On Some Properties of Space w S

Abstract

In this study, first of all we define spaces ( ) d S and ( ) dw S and give examples of these spaces. After wedefine ( ) dw S to be the vector space of 1( ) df Lw such that the fractional Fourier transform F f  belongsto ( ) dw S . We endow this space with the sum normSw 1,w Swf f F f      and then show that it is aBanach space. We show that ( ) dw Sis a Banach algebra and a Banach ideal on   1 dw L if the space( ) dw S is solid. Furthermore, we prove that the space ( ) dw S is translation and character invaryant andalso these operators are continuous. Finally, we discuss inclusion properties of these spaces.

Özet

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.