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Abstract and Applied Analysis

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Abstract and Applied Analysis / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 307404 | https://doi.org/10.1155/2009/307404

Erik Talvila, "Convolutions with the Continuous Primitive Integral", Abstract and Applied Analysis, vol. 2009, Article ID 307404, 18 pages, 2009. https://doi.org/10.1155/2009/307404

Convolutions with the Continuous Primitive Integral

Erik Talvila1

1Department of Mathematics and Statistics, University of the Fraser Valley, Abbotsford, BC, Canada

Academic Editor: H. Bevan Thompson
Received13 May 2009
Accepted07 Sep 2009
Published01 Nov 2009

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‖=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.

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∈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 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 {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∈𝐿1loc, 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  translation 𝜏𝑥 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=𝐹for𝐹∈ℬ𝐶}. If 𝑓∈𝒜𝐶, then ∫𝑏𝑎𝑓=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 in 𝒜𝐶 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 ∫𝑏𝑎𝐹(𝑥)𝑑𝑥=0 but 𝐹∈𝒜𝐶 and ∫𝑏𝑎𝐹=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 ∫𝑏𝑎𝐹=𝐹(𝑏)−𝐹(𝑎) for all 𝑎,𝑏∈ℝ.

The space 𝒜𝐶 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|∫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 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. Let 𝑔∶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).

If 𝑔∈𝐿1loc, 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∈𝐿1loc∣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. 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,𝜏𝑥𝜓⟩; (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 of 𝒜𝐶 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 𝒜𝐶×ℬ𝒱

In this section, we prove basic results for the convolution when 𝑓∈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 of 𝑓∈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. Let 𝑓∈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 ||||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 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 ||𝑠𝑔∗ℎ𝑖𝑡−𝑔∗ℎ𝑖||≤∞−∞||𝑔𝑠𝑖𝑡−𝑦−𝑔𝑖||||||=−𝑦ℎ(𝑦)𝑑𝑦∞−∞||𝑔𝑠𝑖𝑡−𝑦−𝑔𝑖||||||−𝑦ℎ(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∞−∞ℎ(𝑧)∞−∞𝑓(𝑦)𝑔(𝑥−𝑦−𝑧)𝑑𝑦𝑑𝑧=(𝑓∗𝑔)∗ℎ(𝑥).(2.3) 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|∫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−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 for 𝑓∈𝐿𝑝 in [1, Section  8.2].

If we use the equivalent norm ‖𝑓‖=sup𝑥∈ℝ|∫𝑥−∞𝑓|, 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)=𝑥−∞𝑓. 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. If 𝑓∈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 Definition 1.1 are compatible with 𝑓∗𝑔 for 𝑓∈𝒜𝐶 and 𝑔∈ℬ𝒱.

Proof. Let 𝑓∈𝒜𝐶, 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,𝜏𝑥𝜓⟩=∞−∞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∞−∞𝜙(𝑥)∞−∞𝑓(𝑦)𝑔(𝑥−𝑦)𝑑𝑥𝑑𝑥=⟨𝑓∗𝑔,𝜙⟩.(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 𝒜𝐶(loc)={𝑓∈𝒟∣𝑓=𝐹forsome𝐹∈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 ∫𝛽𝛼𝑓(𝑥−𝑦)𝑔(𝑦)𝑑𝑦 exist as 𝛼→−∞ and 𝛽→∞. This gives

𝑓∗𝑔(𝑥)=𝑏𝑎𝑓(𝑥−𝑦)𝑔(𝑦)𝑑𝑦=𝑥−𝑎𝑥−𝑏𝑓(𝑦)𝑔(𝑥−𝑦)𝑑𝑦.(2.7)

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

𝑓∗𝑔(𝑥)=lim𝛼→−∞𝛽→∞𝛽𝛼𝑓(𝑥−𝑦)𝑔(𝑦)𝑑𝑦=lim𝛼→−∞𝛽→∞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 𝒜𝐶, ℬ𝒱 or 𝐿𝑝 for any 1≤𝑝≤∞ but 𝑓∗𝑔 exists on ℝ because 𝑓,𝑔∈𝐿1loc 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∈𝒜𝐶, we have ∫𝑓∗𝑔(𝑥)=𝑥−∞𝑓=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 in 𝒜𝐶×𝐿1

We now extend the convolution 𝑓∗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. Let 𝑓∈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 that ℬ𝒱∩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

𝛽𝛼𝑓∗𝑔𝑛(𝑥)𝑑𝑥=𝛽𝛼∞−∞𝑓(𝑦)𝑔𝑛=(𝑥−𝑦)𝑑𝑦𝑑𝑥∞−∞𝑓(𝑦)𝛽𝛼𝑔𝑛(=𝑥−𝑦)𝑑𝑥𝑑𝑦(3.1)∞−∞𝑓(𝑦)𝛽−𝑦𝛼−𝑦𝑔𝑛(𝑥)𝑑𝑥𝑑𝑦=−∞−∞𝐹(𝑦)𝑑𝛽−𝑦𝛼−𝑦𝑔𝑛=(3.2)∞−∞𝑔𝐹(𝑦)𝑛(𝛽−𝑦)−𝑔𝑛=(𝛼−𝑦)𝑑𝑦∞−∞𝛽−𝑦𝛼−𝑦𝑓𝑔𝑛(𝑦)𝑑𝑦.(3.3) 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→∞∫𝛽−𝑦𝛼−𝑦𝑔𝑛=0. As 𝐹 is continuous and the function ∫𝑦↦𝛽−𝑦𝛼−𝑦𝑔𝑛 is absolutely continuous, we get (3.3). Taking the supremum over 𝛼,𝛽∈ℝ gives

‖‖𝑓∗𝑔𝑛‖‖≤‖‖𝑔‖𝑓‖𝑛‖‖1.(3.4) We now have

‖‖𝑓∗𝑔𝑚−𝑓∗𝑔𝑛‖‖=‖‖𝑔𝑓∗𝑚−𝑔𝑛‖‖‖‖𝑔≤‖𝑓‖𝑚−𝑔𝑛‖‖1(3.5)

and {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. Let 𝑓∈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 show 𝐿1 is dense in 𝒜𝐶.

Proposition 3.3. 𝐿1 is dense in 𝒜𝐶.

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∈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 Definition 3.2, the required sequence {𝑓𝑛}⊂𝐿1 exists. Let [𝛼,𝛽]⊂ℝ be a compact interval. Then, by the usual Fubini-Tonelli theorem in 𝐿1,

𝛽𝛼𝑓𝑛∗𝑔(𝑥)𝑑𝑥=𝛽𝛼∞−∞𝑓𝑛=(𝑥−𝑦)𝑔(𝑦)𝑑𝑦𝑑𝑥∞−∞𝑔(𝑦)𝛽𝛼𝑓𝑛(𝑥−𝑦)𝑑𝑥𝑑𝑦.(3.6)

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

‖‖𝑓𝑛‖‖≤‖‖𝑓∗𝑔𝑛‖‖‖𝑔‖1.(3.7) 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

‖‖𝑓𝑛∗𝑔−𝑓∗𝑔𝑛‖‖=‖‖𝑓𝑛𝑔−𝑓∗𝑔−𝑓∗𝑛‖‖≤‖‖𝑓−𝑔𝑛‖‖+‖‖𝑔−𝑓∗𝑔𝑓∗𝑛‖‖≤‖‖𝑓−𝑔𝑛‖‖−𝑓‖𝑔‖1‖‖𝑔+‖𝑓‖𝑛‖‖−𝑔1(3.8)

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

Theorem 3.4. Let 𝑓∈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, ‖‖‖𝜉−𝜂‖≤(𝑓∗𝑔)∗ℎ𝑛‖‖+‖‖−𝜉𝑓∗𝑝𝑛‖‖+‖‖−𝜂(𝑓∗𝑔)∗ℎ𝑛−𝑓∗𝑔𝑛∗ℎ𝑛‖‖+‖‖𝑓∗𝑔𝑛∗ℎ𝑛−𝑓∗𝑝𝑛‖‖.(3.9) Using (3.4), ‖‖(𝑓∗𝑔)∗ℎ𝑛−𝑓∗𝑔𝑛∗ℎ𝑛‖‖=‖‖𝑓∗𝑔−𝑔𝑛∗ℎ𝑛‖‖‖‖𝑔≤‖𝑓‖𝑛‖‖−𝑔1‖‖ℎ𝑛‖‖1⟶0as𝑛⟶∞.(3.10) Finally, use Theorem 2.1(e) and (3.4) to write ‖‖𝑓∗𝑔𝑛∗ℎ𝑛−𝑓∗𝑝𝑛‖‖=‖‖𝑔𝑓∗𝑛∗ℎ𝑛−𝑝𝑛‖‖‖‖𝑔≤‖𝑓‖𝑛‖‖−𝑔1‖‖ℎ𝑛‖‖1+‖𝑔‖1‖‖ℎ𝑛‖‖−ℎ1+‖‖𝑝𝑛‖‖−𝑔∗ℎ1⟶0as𝑛⟶∞.(3.11)(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 𝑓∗𝑔𝑡(𝑥)=∞−∞𝑦𝑓(𝑥−𝑦)𝑔𝑡𝑑𝑦𝑡=∞−∞𝑓(𝑥−𝑡𝑦)𝑔(𝑦)𝑑𝑦.(3.12) For −∞<𝛼<𝛽<∞, ||||𝛽𝛼𝑓∗𝑔𝑡||||=||||(𝑥)−𝑎𝑓(𝑥)𝑑𝑥𝛽𝛼∞−∞[𝑓]𝑔||||=||||(𝑥−𝑡𝑦)−𝑓(𝑥)(𝑦)𝑑𝑦𝑑𝑥∞−∞𝛽𝛼[]||||≤𝑓(𝑥−𝑡𝑦)−𝑓(𝑥)𝑔(𝑦)𝑑𝑥𝑑𝑦(3.13)∞−∞‖‖𝜏𝑡𝑦‖‖||||𝑓−𝑓𝑔(𝑦)𝑑𝑦≤2‖𝑓‖‖𝑔‖1.(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∗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
‖‖𝑓∗𝑔𝑡‖‖≤‖‖−𝑎𝑓𝑓∗𝑔𝑡(𝑛)−𝑎(𝑛)𝑓‖‖+‖‖𝑓∗𝑔𝑡(𝑛)−𝑓∗𝑔𝑡‖‖+‖‖𝑎(𝑛)‖‖𝑓−𝑎𝑓.(3.15) By the inequality in (a), ‖𝑓∗𝑔𝑡(𝑛)−𝑓∗𝑔𝑡‖≤‖𝑓‖‖𝑔𝑡(𝑛)−𝑔𝑡‖1. Whereas, ‖‖𝑔𝑡(𝑛)−𝑔𝑡‖‖1=∞−∞|||𝑔(𝑛)𝑥𝑡𝑥−𝑔𝑡|||𝑑𝑥𝑡=‖‖𝑔(𝑛)‖‖−𝑔1⟶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∗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
𝛽𝛼∞−∞𝑓(𝑥−𝑡𝑦)𝑔(𝑦)𝑑𝑦𝑑𝑥=𝛽𝛼∞−∞𝑓(𝑦)𝑔𝑥−𝑦𝑡𝑑𝑦𝑡=𝑑𝑥∞−∞𝛽𝛼𝑓(𝑦)𝑔𝑥−𝑦𝑡𝑑𝑥𝑑𝑦𝑡,∞−∞𝛽𝛼𝑓(𝑥−𝑡𝑦)𝑔(𝑦)𝑑𝑥𝑑𝑦=∞−∞𝑓𝑦(𝑥)𝑔𝑡𝜒(𝛼−𝑦,𝛽−𝑦)(𝑥)𝑑𝑥𝑑𝑦𝑡=∞−∞𝑦𝑓(𝑥)𝑔𝑡𝜒(𝛼−𝑦,𝛽−𝑦)(𝑥)𝑑𝑦𝑡=𝑑𝑥∞−∞𝛽𝛼𝑓(𝑥)𝑔𝑦−𝑥𝑡𝑑𝑦𝑡𝑑𝑥.(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∈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 that 𝒜𝐶 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 Definition 1.1(iii) and the aforementioned definitions agree.

Proposition 3.5. Let 𝑓∈𝒜𝐶, 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 {𝑔𝑛}⊂ℬ𝒱∩𝐿1 such that ‖𝑔𝑛−𝑔‖1→0. Since convergence in ‖⋅‖ implies convergence in 𝒟, we have
⟨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.(3.19) 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.

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⊂𝒜𝐶 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=𝑥1−2𝛼∫10𝑦−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, consider 𝑓(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,

𝑓∗𝑔(𝑥)=𝑥𝑥−1sin(𝜋𝑦)=log(𝑦)𝑑𝑦for𝑥≥2cos(𝜋(𝑥−1))−𝜋log(𝑥−1)cos(𝜋𝑥)−1𝜋log(𝑥)𝜋𝑥𝑥−1cos(𝜋𝑦)𝑦log2(𝑦)𝑑𝑦∼−2cos(𝜋𝑥)𝜋log(𝑥)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∉𝐿𝑝.

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∈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 consider 𝑛=1. Let 𝑥∈ℝ. Then (𝑓∗𝑔)(𝑥)=limℎ→0∞−∞𝑓(𝑦)𝑔(𝑥+ℎ−𝑦)−𝑔(𝑥−𝑦)ℎ𝑑𝑦.(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 𝑉𝑦∈ℝ𝑔(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.(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)(