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Omran, Maryam; Kiliçman, Adem

On fractional order Mellin transform and some of its properties. (English) Zbl 1376.44005

Tbil. Math. J. 10, No. 1, 315-324 (2017).
Summary: In this work, we introduce the fractional Mellin transform of order \(\alpha\), \(0 <\alpha\leq 1\) on a function which belongs to the Lizorkin space. Further, some properties and applications of fractional Mellin transform are given.

Cited in 4 Documents

MSC:

44A15 Special integral transforms (Legendre, Hilbert, etc.)
26A33 Fractional derivatives and integrals
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Keywords:

Mellin transform; fractional Fourier transform; fractional Mellin transform; fractional operator
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References:

[1] J. Bertrand, P. Bertrand, and J. Ovarlez, The Mellin transform, in The Transforms and Applications Handbook, D. Alexander, Ed., CRC Press, Boca Raton, Fla, USA, 2nd edition, (2000).; · Zbl 0900.42016
[2] R. Bracewell, The Fourier Transform and its Applications, New York (1965).; · Zbl 0149.08301
[3] P.L. Butzer, S. Jansche, A direct approach to the Mellin transform, Journal Fourier Analysis and Application (1997) 325-376.; · Zbl 0885.44004
[4] P.L. Butzer, S. Jansche, Mellin transform theory and the role of its differential and integral operators. In: Transform Methods and Special Functions, Varna 96 (Proc. 2nd Int. Workshop; P. Rusev, I. Dimovski, V. Kiryakova, Eds.), Bulg. Acad. Sci., So_a (1998), 63-83.; · Zbl 0923.44002
[5] B. Davies, Integral Transforms and their Applications, Springer, (2002).; · Zbl 0996.44001
[6] L. Debnath and D. Bhatta, Integral Transforms and Their Applications, Chapman and Hall/CRC, Boca Raton-London-New York, 2007.; · Zbl 1310.44001
[7] A. Erd_elyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vols 1-2., McGraw-Hill (1953)-(1955).;
[8] P. Flajolet, X. Gourdon, P. Dumas, Mellin transforms and asymptotics: Harmonic sums, Theo. Comp. Sci., 144(1-2), pp 3-58 (1995).; · Zbl 0869.68057
[9] P. Flajolet, M. Regnier and R. Sedgewick, Some uses of the Mellin integral transform in the analysis of algorithms, in: A. Apostolico and Z. Galil, eds., Combinatorial Algorithms on Words (Springer, Berlin, 1985).; · Zbl 0582.68015
[10] A. A. Kilbas, Yu. F. Luchko, H. Martinez, and J.J. Trujillo, Fractional Fourier transform in the framework of fractional calculus operators, Integral Transforms and Special Functions 21, pp. 779-795 (2010).; · Zbl 1205.42010
[11] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 204, Elsevier, Amsterdam, (2006).; · Zbl 1092.45003
[12] A. Kiliçman and M. Omran, Note on fractional Mellin transform and applications, Springer-Plus, 5(1) (2016): 100 .; · Zbl 1438.44002
[13] N. Lebedev, Special functions and their applications, Dover, (1972).; · Zbl 0271.33001
[14] Yu.F. Luchko and V. Kiryakova, The Mellin Integral Transform in Fractional Calculus, Fractional Calculus and Applied Analysis, vol(16)No.(2) pp.405-430 (2013).; · Zbl 1312.26016
[15] Yu.F. Luchko, H. Martínez and J.J. Trujillo, Fractional Fourier Transform and Some of its Applications, Fractional Calculus and Applied Analysis, An International Journal for Theory and Applications, 11, No. 4 (2008). [16] F. Oberhettinger, Tables of Mellin Transforms, Springer-Verlag, New York - Heidelberg - Berlin (1974).;
[16] H.M. Ozaktas, Z. Zalevsky and M.A. Kutay, The Fractional Fourier Transform, Wiley, Chichester, (2001).;
[17] H. Oldham and N. Spanier, The Fractional Calculus, Academic Press, San Diego, CA, (1974).; · Zbl 0292.26011
[18] I. Podlubny, Fractional Di_erential Equations, Academic Press, San Diego (1999); · Zbl 0924.34008
[19] L. G. Romero, R. A. Cerutti and L. L. Luque, A New Fractional Fourier Transform and Convolution Products, Int. Journal of Pure and Applied Mathematics, (66), No. 4 (2011), 397-408.; · Zbl 1218.26004
[20] I. N. Sneddon, Fourier Transforms, Dover Publications, New York, (1995).; · Zbl 0038.26801
[21] E.C. Titchmarsh, Introduction to the theory of Fourier integrals, Vol. 2. Oxford: Clarendon Press, (1948).;
[22] D.V. Widder, An Introduction to Transform Theory, Academic Press, New York, (1971).; · Zbl 0219.44001
[23] X.-J. Yang, D. Baleanu and H. M. Srivastava, Local Fractional Integral Transforms and Their Applications, Academic Press (Elsevier Science Publishers), Amsterdam, Heidelberg, London and New York, (2016).; · Zbl 1336.44001
[24] H. M. Srivastava, A. K. Golmankhaneh, D. Baleanu and X.-J. Yang, Local fractional Sumudu transform with applications to IVPs on Cantor sets, Abstr. Appl. Anal. 2014 (2014), Article ID 620529, 1-7.; · Zbl 1468.26005
[25] H. M. Srivastava and O. Yürekli, A theorem on a Stieltjes-type integral transform and its applications, Complex Variables Theory Appl. 28 (1995), 159-168.; · Zbl 0837.44002
[26] G. Dattoli, H. M. Srivastava and K. Zhukovsky, A new family of integral transforms and their applications, Integral Transforms Spec. Funct. 17 (2006), 31-37.; · Zbl 1086.33009
[27] H. M. Srivastava, M.-J. Luo and R. K. Raina, A new integral transform and its applications, Acta Math. Sci. Ser. B Engl. Ed. 35 (2015), 1386-1400.; · Zbl 1349.44001
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