On Some Properties of Space w S

Öz

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.

Anahtar Kelimeler

• Fractional Fourier transform,
• convolution
• Segal algebras.

On Some Properties of Space S_{w}^{α}

Öz

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.

Anahtar Kelimeler

• Fractional Fourier transform
• convolution
• Segal algebras

Kaynakça

1. Almeida, L. B. 1994. “The fractional Fourier transform and time-frequency representations”, IEEE Transactions on Signal Processing, 42 (11), 3084-3091.
2. Almeida, L. B. 1997. “Product and convolution theorems for the fractional Fourier transform”, IEEE Signal Processing Letters, 4 (1), 15-17.
3. Bultheel, A. and Martinez, H. 2002. “A shattered survey of the fractional Fourier transform”, Department of Computer Science, K.U.Leuveven, Report TW337.
4. Cigler, J. 1969. “Normed ideals in ”, Indagationes Mathematicae, 72(3), 273-282.
5. Doğan, M. and Gürkanlı, A. T. 2000. On functions with Fourier transforms in . Bulletin of Calcutta Mathematical Society, 92(2), 111-120.
6. Feichtinger, H. G. 1977. “On a class of convolution algebras of functions”, Annales de l’institut Fourier, 27(3), 135-162.
7. Feichtinger, H. G., Graham, C. and Lakien, E. 1979. “Nonfactorization in commutative, weakly selfadjoint Banach algebras”, Pacific Journal of Mathematics, 80(1), 117-125.
8. Feichtinger, H. G. ve Gürkanlı A. T. 1990. “On a family of weighted convolution algebras”, International Journal of Mathematics and Mathematical Sciences, 13(3), 517-525.

9. Fischer, R. H., Gürkanlı, A. T. and Liu, T. S. 1996. “On a family of weighted spaces”, Mathematica Slovaca, 46(1), 71-82.
10. Namias, V. 1980. “The fractional order of Fourier transform and its application in quantum mechanics”, Journal of the Institute of Mathematics and its Applications, 25, 241-265.
11. Ozaktas, H. M., Kutay, M. A. and Zalevsky, Z. 2001. “The fractional Fourier transform with applications in optics and signal processing”. John Wiley and Sons, England.
12. Reiter, H. and Stegeman, J. D. 2000. “Classical harmonic analysis and locally compact group”, Clarendon Press, Oxford.
13. Reiter, H. 1971. “ Algebras and Segal Algebras”, Springer-Verlag, New York.
14. Toksoy, E. and Sandıkçı, A. 2015. “On function spaces with fractional Fourier transform in weighted Lebesgue spaces”, Journal of Inequalities and Applications, 2015(1), 87.
15. Wang, H. C. 1977. “Homogeneous Banach Algebras”, Marcel Dekker Inc., New York and Basel.