Mixed Parseval equation and generalized Hankel-type integral transformation of distributions

Fractionalization of Hankel Type Integral Transforms and Their Relevance

“…Various forms of Hankel like integral transforms have been considered in detail in a series of papers [6,7,[10][11][12]22]. We will be concerned here with certain Hankel type transform having relevance among others, in connection with the solution to evolution problems involving the Bessel type differential operators x α+3β−1 ∂ ∂x x 2(α−β)+1 ∂ ∂x x −3α−β and x −3α−β ∂ ∂x x 2(α−β)+1 ∂ ∂x x α+3β−1 with α − β ≥ − 1 2 and −2(α + β) real parameter [6,11].…”

“…relevant K− generator(12) being the radical part of the 2D Laplacian operator. In fact, as noted earlier, the Hankel transform of integer Bessel order can be regarded as the radial part of the 2D Fourier transform of rotationally symmetric function, when polar co-ordinates are adopted.In otherwords we have Fa f (ζ, η) (x, y ) = e −imφ e −imθ Ĥa m ρ 1/2 g(ρ) (r ), m = 0, 1, 2, ...(69)where (ρ, φ) and (r, θ) are polar co-ordinates respectively in the function and transform domain and f is a rotationally symmetric function:f (ζ, η) = g(ρ) e imφ .Evidently (69) can be generalised to the transform (21) and (24) as Fa f (ζ, η) (x, y ) = e −imφ e −imθ r −1+2(α+β) Ĥa 1,m ρ 1−2(α+β) g(ρ) (r ), Fa f (ζ, η) (x, y ) = e −imφ e −imθ r −2(α+β) Ĥa 2,m,−2(α+β) ρ 2(α+β) g(ρ) (r )…”