Fractional Hartley transform on $G$-Boehmian space
Abstract
Using a special type of fractional convolution, a $G$-Boehmian space $\mathcal{B}_\alpha$ containing integrable functions on $\mathbb{R}$ is constructed. The fractional Hartley transform ({\sc frht}) is defined as a linear, continuous injection from $\mathcal{B}_\alpha$ into the space of all continuous functions on $\mathbb{R}$. This extension simultaneously generalizes the fractional Hartley transform on $L^1(\mathbb{R})$ as well as Hartley transform on an integrable Boehmian space.
Downloads
References
Agarwal, P., Al-Omari, S. K. Q., Choi, J., Real covering of the generalized Hankel-Clifford transform of Fox kernel type of a class of Boehmians, Bull. Korean Math. Soc., 52 (2015), 1607-1619. https://doi.org/10.4134/BKMS.2015.52.5.1607
Akila, L., Roopkumar, R., A natural convolution of quaternion valued functions and its applications, Appl. Math. Comput., 242, (2014), 633-642. https://doi.org/10.1016/j.amc.2014.06.007
Akila, L., Roopkumar, R., Quaternionic Stockwell transform, Integral Transforms Spec. Funct., 27 (2016), 484-504. https://doi.org/10.1080/10652469.2016.1155570
Alieva, T., Bastiaans, M. J., Fractional cosine and sine transform in relation to the Fractional Fourier and Hartley transforms, In: Proceedings of the Seventh International Symposium on Signal Processing and its Applications, Paris, France, 1 (2003), 561-564. https://doi.org/10.1109/ISSPA.2003.1224765
Al-Omari, S. K, On Some Variant of a Whittaker Integral Operator and its Representative in a Class of Square Integrable Boehmians, Bol. Soc. Paran. Mat. (3s.) 38 (2020), 173-183. https://doi.org/10.5269/bspm.v38i1.36468
Arteaga, C., Marrero, I., The Hankel transform of tempered Boehmians via the exchange property, Appl. Math. Comp., 219, 810-818. https://doi.org/10.1016/j.amc.2012.06.043
Bracewell, R. N., The Hartley transform, New York: Oxford University Press; (1986).
Burzyk, J., Mikusinski, P., A generalization of the construction of a field of quotients with applications in analysis, Int. J. Math. Sci., 2 (2003), 229-236.
Ganesan, C., Roopkumar, R., Convolution theorems for fractional Fourier cosine and sine transforms and their extensions to Boehmians, Commun. Korean Math. Soc., 31 (2016), 791-809. https://doi.org/10.4134/CKMS.c150244
Ganesan, C., Roopkumar, R., On generalizations of Boehmian space and Hartley transform, Mat. Vesnik., 69 (2017), 133-143.
Hartley, R. V. L., A more symmetrical Fourier analysis applied to transmission problems, Proceedings of the Institute of Radio Engineers, 30 (1942), 144-150. https://doi.org/10.1109/JRPROC.1942.234333
Karunakaran, V., Ganesan, C., Fourier transform on integrable Boehmians, Integral Transform. Spec. Funct., 20 (2009) 937-941. https://doi.org/10.1080/10652460902734116
Katsevich, A., Mikusnski, P., On De Graaf spaces of pseudoquotients, Rocky Mountain J. Math. 45 (2015), 1445-1455. https://doi.org/10.1216/RMJ-2015-45-5-1445
Mikusinski, J., Mikusinski, P., Quotients de suites et leurs applications dans l'anlyse fonctionnelle, C. R. Acad. Sci. Paris, 293 (1981), 463-464.
Mikusinski, P., Convergence of Bohemians, Japan J. Math., 9 (1983), 159-179. https://doi.org/10.4099/math1924.9.159
Mikusinski, P., Fourier transform for integrable Bohemians, Rocky Mountain J. Math., 17 (1987), 577-582. https://doi.org/10.1216/RMJ-1987-17-3-577
Mikusinski, P., On flexibility of Boehmians, Integral Transform. Spec. Funct., 4 (1996), 141-146. https://doi.org/10.1080/10652469608819101
Mikusinski, P., Generalized quotients with applications in analysis, Methods Appl. Anal., 10 (2004), 377-386. https://doi.org/10.4310/MAA.2003.v10.n3.a4
Mikusinski, P., Boehmians and pseudoquotients. Appl. Math. Inf. Sci., 5 ( 2011), 192-204.
Olejniczak, K. J., The Hartley transforms. The Transforms and Applications Handbook, In: Poularikas, A.D., editor, CRC Press, Boca Raton, (2000). https://doi.org/10.1201/9781420036756.ch4
Namias, V., The fractional order Fourier transform and its application to quantum mechanics, J. Inst. Math. Appl., 25 (1980), 241-265. https://doi.org/10.1093/imamat/25.3.241
Nemzer, D., Extending the Stieltjes transform, Sarajevo J. Math., 10 (2014), 197-208. https://doi.org/10.5644/SJM.10.2.06
Nemzer, D., Extending the Stieltjes transform II, Fract. Calc. Appl. Anal., 17 (2014), 1060-1074. https://doi.org/10.2478/s13540-014-0214-0
Pei, S.C., Ding, J. J., Fractional cosine, sine, and Hartley transforms, IEEE Transactions On Signal Processing, 50 (2002), 1661-1680. https://doi.org/10.1109/TSP.2002.1011207
Roopkumar, R., Stockwell transform for Boehmians, Integral Transform. Spec. Funct., 24 (2013), 251-262. https://doi.org/10.1080/10652469.2012.686903
Roopkumar, R., Ripplet transform and its extension to Boehmians. Georgian Math. J., (in press). DOI: 10.1515/gmj2017-0056
Roopkumar, R., Negrin, E. R., A unified extension of Stieltjes and Poisson transforms to Boehmians, Integral Transform. Spec. Funct., 22 (2011), 195-206. https://doi.org/10.1080/10652469.2010.511208
Rudin, W., Real and complex analysis, Third Edition, McGraw-Hill, New York, (1987).
Singh, A., On the exchange property for the Mehler-Fock transform, Appl. Appl. Math., 11 (2016), 828-839.
Sundararajan, N., Fourier and Hartley transforms-a mathematical twin, Indian J. Pure Appl. Math., 28 (1997), 1361-1365.
Copyright (c) 2021 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
