AI Summary & Key Takeaways
AI InsightFinally, we discuss inclusion properties of these spaces.
📌 3 Key Takeaways
Abstract
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 f Lw such that the fractional Fourier transform F f belongs to ( ) d w S . We endow this space with the sum norm Sw 1,w Sw 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 w L 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.
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.