On Some Properties of Space w S

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.

Keywords

• Fractional Fourier transform,
• convolution
• Segal algebras.

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.

Keywords

• Fractional Fourier transform
• convolution
• Segal algebras

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