Boundedness Proerties for Some Integral Transform V. D. Sharma, A. N. Rangari 2 Deartment of Mathematics, Arts, Commerce and Science College, Amravati- 444606(M.S), India Deartment of Mathematics, Adarsh College, Dhamangaon Rly. - 444709 (M.S), India 2 ABSTRACT: There are various integral transforms which have widely used in hysics, astronomy as well as in engineering. In order to solve the differential equations, the integral transform were etensively used and thus there are several works on the theory and alication of integral transform such as the Lalace, Fourier, Mellin and Hankel etc. and these transforms have various roerties which are very alicable which is given in our revious aers. In this aer, we have discussed the boundedness roerties of Fourier-Lalace transform and Fourier-Finite Mellin transform. KEYWORDS: Fourier transform; Lalace transform; Fourier- Lalace transform; Finite Mellin Transform; Fourier- Finite Mellin Transform; Integral Transform. I.INTRODUCTION Various integral transforms have been etensively used in the formulation of electromagnetic scattering, radiation, antennas, and electromagnetic interference related roblems. The integral transform technique is an indisensable tool for reresenting the fields in the unbounded (oen) region []. Integral transforms lay an imortant role in Analytic number theory [2]. An integral transform mas some function onto another one and as a consequence a function sace into or onto another one. Oerations in the original sace are converted in general into oeration in the image sace. Integral transforms are therefore used in the first lace if handling with the oerations in the image sace is easier to do or is better known as in the original sace. As an eamle: the Lalace transforms converts differentiation in the sace of definition into a simle algebraic oeration in the image sace [3]. The most common integral transforms that are used are: Fourier transforms (Joseh Fourier, 768-830), Lalace transforms (Pierre-Simon, marquis de Lalace, 749-827) and Mellin transforms (Robert Hjalmar Mellin, 854-933). The Fourier transform technique has long been used for electromagnetic scattering, diffraction, and antenna alications []. The Lalace transform is also very useful in the solution of many artial differential equations and it can be used as a very effective tool in simlifying the calculations in many fields of Engineering and Mathematics [4]. The Mellin transform is a basic tool for analyzing the behavior of many imortant functions in Mathematics and mathematical hysics. Mellin transform has many alications such as quantum calculus, radar classification of shis, electromagnetic stress distribution, agriculture, medical stream, statistics, robability, signal rocessing, otics, attern recognition, algorithms, correlators, navigation, vowel recognition and crytograhic scheme [5]. It is also used in solution of fractional differential equations [6]. Many authors studied various roerties of integral transforms. Dhunde R. R. et.al. [4] discussed some remarks on the roerties of double Lalace transform, V. D. Sharma and P. B. Deshmukh [5] have also discussed oeration transform formulae for two dimensional fractional Mellin transform. V. D. Sharma and A. N. Rangari have already described oeration transform formulae of Fourier-Lalace transform [7] and oerational calculus on Generalized Fourier- Lalace transform [8]. Motivated by the work here we discussed and roved the boundedness roerties for Fourier- Lalace transform and Fourier-Finite Mellin transform. So these transforms are as follows: 0 FL f ( t, ) F( s, ) K( t, ) f ( t, ) dtd Coyright to IJIRSET DOI:0.5680/IJIRSET.205.0404 759
where, i( st i). Also K( t, ) e a 0 FM f ( t, ) F( s, ) K( t, ) f ( t, ) dtd f ist a where, K( t, ) e. And the develoment of these transforms requires some testing function saces and Distributional Fourier-Lalace transform as well as Distributional Fourier-Finite Mellin transform as follows:.. The sace FLab,, This sace is given by su FL : E / t, 0 t t e D D t, C A k 0 k a l q k k a, b, a, b, k, q, l t lq Where the constants A and.2. The sace FM C lq deend on the testing function. (..) This sace is given by su FM : E / t, 0 t t D D t, C A k 0 a b 0 for each k, l, q 0,,2,3,... where, bc, c a k q l q k k b, c, k, q, l b, c t lq where the constants A and C deend on the testing function. lq.3. Distributional Generalized Fourier-Lalace Transforms (FLT) (.2.) For f ( t, ) FL a,, where FL a, is the dual sace of FL a,. It contains all distributions of comact suort. The distributional Fourier-Lalace transform is a function of f ( t, ) and is defined as where, for each fied t i( st i) (, ) (, ) (, ), 0 t, 0, 0 FL f t F s f t e, (.3.) as an alication of f ( t, ) FL i( sti ) a, to e FL a,. s and 0.4. Distributional Generalized Fourier-Finite Mellin Transforms ( FM f T ), the right hand side of (.3.) has a sense For f ( t, ) FM, where FM is the dual sace of FM. It contains all distributions of comact suort. The Distributional Fourier-Finite Mellin transform is a function of f ( t, ) and is defined as Coyright to IJIRSET DOI:0.5680/IJIRSET.205.0404 760
ist a FM f f ( t, ) F( s, ) f ( t, ), e (.4.) where, for each fied t 0 t, 0, s 0 and 0, the right hand side of (.4.) has a sense as an alication of f ( t, ) FM to ist a e FM. Here we have generalized the integral transform in distributional sense. The main urose of this aer is to give the boundedness roerties of Fourier-Lalace transform and Fourier-Finite Mellin transform. The lanning of this aer is as follows: Boundedness theorem for Fourier-Lalace Transform is given in section 2. Section 3 gives the Boundedness theorem for Fourier-Finite Mellin transform. Section 4 concludes the aer. Notations and terminology as er Zemanian. [9], [0]. 2.. Theorem (Boundedness) f t, FLab,, Let su f t, SA SB II.BOUNDEDNESS THEOREM FOR FOURIER-LAPLACE TRANSFORM and i st i F s, FL f t, f t,, e, a Re b, s 0. Let n such that SA t : t R, t A, A 0 and S : n B R, B, B 0, then for each 0, 0 there eist a constant C 0 and a non-negative integer n such that l s A B F s, C s e e 0 qn Proof: Suose that on a neighbourhood of Since f FLab,, su f t, SA SB and let 0, 0 S and su S. A A n. Choose DR such that t and in view of the boundedness roerty of the generalized functions, there eist a constant C and a non-negative integer n such that F s, f t,, e,, f t i st i t e i sti isti C 0 l n a, b, k, q, l t e 0 qn su k a l q C 0 l n t e DtD t e 0 qn isti Coyright to IJIRSET DOI:0.5680/IJIRSET.205.0404 76
su l k l ist a q C 0 l n t Dt t Dt e e D e 0 qn su k l lv ist a q C 0 l n t Dt tis e e e 0 qn k s A l q a Ct e 0 l n s e 0 qn l k s A q C s t e e a 0 qn l s A B C s e C e, where C t e 0 qn l s A B C s e e, where C CC 0 qn k a q III.BOUNDEDNESS THEOREM FOR FOURIER-FINITE MELLIN TRANSFORM 3.. Theorem (Boundedness) ist a F s, FM f t, f t,, e Let f t, FM and f a Re b, s 0. Let su f t, SA SB such that S : n A t t R, t A, A 0 n S : R, B, B 0, then for each 0, 0 there eist a constant C 0 B F s, C s e B B 0 qn l s A integer n such that Proof: Suose that on a neighbourhood of Since f FM su f t, SA SB and let 0, 0 S and su S. A A n. Choose DR and and a non-negative such that t and in view of the boundedness roerty of the generalized functions, there eist a constant C and a non-negative integer n such that ist a F s, f t,, e, Coyright to IJIRSET DOI:0.5680/IJIRSET.205.0404 762
a ist f t,, t e ist a C 0 l n a, b, k, q, l te 0 qn su k q l q ist a C 0 l n t b, c Dt D te 0 qn su k l l ist q q a C 0 l n t D t tdt e b, c D 0 qn su l C 0 l n t D tis e a P q P q 0 qn k l v ist q q q t b, c where P q is a olynomial in q etc. k s A l a 0 bc, C t e l n s P q P q 0 qn l k s A a bc, C s t e P q P q 0 qn l k s A a bc, C s t e P q P q 0 qn l k s A C s t e bc, a P q 0 qn l k s A C s t e bc, P q 0 qn Coyright to IJIRSET DOI:0.5680/IJIRSET.205.0404 763
l s A l s A C s e C B C s e C B 0 q n 0 q n k k where C t a P q and C t P q bc, l s A C s e l s A B C s e B 0 q n 0 q n C CC where l s A bc, C s e B B 0 qn IV.CONCLUSION This aer discussed and roved Boundedness roerties for Fourier-Lalace transform and Fourier-Finite Mellin transform. REFERENCES [] Joon Eom Hyo, Integral transforms in Electromagnetic Formulation, Journal of Electromagnetic Engineering and science, Vol. 4, No. 3,. 273-277, Set. 204. [2] Aleksandar Ivic, Some alications of Lalace transforms in analytic number theory, NOVI SAD J. MATH., Vol. 45, No.,. 3-44, 205. [3] Adrianus Schuitman, A class of integral transforms and associated function saces, Bibliotheek Technische Universiteit, 985. [4] R. R. Dhunde, N. M. Bhondge, and P. R. Dhongle, Some remarks on the roerties of double Lalace transforms, International Journal of Alied Physics and Mathematics, Vol. 3, No. 4,. 293-295, July 203. [5] V. D. Sharma, and P.B. Deshmukh, Oeration transform formulae for two dimensional fractional mellin transform, International Journal of Science and Research (IJSR), Vol. 3, Issue 9,. 634-637, Set. 204. [6] Malgorzata Klimek and Daniel Dziembowski, On Mellin transform alication to solution of fractional differential equations, Scientific Research of the Institute of Mathematics and Comuter Science, Vol. 7, Issue 2,. 3-42, 2008. [7] V. D. Sharma, and A.N. Rangari, Oeration Transform Formulae of Fourier-Lalace Transform, Int. Journal of Pure and Alied Sciences and Technology, 5(2),. 62-67, (203). [8] V. D. Sharma, and A.N. Rangari, Oerational Calculus on Generalized Fourier-Lalace Transform, Int. Journal of Scientific and Innovative Mathematical Research (IJSIMR), 2(),. 862-867, (204). [9] A. H. Zemanian, Generalized integral transform, Inter science ublisher, New York, 968. [0] A. H. Zemanian, Distribution theory and transform analysis, Mc Graw Hill, New York, 965. [] V. D. Sharma, Oeration Transform Formulae on Generalized Fractional Fourier Transform, Proceedings International Journal of Comuter Alications (IJCA), (0975-8887), PP. 9-22, 202. [2] V. D. Sharma, and P. B. Deshmukh, Oeration Transform formulae for Two-Dimensional Fractional Mellin transform, International Journal of science and research, Vol.3, Issue 9,. 634-637, Set. 204. [3] V. D. Sharma, and A. N. Rangari, Proerties of Generalized Fourier-Lalace Transform, Int. Journal of Mathematical Archive (IJMA), 5(8),. 36-40, 204. [4] Lokenath Debnath and Dambaru Bhatta, Integral Transforms and their Alications, Chaman and Hall/CRC Taylor and Francis Grou Boca Raton London, New York, 2007. [5] R. S. Pathak, A Course in Distribution Theory and Alications, CRC Press 200. [6] S. M. Khairnar, R. M. Pise, and J. N. Salunkhe, Study of the mellin integral transform with alications in statistics and robability, Archives of Alied Science Research, Vol. 4, No. 3,. 294-30, 202. Coyright to IJIRSET DOI:0.5680/IJIRSET.205.0404 764