Abstract and Applied Analysis

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

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

Abstract

If ๐น is a continuous function on the real line and ๐‘“=๐น๎…ž is its distributional derivative, then the continuous primitive integral of distribution ๐‘“ is โˆซ๐‘๐‘Ž๐‘“=๐น(๐‘)โˆ’๐น(๐‘Ž). 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 โˆซ๐‘“โˆ—๐‘”(๐‘ฅ)=โˆžโˆ’โˆž๐‘“(๐‘ฅโˆ’๐‘ฆ)๐‘”(๐‘ฆ)๐‘‘๐‘ฆ for ๐‘“ an integrable distribution and ๐‘” a function of bounded variation or an ๐ฟ1 function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. For ๐‘” of bounded variation, ๐‘“โˆ—๐‘” is uniformly continuous and we have the estimate โ€–๐‘“โˆ—๐‘”โ€–โˆžโ‰คโ€–๐‘“โ€–โ€–๐‘”โ€–โ„ฌ๐’ฑ, where โ€–๐‘“โ€–=sup๐ผ|โˆซ๐ผ๐‘“| is the Alexiewicz norm. This supremum is taken over all intervals ๐ผโŠ‚โ„. When ๐‘”โˆˆ๐ฟ1, the estimate is โ€–๐‘“โˆ—๐‘”โ€–โ‰คโ€–๐‘“โ€–โ€–๐‘”โ€–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 ๐‘“ and ๐‘” on the real line is โˆซ๐‘“โˆ—๐‘”(๐‘ฅ)=โˆžโˆ’โˆž๐‘“(๐‘ฅโˆ’๐‘ฆ)๐‘”(๐‘ฆ)๐‘‘๐‘ฆ. 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 ๐ฟ1 and โˆ—โˆถ๐ฟ1ร—๐ฟ1โ†’๐ฟ1 such that โ€–๐‘“โˆ—๐‘”โ€–1โ‰คโ€–๐‘“โ€–1โ€–๐‘”โ€–1. The convolution is commutative, associative, and commutes with translations. If ๐‘“โˆˆ๐ฟ1 and ๐‘”โˆˆ๐ถ๐‘›, then ๐‘“โˆ—๐‘”โˆˆ๐ถ๐‘› and (๐‘“โˆ—๐‘”)(๐‘›)(๐‘ฅ)=๐‘“โˆ—๐‘”(๐‘›)(๐‘ฅ). Convolutions also have the approximation property that if ๐‘“โˆˆ๐ฟ๐‘ (1โ‰ค๐‘<โˆž) and ๐‘”โˆˆ๐ฟ1, then โ€–๐‘“โˆ—๐‘”๐‘กโˆ’๐‘Ž๐‘“โ€–๐‘โ†’0 as ๐‘กโ†’0, where ๐‘”๐‘ก(๐‘ฅ)=๐‘”(๐‘ฅ/๐‘ก)/๐‘ก and โˆซ๐‘Ž=โˆžโˆ’โˆž๐‘”. When ๐‘“ is bounded and continuous, there is a similar result for ๐‘=โˆž. 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 ๐’Ÿ=๐ถโˆž๐‘(โ„)={๐œ™โˆถโ„โ†’โ„โˆฃ๐œ™โˆˆ๐ถโˆž(โ„)andsupp(๐œ™)iscompact}. The support of function ๐œ™ is the closure of the set on which ๐œ™ does not vanish and is denoted supp(๐œ™). Under usual pointwise operations, ๐’Ÿ is a linear space over field โ„. In ๐’Ÿ, we have a notion of convergence. If {๐œ™๐‘›}โŠ‚๐’Ÿ, then ๐œ™๐‘›โ†’0 as ๐‘›โ†’โˆž if there is a compact set ๐พโŠ‚โ„ such that for each ๐‘›, supp(๐œ™๐‘›)โŠ‚๐พ, and for each ๐‘šโ‰ฅ0, we have ๐œ™๐‘›(๐‘š)โ†’0 uniformly on ๐พ as ๐‘›โ†’โˆž. Theโ€‰โ€‰distributionsโ€‰โ€‰are denoted ๐’Ÿ๎…ž and are the continuous linear functionals on ๐’Ÿ. For ๐‘‡โˆˆ๐’Ÿ๎…ž and ๐œ™โˆˆ๐’Ÿ, we write โŸจ๐‘‡,๐œ™โŸฉโˆˆโ„. For ๐œ™,๐œ“โˆˆ๐’Ÿ and ๐‘Ž,๐‘โˆˆโ„, we have โŸจ๐‘‡,๐‘Ž๐œ™+๐‘๐œ“โŸฉ=๐‘ŽโŸจ๐‘‡,๐œ™โŸฉ+๐‘โŸจ๐‘‡,๐œ“โŸฉ. Moreover, if ๐œ™๐‘›โ†’0 in ๐’Ÿ, then โŸจ๐‘‡,๐œ™๐‘›โŸฉโ†’0 in โ„. Linear operations are defined in ๐’Ÿ๎…ž by โŸจ๐‘Ž๐‘†+๐‘๐‘‡,๐œ™โŸฉ=๐‘ŽโŸจ๐‘†,๐œ™โŸฉ+๐‘โŸจ๐‘‡,๐œ™โŸฉ for ๐‘†,๐‘‡โˆˆ๐’Ÿ๎…ž; ๐‘Ž,๐‘โˆˆโ„ and ๐œ™โˆˆ๐’Ÿ. If ๐‘“โˆˆ๐ฟ1loc, then โŸจ๐‘‡๐‘“โˆซ,๐œ™โŸฉ=โˆžโˆ’โˆž๐‘“(๐‘ฅ)๐œ™(๐‘ฅ)๐‘‘๐‘ฅ defines a distribution ๐‘‡๐‘“โˆˆ๐’Ÿ๎…ž. The integral exists as a Lebesgue integral. All distributions have derivatives of all orders that are themselves distributions. For ๐‘‡โˆˆ๐’Ÿ๎…ž and ๐œ™โˆˆ๐’Ÿ, theโ€‰โ€‰distributional derivative of ๐‘‡ is ๐‘‡๎…ž where โŸจ๐‘‡๎…ž,๐œ™โŸฉ=โˆ’โŸจ๐‘‡,๐œ™๎…žโŸฉ. This is also called theโ€‰โ€‰weak derivative. If ๐‘โˆถโ„โ†’โ„ is a function that is differentiable in the pointwise sense at ๐‘ฅโˆˆโ„, then we write its derivative as ๐‘๎…ž(๐‘ฅ). If ๐‘ is a ๐ถโˆž bijection such that ๐‘๎…ž(๐‘ฅ)โ‰ 0 forโ€‰โ€‰any ๐‘ฅโˆˆโ„, then the compositionโ€‰โ€‰withโ€‰โ€‰distribution ๐‘‡ isโ€‰โ€‰definedโ€‰โ€‰by โŸจ๐‘‡โˆ˜๐‘,๐œ™โŸฉ=โŸจ๐‘‡,(๐œ™โˆ˜๐‘โˆ’1)/(๐‘๎…žโˆ˜๐‘โˆ’1)โŸฉ for all ๐œ™โˆˆ๐’Ÿ. Translations are a special case. For ๐‘ฅโˆˆโ„, define theโ€‰โ€‰translation ๐œ๐‘ฅ onโ€‰โ€‰distribution ๐‘‡โˆˆ๐’Ÿ๎…ž by โŸจ๐œ๐‘ฅ๐‘‡,๐œ™โŸฉ=โŸจ๐‘‡,๐œโˆ’๐‘ฅ๐œ™โŸฉ for test function ๐œ™โˆˆ๐’Ÿ, where ๐œ๐‘ฅ๐œ™(๐‘ฆ)=๐œ™(๐‘ฆโˆ’๐‘ฅ). Allโ€‰โ€‰ofโ€‰โ€‰theโ€‰โ€‰resultsโ€‰โ€‰on distributions we use can be found in [3].

The following Banach space will be of importance: โ„ฌ๐ถ={๐นโˆถโ„โ†’โ„โˆฃ๐นโˆˆ๐ถ0(โ„),๐น(โˆ’โˆž)=0,๐น(โˆž)โˆˆโ„}. We use the notation ๐น(โˆ’โˆž)=lim๐‘ฅโ†’โˆ’โˆž๐น(๐‘ฅ) and ๐น(โˆž)=lim๐‘ฅโ†’โˆž๐น(๐‘ฅ). The extended real line is denoted by โ„=[โˆ’โˆž,โˆž]. The space โ„ฌ๐ถ then consists of functions continuous on โ„ with a limit of 0 at โˆ’โˆž. We denote the functions that are continuous on โ„ that have real limits at ยฑโˆž by ๐ถ0(โ„). Hence, โ„ฌ๐ถ is properly contained in ๐ถ0(โ„), which is itself properly contained in the space of uniformly continuous functions on โ„. The space โ„ฌ๐ถ is a Banach space under the uniform norm; โ€–๐นโ€–โˆž=sup๐‘ฅโˆˆโ„|๐น(๐‘ฅ)|=max๐‘ฅโˆˆโ„|๐น(๐‘ฅ)| for ๐นโˆˆโ„ฌ๐ถ. The continuous primitive integral is defined by taking โ„ฌ๐ถ as the space of primitives. The space of integrable distributions is ๐’œ๐ถ={๐‘“โˆˆ๐’Ÿ๎…žโˆฃ๐‘“=๐น๎…žfor๐นโˆˆโ„ฌ๐ถ}. If ๐‘“โˆˆ๐’œ๐ถ, then โˆซ๐‘๐‘Ž๐‘“=๐น(๐‘)โˆ’๐น(๐‘Ž) for ๐‘Ž,๐‘โˆˆโ„. The distributional differential equation ๐‘‡๎…ž=0 has only constant solutions so the primitive ๐นโˆˆโ„ฌ๐ถ satisfying ๐น๎…ž=๐‘“ is unique. Integrable distributions are then tempered and of order one. This integral, including a discussion of extensions to โ„๐‘›, 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 ๐’œ๐ถ are ๐‘‡๐‘“ for functions ๐‘“ that have a finite Lebesgue, Henstock-Kurzweil, or wide Denjoy integral. We identify function ๐‘“ with the distribution ๐‘‡๐‘“. Pointwise function values can be recovered from ๐‘‡๐‘“ at points of continuity of ๐‘“ by evaluating the limit โŸจ๐‘‡๐‘“,๐œ™๐‘›โŸฉ for a delta sequence converging to ๐‘ฅโˆˆโ„. This is a sequence of test functions {๐œ™๐‘›}โŠ‚๐’Ÿ such that for each ๐‘›, ๐œ™๐‘›โ‰ฅ0, โˆซโˆžโˆ’โˆž๐œ™๐‘›=1, and the support of ๐œ™๐‘› tends to {๐‘ฅ} as ๐‘›โ†’โˆž. Note that if ๐นโˆˆ๐ถ0(โ„) is an increasing function with ๐น๎…ž(๐‘ฅ)=0 for almost all ๐‘ฅโˆˆโ„, then the Lebesgue integral โˆซ๐‘๐‘Ž๐น๎…ž(๐‘ฅ)๐‘‘๐‘ฅ=0 but ๐น๎…žโˆˆ๐’œ๐ถ and โˆซ๐‘๐‘Ž๐น๎…ž=๐น(๐‘)โˆ’๐น(๐‘Ž). For another example of a distribution in ๐’œ๐ถ, let ๐นโˆˆ๐ถ0(โ„) be continuous and nowhere differentiable in the pointwise sense. Then ๐น๎…žโˆˆ๐’œ๐ถ and โˆซ๐‘๐‘Ž๐น๎…ž=๐น(๐‘)โˆ’๐น(๐‘Ž) for all ๐‘Ž,๐‘โˆˆโ„.

The space ๐’œ๐ถ is a Banach space under the Alexiewicz norm; โ€–๐‘“โ€–=sup๐ผโŠ‚โ„|โˆซ๐ผ๐‘“|, where the supremum is taken over all intervals ๐ผโŠ‚โ„. An equivalent norm is โ€–๐‘“โ€–๎…ž=sup๐‘ฅโˆˆโ„|โˆซ๐‘ฅโˆ’โˆž๐‘“|. 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 ๐’œ๐ถ 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 ๐’œ๐ถ is a suitable setting for applications of nonabsolute integration.

We will also need to use functions of bounded variation. Let ๐‘”โˆถโ„โ†’โ„. The variation of ๐‘” is โˆ‘๐‘‰๐‘”=sup|๐‘”(๐‘ฅ๐‘–)โˆ’๐‘”(๐‘ฆ๐‘–)| where the supremum is taken over all disjoint intervals {(๐‘ฅ๐‘–,๐‘ฆ๐‘–)}. The functions of bounded variation are denoted โ„ฌ๐’ฑ={๐‘”โˆถโ„โ†’โ„โˆฃ๐‘‰๐‘”<โˆž}. This is a Banach space under the norm โ€–๐‘”โ€–โ„ฌ๐’ฑ=|๐‘”(โˆ’โˆž)|+๐‘‰๐‘”. Equivalent norms are โ€–๐‘”โ€–โˆž+๐‘‰๐‘” and |๐‘”(๐‘Ž)|+๐‘‰๐‘” for each ๐‘Žโˆˆโ„. Functions of bounded variation have a left and right limit at each point in โ„ and limits at ยฑโˆž, so, as above, we will define ๐‘”(ยฑโˆž)=lim๐‘ฅโ†’ยฑโˆž๐‘”(๐‘ฅ).

If ๐‘”โˆˆ๐ฟ1loc, then the essential variation of ๐‘” is โˆซessvar๐‘”=supโˆžโˆ’โˆž๐‘”๐œ™๎…ž, where the supremum is taken over all ๐œ™โˆˆ๐’Ÿ with โ€–๐œ™โ€–โˆžโ‰ค1. Then โ„ฐโ„ฌ๐’ฑ={๐‘”โˆˆ๐ฟ1locโˆฃessvar๐‘”<โˆž}. This is a Banach space under the norm โ€–๐‘”โ€–โ„ฐโ„ฌ๐’ฑ=esssup|๐‘”|+essvar๐‘”. Let 0โ‰ค๐›พโ‰ค1. For ๐‘”โˆถโ„โ†’โ„, define ๐‘”๐›พ(๐‘ฅ)=(1โˆ’๐›พ)๐‘”(๐‘ฅโˆ’)+๐›พ๐‘”(๐‘ฅ+). For left continuity, ๐›พ=0 and for right continuity ๐›พ=1. The functionsโ€‰โ€‰ofโ€‰โ€‰normalized bounded variation are ๐’ฉโ„ฌ๐’ฑ๐›พ={๐‘”๐›พโˆฃ๐‘”โˆˆโ„ฌ๐’ฑ}. If ๐‘”โˆˆโ„ฐโ„ฌ๐’ฑ, then essvar๐‘”=inf๐‘‰โ„Ž such that โ„Ž=๐‘” almost everywhere. For each 0โ‰ค๐›พโ‰ค1, there is exactly one function โ„Žโˆˆ๐’ฉโ„ฌ๐’ฑ๐›พ such that ๐‘”=โ„Ž almost everywhere. In this case, essvar๐‘”=๐‘‰โ„Ž. Changing ๐‘” 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 ๐œ‡๐‘” where โŸจ๐‘”๎…ž,๐œ™โŸฉ=โˆ’โŸจ๐‘”,๐œ™๎…žโˆซโŸฉ=โˆ’โˆžโˆ’โˆž๐‘”๐œ™๎…ž=โˆซโˆžโˆ’โˆž๐œ™๐‘‘๐œ‡๐‘” for all ๐œ™โˆˆ๐’Ÿ.

We will see that โˆ—โˆถ๐’œ๐ถร—โ„ฌ๐’ฑโ†’๐ถ0(โ„) and that โ€–๐‘“โˆ—๐‘”โ€–โˆžโ‰คโ€–๐‘“โ€–โ€–๐‘”โ€–โ„ฌ๐’ฑ. Similarly for ๐‘”โˆˆโ„ฐโ„ฌ๐’ฑ. Convolutions for ๐‘“โˆˆ๐’œ๐ถ and ๐‘”โˆˆ๐ฟ1 will be defined using sequences in โ„ฌ๐’ฑโˆฉ๐ฟ1 that converge to ๐‘” in the ๐ฟ1 norm. It will be shown that โˆ—โˆถ๐’œ๐ถร—๐ฟ1โ†’๐’œ๐ถ and that โ€–๐‘“โˆ—๐‘”โ€–โ‰คโ€–๐‘“โ€–โ€–๐‘”โ€–1.

Convolutions can be defined for distributions in several different ways.

Definition 1.1. Let ๐‘†,๐‘‡โˆˆ๐’Ÿ๎…ž and ๐œ™,๐œ“โˆˆ๐’Ÿ. Define ๎‚๐œ™(๐‘ฅ)=๐œ™(โˆ’๐‘ฅ): (i) โŸจ๐‘‡โˆ—๐œ“,๐œ™โŸฉ=โŸจ๐‘‡,๐œ™โˆ—๎‚๐œ“โŸฉ, (ii) for each ๐‘ฅโˆˆโ„, let ๐‘‡โˆ—๐œ“(๐‘ฅ)=โŸจ๐‘‡,๐œ๐‘ฅ๎‚๐œ“โŸฉ; (iii) โŸจ๐‘†โˆ—๐‘‡,๐œ™โŸฉ=โŸจ๐‘†(๐‘ฅ),โŸจ๐‘‡(๐‘ฆ),๐œ™(๐‘ฅ+๐‘ฆ)โŸฉโŸฉ.

In (i), โˆ—โˆถ๐’Ÿ๎…žร—๐’Ÿโ†’๐’Ÿ๎…ž. 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), โˆ—โˆถ๐’Ÿ๎…žร—๐’Ÿโ†’๐ถโˆž. In [1], it is shown that definitions (i) and (ii) are equivalent. In (iii), โˆ—โˆถ๐’Ÿ๎…žร—๐’Ÿ๎…žโ†’๐’Ÿ๎…ž. However, this definition requires restrictions on the supports of ๐‘† and ๐‘‡. 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, โŸจ๐‘†โŠ—๐‘‡,ฮฆโŸฉ=โŸจ๐‘†(๐‘ฅ),โŸจ๐‘‡(๐‘ฆ),ฮฆ(๐‘ฅ,๐‘ฆ)โŸฉโŸฉ, where now ฮฆโˆˆ๐’Ÿ(โ„2).

Under (i), ๐‘‡โˆ—๐œ“ is in ๐ถโˆž. It satisfies (๐‘‡โˆ—๐œ“)โˆ—๐œ™=๐‘‡โˆ—(๐œ“โˆ—๐œ™), ๐œ๐‘ฅ(๐‘‡โˆ—๐œ“)=(๐œ๐‘ฅ๐‘‡)โˆ—๐œ“=๐‘‡โˆ—(๐œ๐‘ฅ๐œ“), and (๐‘‡โˆ—๐œ“)(๐‘›)=๐‘‡โˆ—๐œ“(๐‘›)=๐‘‡(๐‘›)โˆ—๐œ“. Under (iii), with appropriate support restrictions, ๐‘†โˆ—๐‘‡ is in ๐’Ÿ๎…ž. It is commutative and associative, commutes with translations, and satisfies (๐‘†โˆ—๐‘‡)(๐‘›)=๐‘†(๐‘›)โˆ—๐‘‡=๐‘†โˆ—๐‘‡(๐‘›). It is weakly continuous in ๐’Ÿ๎…ž, that is, if ๐‘‡๐‘›โ†’๐‘‡ in ๐’Ÿ๎…ž, then ๐‘‡๐‘›โˆ—๐œ“โ†’๐‘‡โˆ—๐œ“ in ๐’Ÿ๎…ž see [1, 3, 6, 7] for additional properties of convolutions of distributions.

Although elements of ๐’œ๐ถ 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 ๐’œ๐ถร—โ„ฌ๐’ฑ

In this section, we prove basic results for the convolution when ๐‘“โˆˆ๐’œ๐ถ and ๐‘”โˆˆโ„ฌ๐’ฑ. Under these conditions, ๐‘“โˆ—๐‘” is commutative, continuous on โ„, and commutes with translations. It can be estimated in the uniform norm in terms of the Alexiewicz and โ„ฌ๐’ฑ norms. There is also an associative property. We first need the result that โ„ฌ๐’ฑ forms the space of multipliers for ๐’œ๐ถ, that is, if ๐‘“โˆˆ๐’œ๐ถ, then ๐‘“๐‘”โˆˆ๐’œ๐ถ for all ๐‘”โˆˆโ„ฌ๐’ฑ. The integral โˆซ๐ผ๐‘“๐‘” is defined using the integration by parts formula in the appendix. The Hรถlder inequality (A.5) shows that โ„ฌ๐’ฑ is the dual space of ๐’œ๐ถ.

We define the convolution of ๐‘“โˆˆ๐’œ๐ถ and ๐‘”โˆˆโ„ฌ๐’ฑ as โˆซ๐‘“โˆ—๐‘”(๐‘ฅ)=โˆžโˆ’โˆž(๐‘“โˆ˜๐‘Ÿ๐‘ฅ)๐‘”, where ๐‘Ÿ๐‘ฅ(๐‘ก)=๐‘ฅโˆ’๐‘ก. We write this as โˆซ๐‘“โˆ—๐‘”(๐‘ฅ)=โˆžโˆ’โˆž๐‘“(๐‘ฅโˆ’๐‘ฆ)๐‘”(๐‘ฆ)๐‘‘๐‘ฆ.

Theorem 2.1. Let ๐‘“โˆˆ๐’œ๐ถ and let ๐‘”โˆˆโ„ฌ๐’ฑ. Then (a)โ€‰โ€‰๐‘“โˆ—๐‘” exists on โ„.โ€‰โ€‰(b) Let ๐‘“โˆ—๐‘”=๐‘”โˆ—๐‘“.โ€‰โ€‰(c)โ€‰โ€‰Letโ€‰โ€‰โ€–๐‘“โˆ—๐‘”โ€–โˆžโˆซโ‰ค|โˆžโˆ’โˆž๐‘“|infโ„|๐‘”|+โ€–๐‘“โ€–๐‘‰๐‘”โ‰คโ€–๐‘“โ€–โ€–๐‘”โ€–โ„ฌ๐’ฑ. โ€‰โ€‰(d) โ€‰โ€‰Assume ๐‘“โˆ—๐‘”โˆˆ๐ถ0(โ„), lim๐‘ฅโ†’ยฑโˆžโˆซ๐‘“โˆ—๐‘”(๐‘ฅ)=๐‘”(ยฑโˆž)โˆžโˆ’โˆž๐‘“. โ€‰โ€‰(e) If โ„Žโˆˆ๐ฟ1, then ๐‘“โˆ—(๐‘”โˆ—โ„Ž)=(๐‘“โˆ—๐‘”)โˆ—โ„Žโˆˆ๐ถ0(โ„). โ€‰โ€‰ (f) Let ๐‘ฅ,๐‘งโˆˆโ„, โ€‰โ€‰then ๐œ๐‘ง(๐‘“โˆ—๐‘”)(๐‘ฅ)=(๐œ๐‘ง๐‘“)โˆ—๐‘”(๐‘ฅ)=(๐‘“โˆ—๐œ๐‘ง๐‘”)(๐‘ฅ). โ€‰โ€‰(g) โ€‰โ€‰For eachโ€‰โ€‰๐‘“โˆˆ๐’œ๐ถ, โ€‰โ€‰define ฮฆ๐‘“โˆถโ„ฌ๐’ฑโ†’๐ถ0(โ„) by ฮฆ๐‘“[๐‘”]=๐‘“โˆ—๐‘”. โ€‰โ€‰Then ฮฆ๐‘“ is a bounded linear operator and โ€–ฮฆ๐‘“โ€–โ‰คโ€–๐‘“โ€–. โ€‰โ€‰There exists a nonzero distributionโ€‰โ€‰๐‘“โˆˆ๐’œ๐ถ such that โ€–ฮฆ๐‘“โ€–=โ€–๐‘“โ€–. For each ๐‘”โˆˆโ„ฌ๐’ฑ, define ฮจ๐‘”โˆถ๐’œ๐ถโ†’๐ถ0(โ„) by ฮจ๐‘”[๐‘“]=๐‘“โˆ—๐‘”. Then ฮจ๐‘” is a bounded linear operator and โ€–ฮจ๐‘”โ€–โ‰คโ€–๐‘”โ€–โ„ฌ๐’ฑ. There exists a nonzero function ๐‘”โˆˆโ„ฌ๐’ฑ such that โ€–ฮจ๐‘”โ€–=โ€–๐‘”โ€–โ„ฌ๐’ฑ. โ€‰โ€‰(h)โ€‰โ€‰supp(๐‘“โˆ—๐‘”)โŠ‚supp(๐‘“)+supp(๐‘”).

Proof. (a) Existence is given via the integration by parts formula (A.1) in the appendix. (b) See [4, Theorem โ€‰11] for a change of variables theorem that can be used with ๐‘ฆโ†ฆ๐‘ฅโˆ’๐‘ฆ. (c) This inequality follows from the Hรถlder inequality (A.5). (d) Let ๐‘ฅ,๐‘กโˆˆโ„. From (c), we have ||||๐‘“โˆ—๐‘”(๐‘ก)โˆ’๐‘“โˆ—๐‘”(๐‘ฅ)โ‰คโ€–๐‘“(๐‘กโˆ’โ‹…)โˆ’๐‘“(๐‘ฅโˆ’โ‹…)โ€–โ€–๐‘”โ€–โ„ฌ๐’ฑ=โ€–๐‘“(๐‘กโˆ’๐‘ฅโˆ’โ‹…)โˆ’๐‘“(โ‹…)โ€–โ€–๐‘”โ€–โ„ฌ๐’ฑโŸถ0as๐‘กโŸถ๐‘ฅ.(2.1) The last line follows from continuity in the Alexiewicz norm [4, Theorem โ€‰22]. Hence, ๐‘“โˆ—๐‘” is uniformly continuous on โ„. Also, it follows that lim๐‘ฅโ†’โˆžโˆซโˆžโˆ’โˆžโˆซ๐‘“(๐‘ฆ)๐‘”(๐‘ฅโˆ’๐‘ฆ)๐‘‘๐‘ฆ=โˆžโˆ’โˆž๐‘“(๐‘ฆ)lim๐‘ฅโ†’โˆžโˆซ๐‘”(๐‘ฅโˆ’๐‘ฆ)๐‘‘๐‘ฆ=๐‘”(โˆž)โˆžโˆ’โˆž๐‘“. The limit ๐‘ฅโ†’โˆž can be taken under the integral sign since ๐‘”(๐‘ฅโˆ’๐‘ฆ) is of uniform bounded variation, that is, ๐‘‰๐‘ฆโˆˆโ„๐‘”(๐‘ฅโˆ’๐‘ฆ)=๐‘‰๐‘”. Theorem โ€‰22 in [4] then applies. Similarly, as ๐‘ฅโ†’โˆ’โˆž. (e) First show ๐‘”โˆ—โ„Žโˆˆโ„ฌ๐’ฑ. Let {(๐‘ ๐‘–,๐‘ก๐‘–)} be disjoint intervals in โ„. Then ๎“||๎€ท๐‘ ๐‘”โˆ—โ„Ž๐‘–๎€ธ๎€ท๐‘กโˆ’๐‘”โˆ—โ„Ž๐‘–๎€ธ||โ‰ค๎“๎€œโˆžโˆ’โˆž||๐‘”๎€ท๐‘ ๐‘–๎€ธ๎€ท๐‘กโˆ’๐‘ฆโˆ’๐‘”๐‘–๎€ธ||||||=๎€œโˆ’๐‘ฆโ„Ž(๐‘ฆ)๐‘‘๐‘ฆโˆžโˆ’โˆž๎“||๐‘”๎€ท๐‘ ๐‘–๎€ธ๎€ท๐‘กโˆ’๐‘ฆโˆ’๐‘”๐‘–๎€ธ||||||โˆ’๐‘ฆโ„Ž(๐‘ฆ)๐‘‘๐‘ฆ.(2.2) Hence, ๐‘‰(๐‘”โˆ—โ„Ž)โ‰ค๐‘‰๐‘”โ€–โ„Žโ€–1. The interchange of sum and integral follows from the Fubini-Tonelli theorem. Now (d) shows ๐‘“โˆ—(๐‘”โˆ—โ„Ž)โˆˆ๐ถ0(โ„). Write ๎€œ๐‘“โˆ—(๐‘”โˆ—โ„Ž)(๐‘ฅ)=โˆžโˆ’โˆž๎€œ๐‘“(๐‘ฆ)โˆžโˆ’โˆž=๎€œ๐‘”(๐‘ฅโˆ’๐‘ฆโˆ’๐‘ง)โ„Ž(๐‘ง)๐‘‘๐‘ง๐‘‘๐‘ฆโˆžโˆ’โˆž๎€œโ„Ž(๐‘ง)โˆžโˆ’โˆž๐‘“(๐‘ฆ)๐‘”(๐‘ฅโˆ’๐‘ฆโˆ’๐‘ง)๐‘‘๐‘ฆ๐‘‘๐‘ง=(๐‘“โˆ—๐‘”)โˆ—โ„Ž(๐‘ฅ).(2.3) We can interchange orders of integration using Proposition A.3. For (ii) in Proposition A.3, the function ๐‘งโ†ฆ๐‘‰๐‘ฆโˆˆโ„๐‘”(๐‘ฅโˆ’๐‘ฆโˆ’๐‘ง)โ„Ž(๐‘ง)=๐‘‰๐‘”โ„Ž(๐‘ง) is in ๐ฟ1 for each fixed ๐‘ฅโˆˆโ„. Since ๐‘” is of bounded variation, it is bounded so |๐‘”(๐‘ฅโˆ’๐‘ฆโˆ’๐‘ง)โ„Ž(๐‘ง)|โ‰คโ€–๐‘”โ€–โˆž|โ„Ž(๐‘ง)| and condition (iii) is satisfied. (f) This follows from a linear change of variables as in (a). (g) From (c), we have โ€–ฮฆ๐‘“โ€–=supโ€–๐‘”โ€–โ„ฌ๐’ฑ=1โ€–๐‘“โˆ—๐‘”โ€–โˆžโ‰คsupโ€–๐‘”โ€–โ„ฌ๐’ฑ=1โ€–๐‘“โ€–โ€–๐‘”โ€–โ„ฌ๐’ฑ=โ€–๐‘“โ€–. Let ๐‘“>0 be in ๐ฟ1. If ๐‘”=1, then โ€–๐‘”โ€–โ„ฌ๐’ฑ=1 and โˆซ๐‘“โˆ—๐‘”(๐‘ฅ)=โˆžโˆ’โˆž๐‘“ so โ€–ฮฆ๐‘“โ€–=โ€–๐‘“โ€–=โ€–๐‘“โ€–1. To prove โ€–ฮจ๐‘”โ€–โ‰คโ€–๐‘”โ€–โ„ฌ๐’ฑ, note that โ€–ฮจ๐‘”โ€–=supโ€–๐‘“โ€–=1โ€–๐‘“โˆ—๐‘”โ€–โˆžโ‰คsupโ€–๐‘“โ€–=1โ€–๐‘“โ€–โ€–๐‘”โ€–โ„ฌ๐’ฑ=โ€–๐‘”โ€–โ„ฌ๐’ฑ. Let ๐‘”=๐œ’(0,โˆž). Then โ€–ฮจ๐‘”โ€–=supโ€–๐‘“โ€–=1โ€–๐‘“โˆ—๐‘”โ€–โˆž=supโ€–๐‘“โ€–=1sup๐‘ฅโˆˆโ„|โˆซ๐‘ฅโˆ’โˆž๐‘“|=1=โ€–๐‘”โ€–โ„ฌ๐’ฑ. (h) Suppose ๐‘ฅโˆ‰supp(๐‘“)+supp(๐‘”). Note that we can write โˆซ๐‘“โˆ—๐‘”(๐‘ฅ)=โˆžโˆ’โˆž๐‘”(๐‘ฅโˆ’๐‘ฆ)๐‘‘๐น(๐‘ฆ) in terms of a Henstock-Stieltjes integral, see [4] for details. This integral is approximated by Riemann sums โˆ‘๐‘๐‘›=1๐‘”(๐‘ฅโˆ’๐‘ง๐‘›)[๐น(๐‘ก๐‘›)โˆ’๐น(๐‘ก๐‘›โˆ’1)] where ๐‘ง๐‘›โˆˆ[๐‘ก๐‘›โˆ’1,๐‘ก๐‘›], โˆ’โˆž=๐‘ก0<๐‘ก1<โ‹ฏ<๐‘ก๐‘=โˆž and there is a gauge function ๐›พ mapping โ„ to the open intervals in โ„ such that [๐‘ก๐‘›โˆ’1,๐‘ก๐‘›]โŠ‚๐›พ(๐‘ง๐‘›). If ๐‘ง๐‘›โˆ‰supp(๐‘“), then since โ„โงตsupp(๐‘“) is open, there is an open interval ๐‘ง๐‘›โŠ‚๐ผโŠ‚โ„โงตsupp(๐‘“). We can take ๐›พ such that [๐‘ก๐‘›โˆ’1,๐‘ก๐‘›]โŠ‚๐ผ for all 1โ‰ค๐‘›โ‰ค๐‘. Also, ๐น is constant on each interval in โ„โงตsupp(๐‘“). Therefore, ๐‘”(๐‘ฅโˆ’๐‘ง๐‘›)[๐น(๐‘ก๐‘›)โˆ’๐น(๐‘ก๐‘›โˆ’1)]=0 and only tags ๐‘ง๐‘›โˆˆsupp(๐‘“) can contribute to the Riemann sum. However, for all ๐‘ง๐‘›โˆˆsupp(๐‘“), we have ๐‘ฅโˆ’๐‘ง๐‘›โˆ‰supp(๐‘”) so ๐‘”(๐‘ฅโˆ’๐‘ง๐‘›)[๐น(๐‘ก๐‘›)โˆ’๐น(๐‘ก๐‘›โˆ’1)]=0. It follows that ๐‘“โˆ—๐‘”(๐‘ฅ)=0.

Similar results are proven for ๐‘“โˆˆ๐ฟ๐‘ in [1, Sectionโ€‰โ€‰8.2].

If we use the equivalent norm โ€–๐‘“โ€–๎…ž=sup๐‘ฅโˆˆโ„|โˆซ๐‘ฅโˆ’โˆž๐‘“|, then โ€–ฮฆ๐‘“โ€–=โ€–๐‘“โ€–๎…ž. Also, integration by parts gives โ€–ฮฆ๐‘“โ€–โ‰คโ€–๐‘“โ€–๎…ž. Now, given ๐‘“โˆˆ๐’œ๐ถ, let ๐‘”=๐œ’(0,โˆž). Then โ€–๐‘”โ€–โ„ฌ๐’ฑ=1,โ€‰โ€‰and โˆซ๐‘“โˆ—๐‘”(๐‘ฅ)=๐‘ฅโˆ’โˆž๐‘“. Hence, โ€–๐‘“โˆ—๐‘”โ€–โˆž=โ€–๐‘“โ€–๎…ž and โ€–ฮฆ๐‘“โ€–=โ€–๐‘“โ€–๎…ž. We can have strict inequality in โ€–ฮจ๐‘”โ€–โ‰คโ€–๐‘”โ€–โ„ฌ๐’ฑ. For example, let ๐‘”=๐œ’{0},โ€‰โ€‰then โ€–๐‘”โ€–โ„ฌ๐’ฑ=2 but integration by parts shows ๐‘“โˆ—๐‘”=0 for each ๐‘“โˆˆ๐’œ๐ถ.

Remark 2.2. If ๐‘“โˆˆ๐’œ๐ถ and ๐‘”โˆˆโ„ฐโ„ฌ๐’ฑ, one can use Definition A.2 to define ๐‘“โˆ—๐‘”(๐‘ฅ)=๐‘“โˆ—๐‘”๐›พ(๐‘ฅ) where ๐‘”๐›พ=๐‘” almost everywhere and ๐‘”๐›พโˆˆ๐’ฉโ„ฌ๐’ฑ๐›พ. All of the results in Theorem 2.1 and the rest of this paper have analogues. Note that ๐‘“โˆ—๐‘”(๐‘ฅ)=๐น(โˆž)๐‘”๐›พ(โˆ’โˆž)+๐นโˆ—๐œ‡๐‘”.

Proposition 2.3. The three definitions of convolution for distributions in Definition 1.1 are compatible with ๐‘“โˆ—๐‘” for ๐‘“โˆˆ๐’œ๐ถ and ๐‘”โˆˆโ„ฌ๐’ฑ.

Proof. Let ๐‘“โˆˆ๐’œ๐ถ, ๐‘”โˆˆโ„ฌ๐’ฑ, and ๐œ™,๐œ“โˆˆ๐’Ÿ. Definition 1.1(i) gives ๎€œโŸจ๐‘“,๎‚๐œ“โˆ—๐œ™โŸฉ=โˆžโˆ’โˆž๎€œ๐‘“(๐‘ฅ)โˆžโˆ’โˆž๎€œ๐œ“(๐‘ฆโˆ’๐‘ฅ)๐œ™(๐‘ฆ)๐‘‘๐‘ฆ๐‘‘๐‘ฅ=โˆžโˆ’โˆž๎€œโˆžโˆ’โˆž๐‘“(๐‘ฅ)๐œ“(๐‘ฆโˆ’๐‘ฅ)๐œ™(๐‘ฆ)๐‘‘๐‘ฅ๐‘‘๐‘ฆ=โŸจ๐‘“โˆ—๐œ“,๐œ™โŸฉ.(2.4) Since ๐œ“โˆˆโ„ฌ๐’ฑ and ๐œ™โˆˆ๐ฟ1, Proposition A.3 justifies the interchange of integrals. Definition 1.1(ii) gives โŸจ๐‘“,๐œ๐‘ฅ๎€œ๎‚๐œ“โŸฉ=โˆžโˆ’โˆž๐‘“(๐‘ฆ)๐œ“(๐‘ฅโˆ’๐‘ฆ)๐‘‘๐‘ฆ=๐‘“โˆ—๐œ“(๐‘ฅ).(2.5) Definition 1.1(iii) gives ๎€œโŸจ๐‘“(๐‘ฆ),โŸจ๐‘”(๐‘ฅ),๐œ™(๐‘ฅ+๐‘ฆ)โŸฉโŸฉ=โˆžโˆ’โˆž๎€œ๐‘“(๐‘ฆ)โˆžโˆ’โˆž=๎€œ๐‘”(๐‘ฅ)๐œ™(๐‘ฅ+๐‘ฆ)๐‘‘๐‘ฅ๐‘‘๐‘ฆโˆžโˆ’โˆž๎€œ๐‘“(๐‘ฆ)โˆžโˆ’โˆž=๎€œ๐‘”(๐‘ฅโˆ’๐‘ฆ)๐œ™(๐‘ฅ)๐‘‘๐‘ฅ๐‘‘๐‘ฆโˆžโˆ’โˆž๎€œ๐œ™(๐‘ฅ)โˆžโˆ’โˆž๐‘“(๐‘ฆ)๐‘”(๐‘ฅโˆ’๐‘ฆ)๐‘‘๐‘ฅ๐‘‘๐‘ฅ=โŸจ๐‘“โˆ—๐‘”,๐œ™โŸฉ.(2.6) The interchange of integrals is accomplished using Proposition A.3 since ๐‘”โˆˆโ„ฌ๐’ฑ and ๐œ™โˆˆ๐ฟ1.

The locally integrable distributions are defined as ๐’œ๐ถ(loc)={๐‘“โˆˆ๐’Ÿ๎…žโˆฃ๐‘“=๐น๎…žforsome๐นโˆˆ๐ถ0(โ„)}. Let ๐‘“โˆˆ๐’œ๐ถ(loc) and let ๐‘”โˆˆโ„ฌ๐’ฑ with support in the compact interval [๐‘Ž,๐‘]. By the Hake theorem [4, Theorem โ€‰25], ๐‘“โˆ—๐‘”(๐‘ฅ) exists if and only if the limits of โˆซ๐›ฝ๐›ผ๐‘“(๐‘ฅโˆ’๐‘ฆ)๐‘”(๐‘ฆ)๐‘‘๐‘ฆ exist as ๐›ผโ†’โˆ’โˆž and ๐›ฝโ†’โˆž. This gives

๎€œ๐‘“โˆ—๐‘”(๐‘ฅ)=๐‘๐‘Ž๎€œ๐‘“(๐‘ฅโˆ’๐‘ฆ)๐‘”(๐‘ฆ)๐‘‘๐‘ฆ=๐‘ฅโˆ’๐‘Ž๐‘ฅโˆ’๐‘๐‘“(๐‘ฆ)๐‘”(๐‘ฅโˆ’๐‘ฆ)๐‘‘๐‘ฆ.(2.7)

There are analogues of the results in Theorem 2.1. For example, โˆซ|๐‘“โˆ—๐‘”(๐‘ฅ)|โ‰ค|๐‘ฅโˆ’๐‘Ž๐‘ฅโˆ’๐‘๐‘“|inf[๐‘Ž,๐‘]|๐‘”|+โ€–๐‘“๐œ’[๐‘ฅโˆ’๐‘,๐‘ฅโˆ’๐‘Ž]โ€–๐‘‰[๐‘Ž,๐‘]๐‘”. There are also versions where the supports are taken to be semi-infinite intervals.

We can also define the distributions with bounded primitive as ๐’œ๐ถ(๐‘๐‘‘)={๐‘“โˆˆ๐’Ÿ๎…žโˆฃ๐‘“=๐น๎…žforsomebounded๐นโˆˆ๐ถ0(โ„)with๐น(0)=0}. Let ๐‘“โˆˆ๐’œ๐ถ(๐‘๐‘‘) and let ๐น be its unique primitive. If ๐‘”โˆˆโ„ฌ๐’ฑ such that ๐‘”(ยฑโˆž)=0, then

๐‘“โˆ—๐‘”(๐‘ฅ)=lim๐›ผโ†’โˆ’โˆž๐›ฝโ†’โˆž๎€œ๐›ฝ๐›ผ๐‘“(๐‘ฅโˆ’๐‘ฆ)๐‘”(๐‘ฆ)๐‘‘๐‘ฆ=lim๐›ผโ†’โˆ’โˆž๐›ฝโ†’โˆž๎‚ธ๎€œ๐น(๐‘ฅโˆ’๐›ผ)๐‘”(๐›ผ)โˆ’๐น(๐‘ฅโˆ’๐›ฝ)๐‘”(๐›ฝ)+๐›ฝ๐›ผ๎‚น=๎€œ๐น(๐‘ฅโˆ’๐‘ฆ)๐‘‘๐‘”(๐‘ฆ)โˆžโˆ’โˆž๎€œ๐น(๐‘ฅโˆ’๐‘ฆ)๐‘‘๐‘”(๐‘ฆ)=โˆžโˆ’โˆž๐น(๐‘ฆ)๐‘‘๐‘”(๐‘ฅโˆ’๐‘ฆ).(2.8)

It follows that โ€–๐‘“โˆ—๐‘”โ€–โˆžโ‰คโ€–๐นโ€–โˆž๐‘‰๐‘”.

It is possible to formulate other existence criteria. For example, if ๐‘“(๐‘ฅ)=log|๐‘ฅ|sin(๐‘ฅ) and ๐‘”(๐‘ฅ)=|๐‘ฅ|โˆ’๐›ผ for some 0<๐›ผ<1, then ๐‘“ and ๐‘” are not in ๐’œ๐ถ, โ„ฌ๐’ฑ or ๐ฟ๐‘ for any 1โ‰ค๐‘โ‰คโˆž but ๐‘“โˆ—๐‘” exists on โ„ because ๐‘“,๐‘”โˆˆ๐ฟ1loc and if โˆซ๐น(๐‘ฅ)=๐‘ฅ0๐‘“, then lim|๐‘ฅ|โ†’โˆž๐น(๐‘ฅ)๐‘”(๐‘ฅ)=0.

The following example shows that ๐‘“โˆ—๐‘” needs not to be of bounded variation and hence not absolutely continuous. Let ๐‘”=๐œ’(0,โˆž). For ๐‘“โˆˆ๐’œ๐ถ, we have โˆซ๐‘“โˆ—๐‘”(๐‘ฅ)=๐‘ฅโˆ’โˆž๐‘“=๐น(๐‘ฅ), where ๐นโˆˆโ„ฌ๐ถ is the primitive of ๐‘“. However, ๐น needs not to be of bounded variation or even of local bounded variation. For example, let ๐‘“(๐‘ฅ)=sin(๐‘ฅโˆ’2)โˆ’2๐‘ฅโˆ’2cos(๐‘ฅโˆ’2) and let ๐น be its primitive in โ„ฌ๐ถ. Finally, although ๐‘“โˆ—๐‘” is continuous, it needs not to be integrable over โ„. For example, let ๐‘”=1, then โˆซ๐‘“โˆ—๐‘”(๐‘ฅ)=โˆžโˆ’โˆž๐‘“ and โˆซโˆžโˆ’โˆž๐‘“โˆ—๐‘” only exists if โˆซโˆžโˆ’โˆž๐‘“=0.

3. Convolution in ๐’œ๐ถร—๐ฟ1

We now extend the convolution ๐‘“โˆ—๐‘” to ๐‘“โˆˆ๐’œ๐ถ and ๐‘”โˆˆ๐ฟ1. Since there are functions in ๐ฟ1 that are not of bounded variation, there are distributions ๐‘“โˆˆ๐’œ๐ถ and functions ๐‘”โˆˆ๐ฟ1 such that the integral โˆซโˆžโˆ’โˆž๐‘“(๐‘ฅโˆ’๐‘ฆ)๐‘”(๐‘ฆ)๐‘‘๐‘ฆ does not exist. The convolution is then defined as the limit in โ€–โ‹…โ€– of a sequence ๐‘“โˆ—๐‘”๐‘› for ๐‘”๐‘›โˆˆโ„ฌ๐’ฑโˆฉ๐ฟ1 such that ๐‘”๐‘›โ†’๐‘” in the ๐ฟ1 norm. This is possible since โ„ฌ๐’ฑโˆฉ๐ฟ1 is dense in ๐ฟ1. We also give an equivalent definition using the fact that ๐ฟ1 is dense in ๐’œ๐ถ. Take a sequence {๐‘“๐‘›}โŠ‚๐ฟ1 such that โ€–๐‘“๐‘›โˆ’๐‘“โ€–โ†’0. Then ๐‘“โˆ—๐‘” is the limit in โ€–โ‹…โ€– of ๐‘“๐‘›โˆ—๐‘”. In this more general setting of convolution defined in ๐’œ๐ถร—๐ฟ1, we now have an Alexiewicz norm estimate for ๐‘“โˆ—๐‘” in terms of estimates of ๐‘“ in the Alexiewicz norm and ๐‘” in the ๐ฟ1 norm. There is associativity with ๐ฟ1 functions and commutativity with translations.

Definition 3.1. Let ๐‘“โˆˆ๐’œ๐ถ and let ๐‘”โˆˆ๐ฟ1. Let {๐‘”๐‘›}โŠ‚โ„ฌ๐’ฑโˆฉ๐ฟ1 such that โ€–๐‘”๐‘›โˆ’๐‘”โ€–1โ†’0. Define ๐‘“โˆ—๐‘” as the unique element in ๐’œ๐ถ such that โ€–๐‘“โˆ—๐‘”๐‘›โˆ’๐‘“โˆ—๐‘”โ€–โ†’0.

To see that the definition makes sense, first note that โ„ฌ๐’ฑโˆฉ๐ฟ1 is dense in ๐ฟ1 since step functions are dense in ๐ฟ1. Hence, the required sequence {๐‘”๐‘›} exists. Let [๐›ผ,๐›ฝ]โŠ‚โ„ be a compact interval. Let ๐นโˆˆโ„ฌ๐ถ be the primitive of ๐‘“. Then

๎€œ๐›ฝ๐›ผ๐‘“โˆ—๐‘”๐‘›๎€œ(๐‘ฅ)๐‘‘๐‘ฅ=๐›ฝ๐›ผ๎€œโˆžโˆ’โˆž๐‘“(๐‘ฆ)๐‘”๐‘›=๎€œ(๐‘ฅโˆ’๐‘ฆ)๐‘‘๐‘ฆ๐‘‘๐‘ฅโˆžโˆ’โˆž๎€œ๐‘“(๐‘ฆ)๐›ฝ๐›ผ๐‘”๐‘›(=๎€œ๐‘ฅโˆ’๐‘ฆ)๐‘‘๐‘ฅ๐‘‘๐‘ฆ(3.1)โˆžโˆ’โˆž๐‘“๎€œ(๐‘ฆ)๐›ฝโˆ’๐‘ฆ๐›ผโˆ’๐‘ฆ๐‘”๐‘›๎€œ(๐‘ฅ)๐‘‘๐‘ฅ๐‘‘๐‘ฆ=โˆ’โˆžโˆ’โˆž๐น๎‚ธ๎€œ(๐‘ฆ)๐‘‘๐›ฝโˆ’๐‘ฆ๐›ผโˆ’๐‘ฆ๐‘”๐‘›๎‚น=๎€œ(3.2)โˆžโˆ’โˆž๎€บ๐‘”๐น(๐‘ฆ)๐‘›(๐›ฝโˆ’๐‘ฆ)โˆ’๐‘”๐‘›๎€ป=๎€œ(๐›ผโˆ’๐‘ฆ)๐‘‘๐‘ฆโˆžโˆ’โˆž๎‚ต๎€œ๐›ฝโˆ’๐‘ฆ๐›ผโˆ’๐‘ฆ๐‘“๎‚ถ๐‘”๐‘›(๐‘ฆ)๐‘‘๐‘ฆ.(3.3) The interchange of orders of integration in (3.1) is accomplished with Proposition A.3 using ๐‘”(๐‘ฅ,๐‘ฆ)=๐‘”๐‘›(๐‘ฅโˆ’๐‘ฆ)๐œ’[๐›ผ,๐›ฝ](๐‘ฅ). Integration by parts gives (3.2) since lim๐‘ฆโ†’โˆžโˆซ๐›ฝโˆ’๐‘ฆ๐›ผโˆ’๐‘ฆ๐‘”๐‘›=0. As ๐น is continuous and the function โˆซ๐‘ฆโ†ฆ๐›ฝโˆ’๐‘ฆ๐›ผโˆ’๐‘ฆ๐‘”๐‘› is absolutely continuous, we get (3.3). Taking the supremum over ๐›ผ,๐›ฝโˆˆโ„ gives

โ€–โ€–๐‘“โˆ—๐‘”๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘”โ€–๐‘“โ€–๐‘›โ€–โ€–1.(3.4) We now have

โ€–โ€–๐‘“โˆ—๐‘”๐‘šโˆ’๐‘“โˆ—๐‘”๐‘›โ€–โ€–=โ€–โ€–๎€ท๐‘”๐‘“โˆ—๐‘šโˆ’๐‘”๐‘›๎€ธโ€–โ€–โ€–โ€–๐‘”โ‰คโ€–๐‘“โ€–๐‘šโˆ’๐‘”๐‘›โ€–โ€–1(3.5)

and {๐‘“โˆ—๐‘”๐‘›} is a Cauchy sequence in ๐’œ๐ถ. Since ๐’œ๐ถ is complete, this sequence has a limit in ๐’œ๐ถ which we denote ๐‘“โˆ—๐‘”. The definition does not depend on the choice of sequence {๐‘”๐‘›}, thus if {โ„Ž๐‘›}โŠ‚โ„ฌ๐’ฑโˆฉ๐ฟ1 such that โ€–โ„Ž๐‘›โˆ’๐‘”โ€–1โ†’0, then โ€–๐‘“โˆ—๐‘”๐‘›โˆ’๐‘“โˆ—โ„Ž๐‘›โ€–โ‰คโ€–๐‘“โ€–(โ€–๐‘”๐‘›โˆ’๐‘”โ€–1+โ€–โ„Ž๐‘›โˆ’๐‘”โ€–1)โ†’0 as ๐‘›โ†’โˆž. The previous calculation also shows that if ๐‘”โˆˆโ„ฌ๐’ฑโˆฉ๐ฟ1, then the integral definition โˆซ๐‘“โˆ—๐‘”(๐‘ฅ)=โˆžโˆ’โˆž๐‘“(๐‘ฅโˆ’๐‘ฆ)๐‘”(๐‘ฆ)๐‘‘๐‘ฆ and the limit definition agree.

Definition 3.2. Let ๐‘“โˆˆ๐’œ๐ถ and let ๐‘”โˆˆ๐ฟ1. Let {๐‘“๐‘›}โŠ‚๐ฟ1 such that โ€–๐‘“๐‘›โˆ’๐‘“โ€–โ†’0. Define ๐‘“โˆ—๐‘” as the unique element in ๐’œ๐ถ such that โ€–๐‘“๐‘›โˆ—๐‘”โˆ’๐‘“โˆ—๐‘”โ€–โ†’0.

To show this definition makes sense, first show ๐ฟ1 is dense in ๐’œ๐ถ.

Proposition 3.3. ๐ฟ1 is dense in ๐’œ๐ถ.

Proof. Let ๐ด๐ถ(โ„) be the functions that are absolutely continuous on each compact interval and which are of bounded variation on the real line. Then, ๐‘“โˆˆ๐ฟ1 if and only if there exists ๐นโˆˆ๐ด๐ถ(โ„) such that ๐น๎…ž(๐‘ฅ)=๐‘“(๐‘ฅ) for almost all ๐‘ฅโˆˆโ„. Let ๐‘“โˆˆ๐’œ๐ถ be given. Let ๐นโˆˆโ„ฌ๐ถ be its primitive. For ๐œ–>0, take ๐‘€>0 such that |F(๐‘ฅ)|<๐œ– for ๐‘ฅ<โˆ’๐‘€ and |๐น(๐‘ฅ)โˆ’๐น(โˆž)|<๐œ– for ๐‘ฅ>๐‘€. Due to the Weierstrass approximation theorem, there is a continuous function ๐‘ƒโˆถโ„โ†’โ„ such that ๐‘ƒ(๐‘ฅ)=๐น(โˆ’๐‘€) for ๐‘ฅโ‰คโˆ’๐‘€, ๐‘ƒ(๐‘ฅ)=๐น(๐‘€) for ๐‘ฅโ‰ฅ๐‘€, |๐‘ƒ(๐‘ฅ)โˆ’๐น(๐‘ฅ)|<๐œ– for |๐‘ฅ|โ‰ค๐‘€ and ๐‘ƒ is a polynomial on [โˆ’๐‘€,๐‘€]. Hence, ๐‘ƒโˆˆ๐ด๐ถ(โ„) and โ€–๐‘ƒ๎…žโˆ’๐‘“โ€–<3๐œ–.

In Definition 3.2, the required sequence {๐‘“๐‘›}โŠ‚๐ฟ1 exists. Let [๐›ผ,๐›ฝ]โŠ‚โ„ be a compact interval. Then, by the usual Fubini-Tonelli theorem in ๐ฟ1,

๎€œ๐›ฝ๐›ผ๐‘“๐‘›๎€œโˆ—๐‘”(๐‘ฅ)๐‘‘๐‘ฅ=๐›ฝ๐›ผ๎€œโˆžโˆ’โˆž๐‘“๐‘›=๎€œ(๐‘ฅโˆ’๐‘ฆ)๐‘”(๐‘ฆ)๐‘‘๐‘ฆ๐‘‘๐‘ฅโˆžโˆ’โˆž๎€œ๐‘”(๐‘ฆ)๐›ฝ๐›ผ๐‘“๐‘›(๐‘ฅโˆ’๐‘ฆ)๐‘‘๐‘ฅ๐‘‘๐‘ฆ.(3.6)

Take the supremum over ๐›ผ,๐›ฝโˆˆโ„ and use the ๐ฟ1โˆ’๐ฟโˆž Hรถlder inequality to get

โ€–โ€–๐‘“๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘“โˆ—๐‘”๐‘›โ€–โ€–โ€–๐‘”โ€–1.(3.7) It now follows that {๐‘“๐‘›โˆ—๐‘”} is a Cauchy sequence. It then converges to an element of ๐’œ๐ถ. However, (3.7) also shows that this limit is independent of the choice of {๐‘“๐‘›}. To see that Definitions 3.1 and 3.2 agree, take {๐‘“๐‘›}โŠ‚๐ฟ1 with โ€–๐‘“๐‘›โˆ’๐‘“โ€–โ†’0 and {๐‘”๐‘›}โŠ‚โ„ฌ๐’ฑโˆฉ๐ฟ1 with โ€–๐‘”๐‘›โˆ’๐‘”โ€–1โ†’0. Then

โ€–โ€–๐‘“๐‘›โˆ—๐‘”โˆ’๐‘“โˆ—๐‘”๐‘›โ€–โ€–=โ€–โ€–๎€ท๐‘“๐‘›๎€ธ๎€ท๐‘”โˆ’๐‘“โˆ—๐‘”โˆ’๐‘“โˆ—๐‘›๎€ธโ€–โ€–โ‰คโ€–โ€–๎€ท๐‘“โˆ’๐‘”๐‘›๎€ธโ€–โ€–+โ€–โ€–๎€ท๐‘”โˆ’๐‘“โˆ—๐‘”๐‘“โˆ—๐‘›๎€ธโ€–โ€–โ‰คโ€–โ€–๐‘“โˆ’๐‘”๐‘›โ€–โ€–โˆ’๐‘“โ€–๐‘”โ€–1โ€–โ€–๐‘”+โ€–๐‘“โ€–๐‘›โ€–โ€–โˆ’๐‘”1(3.8)

Letting ๐‘›โ†’โˆž shows that the limits of ๐‘“๐‘›โˆ—๐‘” in Definition 3.2 and ๐‘“โˆ—๐‘”๐‘› in Definition 3.1 are the same.

Theorem 3.4. Let ๐‘“โˆˆ๐’œ๐ถ and ๐‘”โˆˆ๐ฟ1. Define ๐‘“โˆ—๐‘” as in Definition 3.1. Then (a)โ€–๐‘“โˆ—๐‘”โ€–โ‰คโ€–๐‘“โ€–โ€–๐‘”โ€–1. โ€‰(b) โ€‰Let โ„Žโˆˆ๐ฟ1. โ€‰Then (๐‘“โˆ—๐‘”)โˆ—โ„Ž=๐‘“โˆ—(๐‘”โˆ—โ„Ž)โˆˆ๐’œ๐ถ. โ€‰(c) For each ๐‘งโˆˆโ„, ๐œ๐‘ง(๐‘“โˆ—๐‘”)=(๐œ๐‘ง๐‘“)โˆ—๐‘”=(๐‘“โˆ—๐œ๐‘ง๐‘”). โ€‰(d) For each ๐‘“โˆˆ๐’œ๐ถ, define ฮฆ๐‘“โˆถ๐ฟ1โ†’๐’œ๐ถ by ฮฆ๐‘“[๐‘”]=๐‘“โˆ—๐‘”. Then ฮฆ๐‘“ is a bounded linear operator and โ€–ฮฆ๐‘“โ€–โ‰คโ€–๐‘“โ€–. There exists a nonzero distribution ๐‘“โˆˆ๐’œ๐ถ such that โ€–ฮฆ๐‘“โ€–=โ€–๐‘“โ€–. For each ๐‘”โˆˆ๐ฟ1, define ฮจ๐‘”โˆถ๐’œ๐ถโ†’๐’œ๐ถ by ฮจ๐‘”[๐‘“]=๐‘“โˆ—๐‘”. Then ฮจ๐‘” is a bounded linear operator and โ€–ฮจ๐‘”โ€–โ‰คโ€–๐‘”โ€–1. There exists a nonzero function ๐‘”โˆˆ๐ฟ1 such that โ€–ฮจ๐‘”โ€–=โ€–๐‘”โ€–โ„ฌ๐’ฑ. โ€‰(e) Define ๐‘”๐‘ก(๐‘ฅ)=๐‘”(๐‘ฅ/๐‘ก)/๐‘ก for ๐‘ก>0. We have โˆซ๐‘Ž=โˆžโˆ’โˆž๐‘”๐‘กโˆซ(๐‘ฅ)๐‘‘๐‘ฅ=โˆžโˆ’โˆž๐‘”. Then โ€–๐‘“โˆ—๐‘”๐‘กโˆ’๐‘Ž๐‘“โ€–โ†’0 as ๐‘กโ†’0. โ€‰(f) โ€‰Let supp(๐‘“โˆ—๐‘”)โŠ‚supp(๐‘“)+supp(๐‘”).

Proof. Let {๐‘”๐‘›} be as in Definition 3.1. (a) Since โ€–๐‘“โˆ—๐‘”๐‘›โ€–โ†’โ€–๐‘“โˆ—๐‘”โ€–, (3.4) shows โ€–๐‘“โˆ—๐‘”โ€–โ‰คโ€–๐‘“โ€–โ€–๐‘”โ€–1. (b) Let {โ„Ž๐‘›}โŠ‚โ„ฌ๐’ฑโˆฉ๐ฟ1 such that โ€–โ„Ž๐‘›โˆ’โ„Žโ€–1โ†’0. Then (๐‘“โˆ—๐‘”)โˆ—โ„Žโˆถ=๐œ‰โˆˆ๐’œ๐ถ such that โ€–(๐‘“โˆ—๐‘”)โˆ—โ„Ž๐‘›โˆ’๐œ‰โ€–โ†’0. Since ๐‘”โˆ—โ„Žโˆˆ๐ฟ1, there is {๐‘๐‘›}โŠ‚โ„ฌ๐’ฑโˆฉ๐ฟ1 such that โ€–๐‘๐‘›โˆ’๐‘”โˆ—โ„Žโ€–1โ†’0. Then ๐‘“โˆ—(๐‘”โˆ—โ„Ž)โˆถ=๐œ‚โˆˆ๐’œ๐ถ such that โ€–๐‘“โˆ—๐‘๐‘›โˆ’๐œ‚โ€–โ†’0. Now, โ€–โ€–โ€–๐œ‰โˆ’๐œ‚โ€–โ‰ค(๐‘“โˆ—๐‘”)โˆ—โ„Ž๐‘›โ€–โ€–+โ€–โ€–โˆ’๐œ‰๐‘“โˆ—๐‘๐‘›โ€–โ€–+โ€–โ€–โˆ’๐œ‚(๐‘“โˆ—๐‘”)โˆ—โ„Ž๐‘›โˆ’๎€ท๐‘“โˆ—๐‘”๐‘›๎€ธโˆ—โ„Ž๐‘›โ€–โ€–+โ€–โ€–๎€ท๐‘“โˆ—๐‘”๐‘›๎€ธโˆ—โ„Ž๐‘›โˆ’๐‘“โˆ—๐‘๐‘›โ€–โ€–.(3.9) Using (3.4), โ€–โ€–(๐‘“โˆ—๐‘”)โˆ—โ„Ž๐‘›โˆ’๎€ท๐‘“โˆ—๐‘”๐‘›๎€ธโˆ—โ„Ž๐‘›โ€–โ€–=โ€–โ€–๎€บ๎€ท๐‘“โˆ—๐‘”โˆ’๐‘”๐‘›๎€ธ๎€ปโˆ—โ„Ž๐‘›โ€–โ€–โ€–โ€–๐‘”โ‰คโ€–๐‘“โ€–๐‘›โ€–โ€–โˆ’๐‘”1โ€–โ€–โ„Ž๐‘›โ€–โ€–1โŸถ0as๐‘›โŸถโˆž.(3.10) Finally, use Theorem 2.1(e) and (3.4) to write โ€–โ€–๎€ท๐‘“โˆ—๐‘”๐‘›๎€ธโˆ—โ„Ž๐‘›โˆ’๐‘“โˆ—๐‘๐‘›โ€–โ€–=โ€–โ€–๎€ท๐‘”๐‘“โˆ—๐‘›โˆ—โ„Ž๐‘›โˆ’๐‘๐‘›๎€ธโ€–โ€–๎€ทโ€–โ€–๐‘”โ‰คโ€–๐‘“โ€–๐‘›โ€–โ€–โˆ’๐‘”1โ€–โ€–โ„Ž๐‘›โ€–โ€–1+โ€–๐‘”โ€–1โ€–โ€–โ„Ž๐‘›โ€–โ€–โˆ’โ„Ž1+โ€–โ€–๐‘๐‘›โ€–โ€–โˆ’๐‘”โˆ—โ„Ž1๎€ธโŸถ0as๐‘›โŸถโˆž.(3.11)(c) The Alexiewicz norm is invariant under translation [4, Theorem โ€‰28] so ๐œ๐‘ง(๐‘“โˆ—๐‘”)โˆˆ๐’œ๐ถ. Use Theorem 2.1(f) to write โ€–๐œ๐‘ง(๐‘“โˆ—๐‘”)โˆ’๐œ๐‘ง(๐‘“โˆ—๐‘”๐‘›)โ€–=โ€–๐‘“โˆ—๐‘”โˆ’๐‘“โˆ—๐‘”๐‘›โ€–=โ€–๐œ๐‘ง(๐‘“โˆ—๐‘”)โˆ’(๐œ๐‘ง๐‘“)โˆ—๐‘”๐‘›)โ€–=โ€–๐œ๐‘ง(๐‘“โˆ—๐‘”)โˆ’๐‘“โˆ—(๐œ๐‘ง๐‘”๐‘›)โ€–. Translation invariance of the ๐ฟ1 norm completes the proof. (d) From (a), we have โ€–ฮฆ๐‘“โ€–=supโ€–๐‘”โ€–1=1โ€–๐‘“โˆ—๐‘”โ€–โ‰คsupโ€–๐‘”โ€–1=1โ€–๐‘“โ€–โ€–๐‘”โ€–1=โ€–๐‘“โ€–. We get equality by considering ๐‘“ and ๐‘” to be positive functions in ๐ฟ1. To prove โ€–ฮจ๐‘”โ€–โ‰คโ€–๐‘”โ€–1, note that โ€–ฮจ๐‘”โ€–=supโ€–๐‘“โ€–=1โ€–๐‘“โˆ—๐‘”โ€–โ‰คsupโ€–๐‘“โ€–=1โ€–๐‘“โ€–โ€–๐‘”โ€–1=โ€–๐‘”โ€–1. We get equality by considering ๐‘“ and ๐‘” to be positive functions in ๐ฟ1. (e) First consider ๐‘”โˆˆโ„ฌ๐’ฑโˆฉ๐ฟ1. We have ๐‘“โˆ—๐‘”๐‘ก๎€œ(๐‘ฅ)=โˆžโˆ’โˆž๎‚€๐‘ฆ๐‘“(๐‘ฅโˆ’๐‘ฆ)๐‘”๐‘ก๎‚๐‘‘๐‘ฆ๐‘ก=๎€œโˆžโˆ’โˆž๐‘“(๐‘ฅโˆ’๐‘ก๐‘ฆ)๐‘”(๐‘ฆ)๐‘‘๐‘ฆ.(3.12) For โˆ’โˆž<๐›ผ<๐›ฝ<โˆž, ||||๎€œ๐›ฝ๐›ผ๎€บ๐‘“โˆ—๐‘”๐‘ก๎€ป||||=||||๎€œ(๐‘ฅ)โˆ’๐‘Ž๐‘“(๐‘ฅ)๐‘‘๐‘ฅ๐›ฝ๐›ผ๎€œโˆžโˆ’โˆž[๐‘“]๐‘”||||=||||๎€œ(๐‘ฅโˆ’๐‘ก๐‘ฆ)โˆ’๐‘“(๐‘ฅ)(๐‘ฆ)๐‘‘๐‘ฆ๐‘‘๐‘ฅโˆžโˆ’โˆž๎€œ๐›ฝ๐›ผ[]||||โ‰ค๎€œ๐‘“(๐‘ฅโˆ’๐‘ก๐‘ฆ)โˆ’๐‘“(๐‘ฅ)๐‘”(๐‘ฆ)๐‘‘๐‘ฅ๐‘‘๐‘ฆ(3.13)โˆžโˆ’โˆžโ€–โ€–๐œ๐‘ก๐‘ฆโ€–โ€–||||๐‘“โˆ’๐‘“๐‘”(๐‘ฆ)๐‘‘๐‘ฆโ‰ค2โ€–๐‘“โ€–โ€–๐‘”โ€–1.(3.14) By dominated convergence, we can take the limit ๐‘กโ†’0 inside the integral (3.14). Continuity of ๐‘“ in the Alexiewicz norm then shows โ€–๐‘“โˆ—๐‘”๐‘กโˆ’๐‘Ž๐‘“โ€–โ†’0 as ๐‘กโ†’0.
Now take a sequence {๐‘”(๐‘›)}โŠ‚โ„ฌ๐’ฑโˆฉ๐ฟ1 such that โ€–๐‘”(๐‘›)โˆ’๐‘”โ€–1โ†’0. Define ๐‘”๐‘ก(๐‘›)(๐‘ฅ)=๐‘”(๐‘›)(๐‘ฅ/๐‘ก)/๐‘ก and ๐‘Ž(๐‘›)=โˆซโˆžโˆ’โˆž๐‘”(๐‘›)(๐‘ฅ)๐‘‘๐‘ฅ. We have
โ€–โ€–๐‘“โˆ—๐‘”๐‘กโ€–โ€–โ‰คโ€–โ€–โˆ’๐‘Ž๐‘“๐‘“โˆ—๐‘”๐‘ก(๐‘›)โˆ’๐‘Ž(๐‘›)๐‘“โ€–โ€–+โ€–โ€–๐‘“โˆ—๐‘”๐‘ก(๐‘›)โˆ’๐‘“โˆ—๐‘”๐‘กโ€–โ€–+โ€–โ€–๐‘Ž(๐‘›)โ€–โ€–๐‘“โˆ’๐‘Ž๐‘“.(3.15) By the inequality in (a), โ€–๐‘“โˆ—๐‘”๐‘ก(๐‘›)โˆ’๐‘“โˆ—๐‘”๐‘กโ€–โ‰คโ€–๐‘“โ€–โ€–๐‘”๐‘ก(๐‘›)โˆ’๐‘”๐‘กโ€–1. Whereas, โ€–โ€–๐‘”๐‘ก(๐‘›)โˆ’๐‘”๐‘กโ€–โ€–1=๎€œโˆžโˆ’โˆž|||๐‘”(๐‘›)๎‚€๐‘ฅ๐‘ก๎‚๎‚€๐‘ฅโˆ’๐‘”๐‘ก๎‚|||๐‘‘๐‘ฅ๐‘ก=โ€–โ€–๐‘”(๐‘›)โ€–โ€–โˆ’๐‘”1โŸถ0as๐‘›โŸถโˆž,(3.16) and โ€–๐‘Ž(๐‘›)๐‘“โˆ’๐‘Ž๐‘“โ€–=|๐‘Ž(๐‘›)โˆ’๐‘Ž|โ€–๐‘“โ€–=โ€–๐‘”(๐‘›)โˆ’๐‘”โ€–1โ€–๐‘“โ€–. Given ๐œ–>0 fix ๐‘› large enough so that โ€–๐‘“โˆ—๐‘”๐‘ก(๐‘›)โˆ’๐‘“โˆ—๐‘”๐‘กโ€–+โ€–๐‘Ž(๐‘›)๐‘“โˆ’๐‘Ž๐‘“โ€–<๐œ–. Now let ๐‘กโ†’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
๎€œ๐›ฝ๐›ผ๎€œโˆžโˆ’โˆž๎€œ๐‘“(๐‘ฅโˆ’๐‘ก๐‘ฆ)๐‘”(๐‘ฆ)๐‘‘๐‘ฆ๐‘‘๐‘ฅ=๐›ฝ๐›ผ๎€œโˆžโˆ’โˆž๎‚€๐‘“(๐‘ฆ)๐‘”๐‘ฅโˆ’๐‘ฆ๐‘ก๎‚๐‘‘๐‘ฆ๐‘ก=๎€œ๐‘‘๐‘ฅโˆžโˆ’โˆž๎€œ๐›ฝ๐›ผ๎‚€๐‘“(๐‘ฆ)๐‘”๐‘ฅโˆ’๐‘ฆ๐‘ก๎‚๐‘‘๐‘ฅ๐‘‘๐‘ฆ๐‘ก,๎€œโˆžโˆ’โˆž๎€œ๐›ฝ๐›ผ๐‘“๎€(๐‘ฅโˆ’๐‘ก๐‘ฆ)๐‘”(๐‘ฆ)๐‘‘๐‘ฅ๐‘‘๐‘ฆ=โˆžโˆ’โˆž๐‘“๎‚€๐‘ฆ(๐‘ฅ)๐‘”๐‘ก๎‚๐œ’(๐›ผโˆ’๐‘ฆ,๐›ฝโˆ’๐‘ฆ)(๐‘ฅ)๐‘‘๐‘ฅ๐‘‘๐‘ฆ๐‘ก=๎€โˆžโˆ’โˆž๎‚€๐‘ฆ๐‘“(๐‘ฅ)๐‘”๐‘ก๎‚๐œ’(๐›ผโˆ’๐‘ฆ,๐›ฝโˆ’๐‘ฆ)(๐‘ฅ)๐‘‘๐‘ฆ๐‘ก=๎€œ๐‘‘๐‘ฅโˆžโˆ’โˆž๎€œ๐›ฝ๐›ผ๐‘“๎‚€(๐‘ฅ)๐‘”๐‘ฆโˆ’๐‘ฅ๐‘ก๎‚๐‘‘๐‘ฆ๐‘ก๐‘‘๐‘ฅ.(3.17) Note that โˆซ๐›ฝ๐›ผโˆซโˆžโˆ’โˆžโˆซ๐‘“(๐‘ฅ)๐‘”(๐‘ฆ)๐‘‘๐‘ฆ๐‘‘๐‘ฅ=โˆžโˆ’โˆžโˆซ๐›ฝ๐›ผ๐‘“(๐‘ฅ)๐‘”(๐‘ฆ)๐‘‘๐‘ฅ๐‘‘๐‘ฆ 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 โ€‰5.4-2 and 5.3-1].

Young's inequality states that โ€–๐‘“โˆ—๐‘”โ€–๐‘โ‰คโ€–๐‘“โ€–๐‘โ€–๐‘”โ€–1 when ๐‘“โˆˆ๐ฟ๐‘ for some 1โ‰ค๐‘โ‰คโˆž and ๐‘”โˆˆ๐ฟ1. Part (a) of Theorem 3.4 extends this to ๐‘“โˆˆ๐’œ๐ถ. see [1] for other results when ๐‘“โˆˆ๐ฟ๐‘.

The fact that convolution is linear in both arguments, together with (b), shows that ๐’œ๐ถ is an ๐ฟ1-module over the ๐ฟ1 convolution algebra, see [8] for the definition. It does not appear that ๐’œ๐ถ is a Banach algebra under convolution.

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

Proposition 3.5. Let ๐‘“โˆˆ๐’œ๐ถ, ๐‘”โˆˆ๐ฟ1, and ๐œ™โˆˆ๐’Ÿ. Define โˆซ๐น(๐‘ฆ)=๐‘ฆโˆ’โˆž๐‘“ and โˆซ๐บ(๐‘ฅ)=๐‘ฅโˆ’โˆž๐‘”. Definitions 1.1 and 3.1 both give ๎€œโŸจ๐‘“โˆ—๐‘”,๐œ™โŸฉ=โˆžโˆ’โˆž๎€œ๐‘“(๐‘ฆ)โˆžโˆ’โˆž=๎€๐‘”(๐‘ฅ)๐œ™(๐‘ฅ+๐‘ฆ)๐‘‘๐‘ฅ๐‘‘๐‘ฆโˆžโˆ’โˆž๐น(๐‘ฆ)๐บ(๐‘ฅ)๐œ™๎…ž(๐‘ฅ+๐‘ฆ)๐‘‘๐‘ฅ๐‘‘๐‘ฆ.(3.18)

Proof. Let โˆซฮฆ(๐‘ฆ)=โˆžโˆ’โˆž๐‘”(๐‘ฅ)๐œ™(๐‘ฅ+๐‘ฆ)๐‘‘๐‘ฅ. Then ฮฆโˆˆ๐ถโˆž(โ„) and ฮฆ๎…žโˆซ(๐‘ฆ)=โˆžโˆ’โˆž๐‘”(๐‘ฅ)๐œ™๎…ž(๐‘ฅ+๐‘ฆ)๐‘‘๐‘ฅ. Also, โˆซโˆžโˆ’โˆž|ฮฆ๎…žโˆซ(๐‘ฆ)|๐‘‘๐‘ฆโ‰คโˆžโˆ’โˆžโˆซ|๐‘”(๐‘ฅ)|โˆžโˆ’โˆž|๐œ™๎…ž(๐‘ฅ+๐‘ฆ)|๐‘‘๐‘ฆ๐‘‘๐‘ฅโ‰คโ€–๐‘”โ€–1โ€–๐œ™๎…žโ€–1, so ฮฆโˆˆ๐ด๐ถ(โ„). Dominated convergence then shows lim|๐‘ฆ|โ†’โˆžฮฆ(๐‘ฆ)=0. Integration by parts now gives (3.18).
Let {๐‘”๐‘›}โŠ‚โ„ฌ๐’ฑโˆฉ๐ฟ1 such that โ€–๐‘”๐‘›โˆ’๐‘”โ€–1โ†’0. Since convergence in โ€–โ‹…โ€– implies convergence in ๐’Ÿ๎…ž, we have
โŸจ๐‘“โˆ—๐‘”,๐œ™โŸฉ=lim๐‘›โ†’โˆžโŸจ๐‘“โˆ—๐‘”๐‘›,๐œ™โŸฉ=lim๐‘›โ†’โˆž๎€โˆžโˆ’โˆž๐‘“(๐‘ฆ)๐‘”๐‘›(๐‘ฅโˆ’๐‘ฆ)๐œ™(๐‘ฅ)๐‘‘๐‘ฆ๐‘‘๐‘ฅ=lim๐‘›โ†’โˆž๎€œโˆžโˆ’โˆž๐‘“๎€œ(๐‘ฆ)โˆžโˆ’โˆž๐‘”๐‘›(๐‘ฅโˆ’๐‘ฆ)๐œ™(๐‘ฅ)๐‘‘๐‘ฅ๐‘‘๐‘ฆ.(3.19) Proposition A.3 allows interchange of the iterated integrals. Define ฮฆ๐‘›โˆซ(๐‘ฆ)=โˆžโˆ’โˆž๐‘”๐‘›(๐‘ฅ)๐œ™(๐‘ฅ+๐‘ฆ)๐‘‘๐‘ฅ. Then, ๐‘‰ฮฆ๐‘›โ‰คโ€–๐‘”๐‘›โ€–1โ€–๐œ™๎…žโ€–1โ‰ค(โ€–๐‘”โ€–1+1)โ€–๐œ™๎…žโ€–1 for large enough ๐‘›. Hence, ฮฆ๐‘› is of uniform bounded variation. Theorem โ€‰22 in [4], then gives โˆซโŸจ๐‘“โˆ—๐‘”,๐œ™โŸฉ=โˆžโˆ’โˆž๐‘“(๐‘ฆ)lim๐‘›โ†’โˆžฮฆ๐‘›โˆซ(๐‘ฆ)๐‘‘๐‘ฆ=โˆžโˆ’โˆžโˆซ๐‘“(๐‘ฆ)โˆžโˆ’โˆž๐‘”(๐‘ฅโˆ’๐‘ฆ)๐œ™(๐‘ฅ)๐‘‘๐‘ฅ๐‘‘๐‘ฆ. The last step follows since โ€–๐‘”๐‘›โˆ’๐‘”โ€–1โ†’0.

If ๐‘”โˆˆ๐ฟ1โงตโ„ฌ๐’ฑ, then ๐‘“โˆ—๐‘” needs not to be continuous or bounded. For example, take 1/2โ‰ค๐›ผ<1 and let ๐‘“(๐‘ฅ)=๐‘”(๐‘ฅ)=๐‘ฅโˆ’๐›ผ๐œ’(0,1)(๐‘ฅ). Then, ๐‘“โˆˆ๐ฟ1โŠ‚๐’œ๐ถ and ๐‘”โˆˆ๐ฟ1โงตโ„ฌ๐’ฑ. We have ๐‘“โˆ—๐‘”(๐‘ฅ)=0 for ๐‘ฅโ‰ค0. For 0<๐‘ฅโ‰ค1, we have โˆซ๐‘“โˆ—๐‘”(๐‘ฅ)=๐‘ฅ0๐‘ฆโˆ’๐›ผ(๐‘ฅโˆ’๐‘ฆ)โˆ’๐›ผ๐‘‘๐‘ฆ=๐‘ฅ1โˆ’2๐›ผโˆซ10๐‘ฆโˆ’๐›ผ(1โˆ’๐‘ฆ)โˆ’๐›ผ๐‘‘๐‘ฆ=๐‘ฅ1โˆ’2๐›ผฮ“2(1โˆ’๐›ผ)/ฮ“(2โˆ’2๐›ผ). Hence, ๐‘“โˆ—๐‘” is not continuous at 0. If 1/2<๐›ผ<1, then ๐‘“โˆ—๐‘” is unbounded at 0.

As another example, consider ๐‘“(๐‘ฅ)=sin(๐œ‹๐‘ฅ)/log|๐‘ฅ| and ๐‘”(๐‘ฅ)=๐œ’(0,1)(๐‘ฅ). Then ๐‘“โˆˆ๐’œ๐ถ and for each 1โ‰ค๐‘โ‰คโˆž, we have ๐‘”โˆˆโ„ฌ๐’ฑโˆฉ๐ฟ๐‘. And,

๎€œ๐‘“โˆ—๐‘”(๐‘ฅ)=๐‘ฅ๐‘ฅโˆ’1sin(๐œ‹๐‘ฆ)=log(๐‘ฆ)๐‘‘๐‘ฆfor๐‘ฅโ‰ฅ2cos(๐œ‹(๐‘ฅโˆ’1))โˆ’๐œ‹log(๐‘ฅโˆ’1)cos(๐œ‹๐‘ฅ)โˆ’1๐œ‹log(๐‘ฅ)๐œ‹๎€œ๐‘ฅ๐‘ฅโˆ’1cos(๐œ‹๐‘ฆ)๐‘ฆlog2(๐‘ฆ)๐‘‘๐‘ฆโˆผโˆ’2cos(๐œ‹๐‘ฅ)๐œ‹log(๐‘ฅ)as๐‘ฅโŸถโˆž.(3.20)

Therefore, by Theorem 2.1(d), ๐‘“โˆ—๐‘”โˆˆ๐ถ0(โ„), and lim|๐‘ฅ|โ†’โˆž๐‘“โˆ—๐‘”(๐‘ฅ)=0 but for each 1โ‰ค๐‘<โˆž, we have ๐‘“โˆ—๐‘”โˆ‰๐ฟ๐‘.

4. Differentiation and Integration

If ๐‘” is sufficiently smooth, then the pointwise derivative is (๐‘“โˆ—๐‘”)๎…ž(๐‘ฅ)=๐‘“โˆ—๐‘”๎…ž(๐‘ฅ). Recall the definition ๐ด๐ถ(โ„) of primitives of ๐ฟ1 functions given in the proof of Proposition 3.3. In the following theorem, we require pointwise derivatives of ๐‘” to exist at each point in โ„.

Theorem 4.1. Let ๐‘“โˆˆ๐’œ๐ถ, ๐‘›โˆˆโ„•, and ๐‘”(๐‘˜)โˆˆ๐ด๐ถ(โ„) for each 0โ‰ค๐‘˜โ‰ค๐‘›. Then ๐‘“โˆ—๐‘”โˆˆ๐ถ๐‘›(โ„) and (๐‘“โˆ—๐‘”)(๐‘›)(๐‘ฅ)=๐‘“โˆ—๐‘”(๐‘›)(๐‘ฅ) for each ๐‘ฅโˆˆโ„.

Proof. First consider ๐‘›=1. Let ๐‘ฅโˆˆโ„. Then (๐‘“โˆ—๐‘”)๎…ž(๐‘ฅ)=limโ„Žโ†’0๎€œโˆžโˆ’โˆž๎‚ธ๐‘“(๐‘ฆ)๐‘”(๐‘ฅ+โ„Žโˆ’๐‘ฆ)โˆ’๐‘”(๐‘ฅโˆ’๐‘ฆ)โ„Ž๎‚น๐‘‘๐‘ฆ.(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<|โ„Ž|โ‰ค1. Let โ„Žโ‰ 0. Since ๐‘”โˆˆ๐ด๐ถ(โ„) it follows that the variation is given by the Lebesgue integrals ๐‘‰๐‘ฆโˆˆโ„๎‚ธ๐‘”(๐‘ฅ+โ„Žโˆ’๐‘ฆ)โˆ’๐‘”(๐‘ฅโˆ’๐‘ฆ)โ„Ž๎‚น=๎€œโˆžโˆ’โˆž||||๐‘”๎…ž(๐‘ฅ+โ„Žโˆ’๐‘ฆ)โˆ’๐‘”๎…ž(๐‘ฅโˆ’๐‘ฆ)โ„Ž||||โ‰ค๎€œ๐‘‘๐‘ฆโˆžโˆ’โˆž||๐‘”๎…ž๎…ž||๎€œ(๐‘ฆ)๐‘‘๐‘ฆ+โˆžโˆ’โˆž||||๐‘”๎…ž(๐‘ฅ+โ„Žโˆ’๐‘ฆ)โˆ’๐‘”๎…ž(๐‘ฅโˆ’๐‘ฆ)โ„Žโˆ’๐‘”๎…ž๎…ž||||(๐‘ฅโˆ’๐‘ฆ)๐‘‘๐‘ฆ.(4.2) Since ๐‘”๎…žโˆˆ๐ด๐ถ(โ„), we have ๐‘”๎…ž๎…žโˆˆ๐ฟ1. The second integral on the right of (4.2) gives the ๐ฟ1 derivative of ๐‘”๎…ž in the limit โ„Žโ†’0; see [1, page 246]. Hence, in (4.1), we can use [4, Theorem โ€‰22] to take the limit under the integral sign. This then gives