In this study, first of all we define spaces ( ) d S and ( ) d w S and give examples of these spaces. After we define ( ) d w S to be the vector space of 1 ( ) d w f L such that the fractional Fourier transform F f belongs to ( ) d w S . We endow this space with the sum norm 1, w w S w S f f F f and then show that it is a Banach space. We show that ( ) d w S is a Banach algebra and a Banach ideal on 1 d L w if the space ( ) d w S is solid. Furthermore, we prove that the space ( ) d w S is translation and character invaryant and also these operators are continuous. Finally, we discuss inclusion properties of these spaces.