On Some Properties of Space S_{w}^{α}

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.