Fractional Hartley transform on $G$-Boehmian space
DOI:
https://doi.org/10.5269/bspm.43828Abstract
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.
References
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2. 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
3. Akila, L., Roopkumar, R., Quaternionic Stockwell transform, Integral Transforms Spec. Funct., 27 (2016), 484-504. https://doi.org/10.1080/10652469.2016.1155570
4. 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
5. 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
6. 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
7. Bracewell, R. N., The Hartley transform, New York: Oxford University Press; (1986).
8. 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.
9. 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
10. Ganesan, C., Roopkumar, R., On generalizations of Boehmian space and Hartley transform, Mat. Vesnik., 69 (2017), 133-143.
11. 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
12. Karunakaran, V., Ganesan, C., Fourier transform on integrable Boehmians, Integral Transform. Spec. Funct., 20 (2009) 937-941. https://doi.org/10.1080/10652460902734116
13. 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
14. Mikusinski, J., Mikusinski, P., Quotients de suites et leurs applications dans l'anlyse fonctionnelle, C. R. Acad. Sci. Paris, 293 (1981), 463-464.
15. Mikusinski, P., Convergence of Bohemians, Japan J. Math., 9 (1983), 159-179. https://doi.org/10.4099/math1924.9.159
16. 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
17. Mikusinski, P., On flexibility of Boehmians, Integral Transform. Spec. Funct., 4 (1996), 141-146. https://doi.org/10.1080/10652469608819101
18. 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
19. Mikusinski, P., Boehmians and pseudoquotients. Appl. Math. Inf. Sci., 5 ( 2011), 192-204.
20. 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
21. 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
22. Nemzer, D., Extending the Stieltjes transform, Sarajevo J. Math., 10 (2014), 197-208. https://doi.org/10.5644/SJM.10.2.06
23. Nemzer, D., Extending the Stieltjes transform II, Fract. Calc. Appl. Anal., 17 (2014), 1060-1074. https://doi.org/10.2478/s13540-014-0214-0
24. 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
25. Roopkumar, R., Stockwell transform for Boehmians, Integral Transform. Spec. Funct., 24 (2013), 251-262. https://doi.org/10.1080/10652469.2012.686903
26. Roopkumar, R., Ripplet transform and its extension to Boehmians. Georgian Math. J., (in press). DOI: 10.1515/gmj2017-0056
27. 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
28. Rudin, W., Real and complex analysis, Third Edition, McGraw-Hill, New York, (1987).
29. Singh, A., On the exchange property for the Mehler-Fock transform, Appl. Appl. Math., 11 (2016), 828-839.
30. Sundararajan, N., Fourier and Hartley transforms-a mathematical twin, Indian J. Pure Appl. Math., 28 (1997), 1361-1365.
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2021-12-17
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