Abstract
Using a special type of fractional convolution, a $G$-Boehmian space $\mathcal{B}_\alpha$ containing integrable functions on $\mathbb{R}$ is constructed. The fractional Hartley transform ({\sc frht}) is defined as a linear, continuous injection from $\mathcal{B}_\alpha$ into the space of all continuous functions on $\mathbb{R}$. This extension simultaneously generalizes the fractional Hartley transform on $L^1(\mathbb{R})$ as well as Hartley transform on an integrable Boehmian space.
Highlights
Denoting the equivalence class gn sn containing ((gn),), we define the G-Boehmian space B as the set of all equivalence classes gn sn induced by the equivalence relation ∼ on A
The major difference between a Boehmian space and a G-Boehmian space is that ⋆ should be commutative on S for a Boehmian space, which is not required for a G-Boehmian space
We introduce a new type of fractional convolution #α, using which, we establish all the results required for constructing the fractional Hartley transformable G-Boehmian space Bα
Summary
Denoting the equivalence class gn sn containing ((gn), (sn)), we define the G-Boehmian space B as the set of all equivalence classes gn sn induced by the equivalence relation ∼ on A. Frht of an arbitrary integrable function f was defined by Where Fα(f ) is the fractional Fourier transform of f , which is defined by We introduce a new type of fractional convolution #α, using which, we establish all the results required for constructing the fractional Hartley transformable G-Boehmian space Bα. +f (x − y)g(y − z)e2iaα(y2−xy−yz)dy dz (by Fubini’s theorem) Since x ∈ R is arbitrary, the proof follows.
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