Hartley transforms on a certain space of generalized functions

Abstract

The idea of the construction of Boehmians was initiated by the concept of regular operators. The construction of Boehmians is similar to the construction of the field of quotients and, in some cases, it just gives the field of quotients. In this article we consider two spaces of Boehmians. The strong space of Boehmians is continuously viewed in the space of general Boehmians. The Hartley transform is extended and obtained as a welldefined continuous mapping with respect to the ı convergence for which certain theorems have been proved. The article is ended up in defining the inverse Hartley transform and discussing some of its properties in detail.

A Note on F Transformation of Generalized Functions

“…Assume that there are Proof. As in [1] , given β n ∆ → β as n → ∞ we can find f 1n,k and f 1n,k iń E (R) such that β n = f 1n,k Ψ 1n , β = f 1n,k Ψ 1n,k and that f 1n,k → f 1,k as n → ∞.…”

“…Detailed proofs carry routine techniques similar to that of the corresponding ones of [1]. We prefer to omit repeated proofs.…”

“…For a = 1 and b = 1, we obtain the Hartley transform pair [17,1,3,11,[13][14] Hf (ξ) = 1 √ 2π R f (y) (cos ξy + sin ξy) dy (6) and f (y) = 1 √ 2π R Hf (ξ) (cos ξy + sin ξy) dξ.…”

Some extensions of a certain integral transform to a quotient space of generalized functions

“…For a somehow much more detailed account of Boehmian spaces we refer to [1][2][3][4][5][6][7][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”

“…However, we deem it proper to recall in this article some of integral transforms that are represented in the space of Boehmians, but not all, such as : the Fourier transform [14]; the Radon transform [6,7]; the Hilbert transform [11]; the Hartley-Hilbert and Fourier-Hilbert transforms [3]; the diffraction Fresnel transform [5]; the optical Fresnel wavelet transform [2]; the Hartley transform [1] to mention but a few.…”