On a Weighted Algebra Under Fractional Convolution

Abstract

In this study, we describe a linear space A_(α,p(.))^(w,ν) (R^d ) of functions f∈L_w^1 (R^d ) whose fractional Fourier transforms F_α f belong to L_ν^p(.) (R^d ) for p^+<∞. We show that A_(α,p(.))^(w,ν) (R^d ) becomes a Banach algebra with the sum norm ‖f‖_(A_(α,p(.))^(w,ν) )=‖f‖_(1,w)+‖F_α f‖_(p(.),ν) and under Θ (fractional convolution) convolution operation. Besides, we indicate that the space A_(α,p(.))^(w,ν) (R^d ) is an abstract Segal algebra, where w is weight function of regular growth. Moreover, we find an approximate identity for A_(α,p(.))^(w,ν) (R^d ). We also discuss some other properties of A_(α,p(.))^(w,ν) (R^d ). Finally, we investigate some inclusions of this space.