦)||𝑑𝑦.(3.14) 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∗𝑔_𝑡−𝑎𝑓‖→0 as 𝑡→0.
Now take a sequence {𝑔^(𝑛)}⊂ℬ𝒱∩𝐿¹ such that ‖𝑔^(𝑛)−𝑔‖₁→0. Define 𝑔^𝑡(𝑛)(𝑥)=𝑔^(𝑛)(𝑥/𝑡)/𝑡 and 𝑎^(𝑛)=∫^∞−∞𝑔^(𝑛)(𝑥)𝑑𝑥. We have
‖‖𝑓∗𝑔_𝑡−𝑎𝑓‖‖≤‖‖𝑓∗𝑔^𝑡(𝑛)−𝑎^(𝑛)𝑓‖‖+‖‖𝑓∗𝑔^𝑡(𝑛)−𝑓∗𝑔_𝑡‖‖+‖‖𝑎^(𝑛)𝑓−𝑎𝑓‖‖.(3.15) By the inequality in (a), ‖𝑓∗𝑔^𝑡(𝑛)−𝑓∗𝑔_𝑡‖≤‖𝑓‖‖𝑔^𝑡(𝑛)−𝑔_𝑡‖₁. Whereas, ‖‖𝑔^𝑡(𝑛)−𝑔_𝑡‖‖₁=^∞−∞|||𝑔_(𝑛)𝑥𝑡−𝑔𝑥𝑡|||𝑑𝑡𝑥=‖‖𝑔^(𝑛)−𝑔‖‖₁⟶0as𝑛⟶∞,(3.16) and ‖𝑎^(𝑛)𝑓−𝑎𝑓‖=|𝑎^(𝑛)−𝑎|‖𝑓‖=‖𝑔^(𝑛)−𝑔‖₁‖𝑓‖. Given 𝜖>0 fix 𝑛 large enough so that ‖𝑓∗𝑔^𝑡(𝑛)−𝑓∗𝑔_𝑡‖+‖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
^𝛽𝛼^∞−∞𝑓(𝑥−𝑡𝑦)𝑔(𝑦)𝑑𝑦𝑑𝑥=^𝛽𝛼^∞−∞𝑓(𝑦)𝑔𝑥−𝑦=𝑡𝑑𝑡𝑦𝑑𝑥_∞−∞^𝛽𝛼𝑓(𝑦)𝑔𝑥−𝑦_∞−∞𝑡𝑑𝑥𝑑𝑡𝑦,^𝛽𝛼𝑓(𝑥−𝑡𝑦)𝑔(𝑦)𝑑𝑥𝑑𝑦=^∞−∞𝑓(𝑥)𝑔𝑦𝑡𝜒^{(𝛼−𝑦,𝛽−𝑦)}(𝑥)𝑑𝑥𝑑𝑦=_∞−∞𝑡𝑓(𝑥)𝑔𝑦𝑡𝜒^{(𝛼−𝑦,𝛽−𝑦)}(𝑥)𝑑𝑦=𝑡𝑑𝑥_∞−∞^𝛽𝛼𝑓(𝑥)𝑔𝑦−𝑥𝑡𝑑𝑡𝑦𝑑𝑥.(3.17) Note that ∫^𝛽𝛼∫^∞−∞𝑓(𝑥)𝑔(𝑦)𝑑𝑦𝑑𝑥=∫^∞−∞∫^𝛽𝛼𝑓(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∈𝐿^𝑝 for some 1≤𝑝≤∞ and 𝑔∈𝐿¹. Part (a) of Theorem 3.4 extends this to 𝑓∈𝒜_𝐶. see [1] for other results when 𝑓∈𝐿^𝑝.

The fact that convolution is linear in both arguments, together with (b), shows that 𝒜_𝐶 is an 𝐿¹-module over the 𝐿¹ convolution algebra, see [8] for the definition. It does not appear that 𝒜_𝐶 is a Banach algebra under convolution.

We now show that Definition 1.1(iii) and the aforementioned definitions agree.

Proposition 3.5. Let 𝑓∈𝒜_𝐶, 𝑔∈𝐿¹, and 𝜙∈𝒟. Define 𝐹(𝑦)=∫^𝑦−∞𝑓 and 𝐺(𝑥)=∫^𝑥−∞𝑔. Definitions 1.1 and 3.1 both give ⟨𝑓∗𝑔,𝜙⟩==^∞−∞𝑓(𝑦)^∞−∞𝑔(𝑥)𝜙(𝑥+𝑦)𝑑𝑥𝑑𝑦_∞−∞𝐹(𝑦)𝐺(𝑥)𝜙^(𝑥+𝑦)𝑑𝑥𝑑𝑦.(3.18)

Proof. Let Φ(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 {𝑔_𝑛}⊂ℬ𝒱∩𝐿¹ such that ‖𝑔_𝑛−𝑔‖₁→0. Since convergence in ‖⋅‖ implies convergence in 𝒟^, we have
⟨𝑓∗𝑔,𝜙⟩==lliimm_{𝑛→∞𝑛→∞}⟨𝑓∗𝑔_𝑛,𝜙⟩^∞−∞=lim_𝑛→∞𝑓(𝑦)𝑔_𝑛(𝑥−𝑦)𝜙(𝑥)𝑑𝑦𝑑𝑥^∞−∞𝑓(𝑦)^∞−∞𝑔_𝑛(𝑥−𝑦)𝜙(𝑥)𝑑𝑥𝑑𝑦.(3.19) Proposition A.3 allows interchange of the iterated integrals. Define Φ_𝑛(𝑦)=∫^∞−∞𝑔_𝑛(𝑥)𝜙(𝑥+𝑦)𝑑𝑥. Then, 𝑉Φ_𝑛≤‖𝑔_𝑛‖₁‖𝜙^‖₁≤(‖𝑔‖₁+1)‖𝜙^‖₁ for large enough 𝑛. Hence, Φ_𝑛 is of uniform bounded variation. Theorem  22 in [4], then gives ⟨𝑓∗𝑔,𝜙⟩=∫^∞−∞𝑓(𝑦)lim_𝑛→∞Φ_𝑛(𝑦)𝑑𝑦=∫^∞−∞𝑓(𝑦)∫^∞−∞𝑔(𝑥−𝑦)𝜙(𝑥)𝑑𝑥𝑑𝑦. The last step follows since ‖𝑔_𝑛−𝑔‖₁→0.

If 𝑔∈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⊂𝒜_𝐶 and 𝑔∈𝐿¹⧵ℬ𝒱. We have 𝑓∗𝑔(𝑥)=0 for 𝑥≤0. For 0<𝑥≤1, we have 𝑓∗𝑔(𝑥)=∫^𝑥0𝑦^−𝛼(𝑥−𝑦)^−𝛼𝑑𝑦=𝑥^1−2𝛼∫¹⁰𝑦^−𝛼(1−𝑦)^−𝛼𝑑𝑦=𝑥^1−2𝛼Γ²(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, consider 𝑓(𝑥)=sin(𝜋𝑥)/log|𝑥| and 𝑔(𝑥)=𝜒_(0,1)(𝑥). Then 𝑓∈𝒜_𝐶 and for each 1≤𝑝≤∞, we have 𝑔∈ℬ𝒱∩𝐿^𝑝. And,

𝑓∗𝑔(𝑥)=^𝑥𝑥−1sin(𝜋𝑦)=cos(𝜋lo(g𝑥(−𝑦)1𝑑))𝑦for𝑥≥2𝜋log(𝑥−1)−𝜋colsog(𝜋(𝑥𝑥))−𝜋1^𝑥𝑥−1cos(𝜋𝑦)𝑦log²(𝑦)𝑑𝑦∼−2𝜋cloosg((𝜋𝑥𝑥))as𝑥⟶∞.(3.20)

Therefore, by Theorem 2.1(d), 𝑓∗𝑔∈𝐶⁰(ℝ), and lim_|𝑥|→∞𝑓∗𝑔(𝑥)=0 but for each 1≤𝑝<∞, we have 𝑓∗𝑔∉𝐿^𝑝.

4. Differentiation and Integration

If 𝑔 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. Let 𝑓∈𝒜_𝐶, 𝑛∈ℕ, and 𝑔^(𝑘)∈𝐴𝐶(ℝ) for each 0≤𝑘≤𝑛. Then 𝑓∗𝑔∈𝐶^𝑛(ℝ) and (𝑓∗𝑔)^(𝑛)(𝑥)=𝑓∗𝑔^(𝑛)(𝑥) for each 𝑥∈ℝ.

Proof. First consider 𝑛=1. Let 𝑥∈ℝ. Then (𝑓∗𝑔)^(𝑥)=lim_ℎ→0^∞−∞𝑓(𝑦)𝑔(𝑥+ℎ−𝑦)−𝑔(𝑥−𝑦)ℎ𝑑𝑦.(4.1) 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 𝑉_𝑦∈ℝ=𝑔(𝑥_∞−∞+ℎ−𝑦ℎ)−𝑔(𝑥−𝑦)||||𝑔^(𝑥+ℎ−𝑦)−𝑔^(𝑥−𝑦)≤ℎ||||𝑑𝑦_∞−∞||𝑔^(𝑦)||𝑑𝑦+^∞−∞||||𝑔^(𝑥+ℎ−𝑦)−𝑔^(𝑥−𝑦)ℎ−𝑔^(𝑥−𝑦)||||𝑑𝑦.(4.2) 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 when 𝑓∈𝐿¹, see [1, Proposition  8.10].

Note that 𝑔^∈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)+∫^𝑥0𝑔^(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 when 𝑔∈ℬ𝒱∩𝐿¹, then 𝑓∗𝑔∈𝒜_𝐶. We can compute the distributional derivative (𝐹∗𝑔)^=𝑓∗𝑔, where 𝐹 is a primitive of 𝑓.

Proposition 4.2. Let 𝐹∈𝐶⁰(ℝ) and write 𝑓=𝐹^∈𝒜_𝐶. Let 𝑔∈ℬ𝒱∩𝐿¹. Then 𝐹∗𝑔∈𝐶⁰(ℝ) and (𝐹∗𝑔)^=𝑓∗𝑔∈𝒜_𝐶.

Proof. Let 𝑥,𝑡∈ℝ. Then by the usual Hölder inequality, ||𝐹∗=𝑔(||||𝑥)−𝐹∗𝑔(𝑡)_∞−∞||≤‖𝐹(𝑥[𝐹−(⋅𝑥)−−𝐹𝑦)(−𝑡−𝐹⋅()𝑡‖−_∞𝑦)]𝑔(𝑦)𝑑𝑦||||‖𝑔‖₁⟶0as𝑡⟶𝑥since𝐹isuniformlycontinuousonℝ.(4.3) Hence, 𝐹∗𝑔 is continuous on ℝ. Dominated convergence shows that lim_𝑥→±∞𝐹∗𝑔(𝑥)=𝐹(±∞)∫^∞−∞𝑔. Therefore, 𝐹∗𝑔∈𝐶⁰(ℝ).
Let 𝜙∈𝒟. Then
(𝐹∗𝑔)^,𝜙==−−𝐹_∞−∞∗𝑔,𝜙^=−_∞−∞𝐹(𝑥−𝑦)𝑔(𝑦)𝜙^(𝑥)𝑑𝑦𝑑𝑥𝑔(𝑦)^∞−∞𝐹(𝑥−𝑦)𝜙^(𝑥)𝑑𝑥𝑑𝑦(Fubini-Tonellitheorem).(4.4) Integrate by parts and use the change of variables 𝑥↦𝑥+𝑦 to get (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_∞−∞=⟨𝑓∗𝜙𝑔,(𝜙𝑦)⟩.^∞−∞𝑓(𝑥)𝑔(𝑦−𝑥)𝑑𝑥𝑑𝑦(byPropositionA.3)(4.5)

This gives an alternate definition of 𝑓∗𝑔 for 𝑓∈𝒜_𝐶 and 𝑔∈𝐿¹.

Theorem 4.3. Let 𝑓∈𝒜_𝐶, let 𝐹∈ℬ_𝐶 be the primitive of 𝑓 and let 𝑔∈𝐿¹. Define 𝑓∗𝑔 as in Definition 3.1. Then (𝐹∗𝑔)^=𝑓∗𝑔∈𝒜_𝐶.

Proof. Let −∞<𝛼<𝛽<∞. Let {𝑔_𝑛}⊂ℬ𝒱∩𝐿¹ such that ‖𝑔_𝑛−𝑔‖₁→0. By Proposition 4.2, we have ^𝛽𝛼(𝐹∗𝑔)^=𝐹∗𝑔(𝛽)−𝐹∗𝑔(𝛼)=^∞−∞𝐹(𝑦)[𝑔(𝛽−𝑦)−𝑔(]