On Some Properties of Space S_{w}^{α}

Authors : Erdem TOKSOY, Ayşe SANDIKÇI

Pages : 923-934

Doi:10.18185/erzifbed.635545

View : 15 | Download : 12

Publication Date : 2020-08-31

Article Type : Research Paper

Abstract :In this study, first of all we define spaces  S^{Θ}(ℝ^{d})  and  S_{w}^{Θ}(ℝ^{d})  and give examples of these spaces. After we define  S_{w}^{α}(ℝ^{d})  to be the vector space of  f∈L_{w}¹(ℝ^{d})  such that the fractional Fourier transform  F_{α}f  belongs to  S_{w}^{Θ}(ℝ^{d}).  We endow this space with the sum norm  ‖f‖_{S_{w}^{α}}=‖f‖_{1,w}+‖F_{α}f‖_{S_{w}^{Θ}}   and then show that it is a Banach space. We show that  S_{w}^{α}(ℝ^{d})   is a Banach algebra and a Banach ideal on  L_{w}¹(ℝ^{d})   if the space     S_{w}^{Θ}(ℝ^{d})   is solid. Furthermore, we proof that the space  S_{w}^{α}(ℝ^{d})  is translation and character invaryant and also these operators are continuous. Finally, we discuss inclusion properties of these spaces.
Keywords : Fractional Fourier transform, convolution, Segal algebras