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Unified Treatment of the Krätzel Transformation for Generalized Functions
| Content Provider | Semantic Scholar |
|---|---|
| Author | Al-Omari, S. K. Kaufman A. Çman |
| Copyright Year | 2014 |
| Abstract | and Applied Analysis 3 Proof. It is clear that K V f and K p V g are continuous and bounded on R+. Moreover, the Fubini’s theorem allows us to interchange the order of integration: ∫ R + f (x) (K V g) (x) dx = ∫ R + (∫ R + f (x) Z p (xy) dx)g (y) dy. (15) Equation (15) follows since the Krätzel kernel Z p (xy) applies for the functions f and g, when the order of integration is interchanged. This completes the proof of the theorem. Now, in consideration of Theorem 2, the adjoint method of extending the Krätzel transform can be read as ⟨K V f, φ⟩ = ⟨f,K p V φ⟩ , (16) where f ∈ S + (L, α, (ai), a) and φ ∈ S+(L , α, (ai), a). Theorem 3. Given that f ∈ S + (L, α, (ai), a) then K V f ∈ S + (L, α, (ai), a). Proof. Consider a zero convergent sequence (φn) in S+(L , α, (ai), a) then certainly (K p V φn) is a zero-convergent sequence in the same space. It follows from (16) that ⟨K V f, φn⟩ = ⟨f,K p V φn⟩ → 0 as n → ∞. (17) Linearity is obvious. This completes the proof. From the above theorem we deduce that the Krätzel transformof a tempered ultradistribution is a tempered ultradistribution. Moreover, the boundedness property of K V f, f ∈ S + (L, α, (ai), a) follows from the following theorem. Theorem 4. Let f ∈ S + (L, α, (ai), a) and then K V f is bounded. Proof. See [4, Proposition 2.3]. It is interesting to know that the Krätzel transform can be defined in an alternative way, namely, by the kernel method. Let f ∈ S + (L, α, (ai), a), and then (K V f) (x) = ⟨f (y) , Z V p (xy)⟩ . (18) In fact, (18) is a straightforward consequence of Lemma 1. Theorem 5. Let f ∈ S + (L, α, (ai), a) then K V f is infinitely differentiable and D k x (K V f) (x) = ⟨f (t) ,D k x Z V p (xt)⟩ (19) for every k ∈ N and x ∈ R+. Proof. See [4, Proposition 2.2]. 4. Boehmian Spaces Boehmians were first constructed as a generalization of regular Mikusinski operators [17]. The minimal structure necessary for the construction of Boehmians consists of the following elements: (i) a nonempty set A, (ii) a commutative semigroup (B, ∗), (iii) an aperation ⊙ : A × B → A such that for each x ∈ A and s1, s2 ∈ B, x ⊙ (s1 ∗ s2) = (x ⊙ s1) ⊙ s2, (iv) a collection Δ ⊂ B such that (a) if x, y ∈ A, (sn) ∈ Δ, x ⊙ sn = y ⊙ sn for all n, then x = y, (b) if (sn), (tn) ∈ Δ, then (sn ∗ tn) ∈ Δ. Elements of Δ are called delta sequences. Consider g = {(xn, sn) : xn ∈A, (sn)∈Δ, xn⊙sm = xm⊙sn, ∀m, n ∈ N} . (20) If (xn, sn), (yn, tn) ∈ g, xn ⊙ tm = ym ⊙ sn, for all m, n ∈ N, thenwe say (xn, sn) ∼ (yn, tn).The relation∼ is an equivalence relation in g. The space of equivalence classes in g is denoted by β. Elements of β are called Boehmians. Between A and β there is a canonical embedding expressed as x → x ⊙ sn sn . (21) The operation ⊙ can be extended to β × A by xn sn ⊙ t = xn ⊙ t sn . (22) In β, there are two types of convergence: (δ convergence) a sequence (hn) in β is said to be δ convergent to h in β, denoted by hn δ → h, if there exists a delta sequence (sn) such that (hn⊙sn), (h⊙sn) ∈ A, for all k, n ∈ N, and (hn⊙sk) → (h⊙sk) as n → ∞, in A, for every k ∈ N, (Δ convergence) a sequence (hn) in β is said to be Δ convergent to h in β, denoted by hn Δ → h, if there exists a (sn) ∈ Δ such that (hn − h) ⊙ sn ∈ A, for all n ∈ N, and (hn − h) ⊙ sn → 0 as n → ∞ in A. For further discussion see [17–21]. 5. The Ultra-Boehmian Space βs+ Denote by D+, or D (R+), the Schwartz space of C functions of bounded support. Let Δ+ be the family of sequences (sn) ∈ D (R+) such that the following holds: (Δ 1) ∫R + sn(x)dx = 1, for all n ∈ N, (Δ 2) sn(x) ≥ 0, for all n ∈ N, (Δ 3) supp sn ⊂ (0, εn), εn → 0 as n → ∞. It is easy to see that each (sn) in Δ+ forms a delta sequence. 4 Abstract and Applied Analysis Let f ∈ S + (L, α, (ai), a) and σ ∈ D (R+) be related by the expression: (f ⋅ σ) υ = f (σ ⊛ υ) , (23) |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.maths.tcd.ie/EMIS/journals/HOA/AAA/Volume2013/750524.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Aortic Valve Insufficiency Arabic numeral 0 C standard library Class DNA Sequence Dirac delta function Embedding Emoticon Generalization (Psychology) Kernel method R language Spaces Suppository Turing completeness Vergence Yoctometer |
| Content Type | Text |
| Resource Type | Article |
