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• W2080452908 abstract "The generalized hankel-clifford transformations defined by F1(y) = (h1,α,βf) (y) = y-(α+β) ∫0∞ ℐj,β(xy)f(x) dx, with h1,α,β = h1,α,β·-1 = ∫0∞ (y/x)-(α+β)/2 Jα-β(2(√xy))f(x)dx and F2(y) = (h2α,β f) (y) = ∫0∞ x-(α+β) ℐα,β(xy)f (x)dx, with h2α,β = h2,α,β-1 = ∫0∞(x/y)-(α+β)/2 Jα-β(2√(xy))f(x)dx where ℐα,β(z) =z(α,β)/2 Jα-β(2√(z)), Jα-β(z) being the Bessel function of the first kind and order (α - β) ≥ -1/ 2 were extended by Malgonde [8] to certain generalized functions and by Malgonde and Bandewar [7] to certain generalized functions of slow growth through a generalization of mixed Parseval's equation as 〈h'1,α,β f,φ〉 = 〈f, h'2,α,βφ〉, where f ∈ H'α(I) and φ ∈ Hα(I). Note that by setting φ=h2,α,βΦ where Φ, h2,α,β Φ ∈ Hα (I). Analogously, we can define the distributional transformation h'2,α,β on H'β(I) as the adjoint of h'1,α,β on H'β(I) as. Thus equation takes the form 〈h'1,α,β f, h'2,α,βφ〉 = 〈f, Φ〉, where f ∈ H'α, Φ ∈ Hα and in this form equation appears as a generalization of the mixed Parseval equation ∫0∞ f(x)g(x)dx = ∫0∞ F1(y)G2(y)dy, where F1(y) = (h1,α,βf)(y) and g(x) = (h2,α,β-1 G2(y))(x), as it happens in the extension of the Hankel-Clifford transform to a certain spaces of generalized functions studied in Malgonde [8] and Mendez [5]. Letter, Malgonde [8] defined the distributional generalized Hankel-Clifford transformation of the conventional transformation (I) for complex y on Hα,β = Hα,β(σ) = ∪n=1∞ Hα,β,αn, where Hα,β,αn is the testing function space whick contains the kernel function, y-α-β(xy)(α+β)/2 Jα-β(2√(xy)). The distributional complex generalized -Hankel-Clifford transformation of the conventional generalized-Hankel-Clifford transformation (I) on Hα,β = Hα,β(σ) = ∪n=1∞ Hα,β,αn, where Hα,β is the testing space which contains the kernel function, y-α-β(xy)(α+β)/2 Jα-β(2√(xy)), is possible for equatuib (I) and (II) similar to Koh and Zemanian [2] and Malgonde and Lakshmi Gorty [9]. The distributional transformation was defined directly as the application of a generalized function to the kernel function, i.e., for f ∈ H'α,β, F'1(y) = (h'1,α,βf)(y) = 〈f(x),y-α-β(xy)(α+β)/2 Jα-β(2√(xy))〉. In this paper, we extend definition (IV) to a larger space of generalized functions. We first introduce the test function space Mα, a which contains the kernel function and show that Hα ⊂ Mα,a ⊂ Hα,β,a. We then form the countable union space Mα = ∪n=1∞ Mα,an whose dual M'α has H'α,β as a subsapce. Our main result is an inversion theorem states as follows. Let F'1 (y) = (h'1,α,β f)(y) = 〈f(x), y-α-β(xy)(α+β)/2 Jα-β (2√(xy))〉, f ∈ M'α where y is restrcited to the positive real axis. Let (α-β) ≥ -½. Then, in the sense of convergence in H'α, f(x) = limr→∞ ∫0∞ F'1(y) y-α-β ℐα,β(xy)dy. This convergence gives a stronger result than the one obtained by Malgonde [8]. Secondly, we prove that every generalized function belonging to M'α,a can be represented by a finite sum of derivatives of measurable functions. This proof is analogous to the method employed in structure theorems for Schwartz distributions [6] and similar to one by Koh [3]." @default.
• W2080452908 created "2016-06-24" @default.
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• W2080452908 date "2013-01-01" @default.
• W2080452908 modified "2023-09-23" @default.
• W2080452908 title "The generalized Hankel-Clifford transformation on M&#x2032;<inf>&#x03B1;</inf> and its representation" @default.
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• W2080452908 doi "https://doi.org/10.1109/icadte.2013.6524712" @default.
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