A Note on Function Spaces with Fractional Fourier Transforms in Wiener-type Spaces

The purpose of this paper is to introduce and study a function space A_insert ignore into journalissuearticles values(α,w);^insert ignore into journalissuearticles values(B,Y); insert ignore into journalissuearticles values(R^d ); to be a linear space of functions h∈L_w^1 insert ignore into journalissuearticles values(R^d ); whose fractional Fourier transforms F_α h belong to the Wiener-type space Winsert ignore into journalissuearticles values(B,Y);insert ignore into journalissuearticles values(R^d );, where w is a Beurling weight function on R^d. We show that this space becomes a Banach algebra with the sum norm 〖‖h‖〗_insert ignore into journalissuearticles values(1,w);+〖‖F_α h‖〗_insert ignore into journalissuearticles values(Winsert ignore into journalissuearticles values(B,Y);); and Θ convolution operation under some conditions. We find an approximate identity in this space and show that this space is an abstract Segal algebra with respect to L_w^1 insert ignore into journalissuearticles values(R^d ); under some conditions.