Convolutions with the Continuous Primitive Integral

and Applied Analysis 3 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 ∈ Lloc, then the essential variation of g is ess var g sup ∫∞ −∞gφ ′, where the supremum is taken over all φ ∈ D with ‖φ‖∞ ≤ 1. Then EBV {g ∈ Lloc | ess var g < ∞}. This is a Banach space under the norm ‖g‖EBV ess sup |g| ess var g. 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 ess var g inf Vh 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, ess var g 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 φ̃ x φ −x : i 〈T ∗ ψ, φ〉 〈T, φ ∗ ψ̃〉, ii for each x ∈ R, let T ∗ ψ x 〈T, τxψ̃〉; iii 〈S ∗ T, φ〉 〈S x , 〈T y , φ x y 〉〉. In i , ∗ : D′ × D → D′. This definition also applies to other spaces of test functions and their duals, such as the Schwartz space of rapidly decreasing functions or the compactly supported distributions. In ii , ∗ : D′ × D → C∞. In 1 , it is shown that definitions i and ii are equivalent. In iii , ∗ : D′ × D′ → D′. However, this definition requires restrictions on the supports of S and T . It suffices that one of these distributions has compact support. Other conditions on the supports can be imposed see 3, 6 . This definition is an instance of the tensor product, 〈S ⊗ T,Φ〉 〈S x , 〈T y ,Φ x, y 〉〉, where now Φ ∈ D R2 . Under i , T∗ψ is inC∞. 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. 4 Abstract and Applied Analysis 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 ofAC. 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 byΨg f f∗g. ThenΨg 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 −→ 0 as t −→ 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 ∣g ∗ h si − g ∗ h ti ∣ ≤ ∑∫∞ −∞ ∣g ( si − y ) − gti − y ∣∣h ( y ∣dy ∫∞ −∞ ∣g ( si − y ) − gti − y ∣∣h ( y ∣dy. 2.2 Abstract and Applied Analysis 5and Applied Analysis 5 Hence, V g ∗ h ≤ Vg ‖h‖1. The interchange of sum and integral follows from the FubiniTonelli theorem. Now d shows f ∗ g ∗ h ∈ C0 R . Write f ∗ g ∗ h x ∫∞ −∞ f ( y ) ∫∞ −∞ g ( x − y − zh z dzdy ∫∞ −∞ h z ∫∞ −∞ f ( y ) g ( x − y − zdy dz ( 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 Vg h 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 L 1. 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 ∑N n 1 g 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 ∈ L 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, ψ̃ ∗ φ ∫∞ −∞ f x ∫∞ −∞ ψ ( y − xφydy dx ∫∞ −∞ ∫∞ −∞ f x ψ ( y − xφydx dy 〈f ∗ ψ, φ〉. 2.4 6 Abstract and Applied Analysis Since ψ ∈ BV and φ ∈ L1, Proposition A.3 justifies the interchange of integrals. Definition 1.1 ii gives 〈 f, τxψ̃ 〉 ∫∞ −∞ f ( y ) ψ ( x − ydy f ∗ ψ x . 2.5 Definition 1.1 iii gives 〈 f ( y ) , 〈 g x , φ ( x y )〉〉 ∫∞ −∞ f ( y ) ∫∞ −∞ g x φ ( x y ) dx dy ∫∞ −∞ f ( y ) ∫∞ −∞ g ( x − yφ x dx dy ∫∞ −∞ φ x ∫∞ −∞ f ( y ) g ( x − ydx dx 〈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 F ′ for someF ∈ 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 ∫β αf x − y g y dy exist as α → −∞ and β → ∞. This gives