# Article ID: MTJPAM-D-21-00034

## Title: On an abstract Segal algebra under fractional convolution

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-21-00034; Volume 4 / Issue 1 / Year 2022, Pages 1-22

Document Type: Research Paper

Author(s): Ayşe Sandıkçı a , Erdem Toksoy b

aOndokuz Mayıs University, Faculty of Art and Sciences, Departman of Mathematics, Kurupelit, Samsun, 55139, Turkey

bOndokuz Mayıs University, Faculty of Art and Sciences, Departman of Mathematics, Kurupelit, Samsun, 55139, Turkey

Received: 19 May 2021, Accepted: 18 June 2021, Published: 30 June 2021.

Corresponding Author: Ayşe Sandıkçı (Email address: ayses@omu.edu.tr)

Full Text: PDF

Abstract

In this work, we find approximate identities for the spaces  ${L}^{{1}}{\left(\mathbb{R}^{d}\right)}$,  ${L_{w}^{1}\left(\mathbb{R}^{d}\right)}$ and  $S_{w}^{\alpha }(\mathbb{R}{^{d}})$  under $\Theta$ convolution. Furthermore, we determine approximate identities with compactly supported fractional Fourier transforms in the spaces  ${L_{w}^{1}\left(\mathbb{R}^{d}\right)}$  and  $S_{w}^{\alpha }(\mathbb{R}{^{d}})$, where  $w$  is weight function of regular growth. We give definitions of multipliers of these Banach algebras under  $\Theta$  convolution. Also, we show that the space  $S_{w}^{\alpha }(\mathbb{R}{^{d}})$  under some conditions is an abstract Segal algebra with recpect to  ${L_{w}^{1}\left(\mathbb{R}^{d}\right)}$.

Keywords: Fractional Fourier transform, approximate identity, Segal algebras

References:
1. L. B. Almeida, The fractional Fourier transform and time-frequency representations, IEEE Trans. Signal Process. 42 (11), 3084–3091, 1994.
2. L. B. Almeida, Product and convolution theorems for the fractional fourier transform, IEEE Signal Process. Lett. 4 (1), 15–17, 1997.
3. A. Bourouihiya, Beurling weighted spaces, product-convolution operators, and the tensor product of frames, Ph.D. thesis, University of Maryland,College Park, Maryland, 2006.
4. A. Bultheel and H. Martinez, A shattered survey of the fractional Fourier transform, Report TW 337, 2002.
5. J. T. Burnham, Closed ideals in subalgebras of Banach algebras. I, Proceedings of the American Mathematical Society 32 (2), 551–555, 1972.
6. J. Cigler, Normed ideals in L1(G), Indagationes Mathematicae 72 (3), 273–282, 1969.
7. M. Doğan and A. T. Gürkanli, On functions with Fourier tansforms in Sω, Bulletin of Calcutta Mathematical Society 92 (2), 111–120, 2000.
8. R. S. Doran and J. Wichmann, Approximate identities and factorization in Banach modules, 768, Springer-Verlag, New York, 1979.
9. H. G. Feichtinger, On a class of convolution algebras of functions, Annales de l’institu Fourier 27 (3), 135–162, 1977.
10. H. G. Feichtinger, C. Graham and E. Lakien, Nonfactorization in commutative, weakly selfadjoint Banach algebras, Pacific Journal of Mathematics 80 (1), 117–125, 1979.
11. R. H. Fischer, A. T. Gürkanli and T. S. Liu, On a family of weighted spaces, Math. Slovaca 46 (1), 71–82, 1996.
12. K. Gröchenig, Weight functions in time-frequency analysis, In: L. Rodino, B. W. Schulze, M. W. Wong, (eds.) Pseudodifferential Operators: Partial Differential Equations and Time-Frequency Analysis, volume 52 of Fields Inst. Commun., pp. 343–366. Amer. Math. Soc., Providence, RI, 2007.
13. R Larsen, An introduction to the theory of multipliers, Springer-Verlag, Berlin Heidelberg, New York, 1971.
14. V. Namias, The fractional order of Fourier transform and its application in quantum mechanics, J. Inst. Math. Appl. 25, 241–265, 1980.
15. H. M. Ozaktas, M. A. Kutay and Z. Zalevsky, The fractional Fourier transform with applications in optics and signal processing, John Wiley and Sons, Chichester, 2001.
16. H. Reiter and J. D. Stegeman, Classical harmonic analysis and locally compact group, Clarendon Press, Oxford, 2000.
17. W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1966.
18. W. Rudin, Functional analysis, Mc Graw-Hill, New York, 1973.
19. A. K. Singh and R. Saxena, On convolution and product theorems for FRFT, Wireless Personal Communications 65 (1), 189–201, 2012.
20. E. Toksoy and A. Sandikçi, On function spaces with fractional Fourier transform in weighted Lebesgue spaces, J. Inequal. Appl. 87, 2015.
21. E. Toksoy and A. Sandikçi, On some properties of space Swα, Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi 13 (2), 923–934, 2020.
22. H. C. Wang, Homogeneous Banach algebras, Marcel Dekker Inc., New York, 1977.
23. A. I. Zayed, On the relationship between the Fourier and fractional Fourier transforms, IEEE Signal Proc. Let. 3, 310–311, 1996.