ON THE FRESNEL SINE INTEGRAL AND THE CONVOLUTION

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1 IJMMS 3:37, PII. S Hndaw Publshng Cop. ON THE FRESNEL SINE INTEGRAL AND THE CONVOLUTION ADEM KILIÇMAN Receved 19 Novembe and n evsed fom 7 Mach 3 The Fesnel sne ntegal Sx), the Fesnel cosne ntegal Cx), and the assocated functons S + x), S x), C + x), andc x) ae defned as locally summable functons on the eal lne. Some convolutons and neutx convolutons of the Fesnel sne ntegal and ts assocated functons wth x +, x ae evaluated. Mathematcs Subject Classfcaton: 33B1, 46F1. 1. Intoducton. The Fesnel ntegals occu n the dffacton theoy and they ae of two knds: the Fesnel ntegal Sx) wth a sne n the ntegal and the Fesnel ntegal Cx) wth a cosne n the ntegal. The Fesnel sne ntegal Sx) s defned by Sx) = x snu du 1.1) π see [5]) and the assocated functons S + x) and S x) ae defned by S + x) = Hx)Sx), S x) = H x)sx). 1.) The Fesnel cosne ntegal Cx) s defned by Cx) = x cosu du 1.3) π see [5]) and the assocated functons C + x) and C x) ae defned by C + x) = Hx)Cx), C x) = H x)cx), 1.4) whee H denotes Heavsde s functon. We defne the functon L x) by x L x) = u snu du 1.5)

2 38 ADEM KILIÇMAN fo =,1,,... In patcula, we have π L x) = Sx), L 1 x) = 1 1 cosx, 1.6) L x) = πcx) cosx ) x. We defne the functons sn + x,sn x, cos + x, and cos x by sn + x = Hx)snx, cos + x = Hx)cosx, sn x = H x)snx, cos x = H x)cosx. 1.7). Convoluton poducts. The classcal defnton fo the convoluton poduct of two functons f and g s as follows. Defnton.1. Let f and g be functons. Then the convoluton f g s defned by fo all ponts x fo whch the ntegal exsts. f g)x) = ft)gx t)dt.1) If the classcal convoluton f g of two functons f and g exsts, then g f exsts and f g = g f..) Futhe, f f g) and f g o f g) exst, then f g) = f g o f g)..3) The classcal defnton of the convoluton can be extended to defne the convoluton f g of two dstbutons f and g n wth the followng defnton, see [4]. Defnton.. Let f and g be dstbutons n. Then the convoluton f g s defned by the equaton f g)x),ϕx) = fy), gx),ϕx+y).4) fo abtay ϕ n, povded that f and g satsfy ethe of the followng condtons: a) ethe f o g has bounded suppot, b) the suppots of f and g ae bounded on the same sde.

3 ON THE FRESNEL SINE INTEGRAL AND THE CONVOLUTION 39 It follows that f the convoluton f g exsts by ths defnton, then.) and.3) ae satsfed. Theoem.3. The convoluton sn + x ) x + exsts and sn+ x ) x + = = ) 1) L x)x+.5) fo =,1,,... Poof. It s obvous that sn + x ) x + = fx<. When x>, we have sn+ x ) x x+ = snt x t) dt ) = 1) L x)x, =.6) thus povng.5). Coollay.4. The convoluton sn x ) x exsts and sn x ) x = = ) L x)x.7) fo =,1,,... Poof. Equaton.7) follows on eplacng x by x n.5) and notng that L x) = 1) +1 L x)..8) Theoem.5. The convoluton S + x) x+ exsts and fo =,1,,... ) S + x) x+ = 1) +1 L +1 x)x π +1) +.9) = Poof. It s obvous that S + x) x + = fx<. When x>, we have π x t S +x) x+ = x t) snu dudt = 1 ) ) +1 L +1 x)x =.1) Thus equaton.9) follows.

4 33 ADEM KILIÇMAN Coollay.6. fo =,1,,... The convoluton S x) x exsts and ) S x) x = L +1 x)x π +1).11) Poof. Equaton.11) follows on eplacng x by x n.9). = 3. Exstence of neutx convoluton poduct. In ode to extend the convoluton poduct to a lage class of dstbutons, the neutx convoluton poduct was ntoduced n [1] and was late extended n [, 3]. Fo the futhe extenson, fst of all, we let τ be a functon n havng the followng popetes: ) τx) = τ x), ) τx) 1, ) τx) = 1fo x 1/, v) τx) = fo x 1. The functon τ ν s now defned fo ν>by 1, x ν, τ ν x) = τ ν ν x ν ν+1), x >ν, τ ν ν x +ν ν+1), x < ν. 3.1) Defnton 3.1. Let f and g be dstbutons n and let f ν = fτ ν fo ν >. The neutx convoluton poduct f g s defned as the neutx lmt of the sequence {f ν g}, povded that the lmt h exsts n the sense that N-lm fν g,ϕ = h,ϕ, 3.) ν fo all ϕ n, whee N s the neutx, see van de Coput [7], havng doman N, the postve eal numbes, wth neglgble functons fnte lnea sums of the functons ν λ ln 1 ν,ln ν, ν snν, and ν snν λ, = 1,,...) and all functons whch convege to zeo n the nomal sense as ν tends to nfnty. Note that n ths defnton the convoluton poduct f ν g s defned n Gel fand and Shlov s sense, wth the dstbuton f ν havng bounded suppot. It was poved n [1] that f f g exsts n the classcal sense o by Defnton.1, then f g exsts and The followng theoem was also poved n [1]. f g = f g. 3.3) Theoem 3.. Let f and g be dstbutons n and suppose that the neutx convoluton poduct f g exsts. Then the neutx convoluton poduct f g

5 ON THE FRESNEL SINE INTEGRAL AND THE CONVOLUTION 331 exsts and f g) = f g. 3.4) Now f we let L = N-lm ν L ν) and note that S ) = C ) = 1, 3.5) see Olve [6], then we have the followng theoem. Theoem 3.3. The neutx convoluton sn + x ) x exsts and ) sn+ x ) x = 1) L x 3.6) = fo =,1,,... Poof. We set sn+ x ) ν = sn + x ) τ ν x). 3.7) Then the convoluton sn + x ) ν x exsts and sn+ x ) ν ν+ν ν ν x = snt x t) dt + τ ν t)snt x t) dt. 3.8) ν Now ν ) ν snt x t) dt = x t) snt dt = ) 3.9) = 1) L ν)x, = and t follows that ) ν N-lm snt x t) dt = 1) L x. 3.1) ν = Futhe, t can easly be seen that fo each fxed x, ν+ν ν lm ν ν τ ν t)snt x t) dt =, 3.11) and 3.6) follows fom 3.9), 3.1), and 3.11). Theoem 3.4. The neutx convoluton S + x) x exsts and ) +1 S + x) x = 1) +1 L +1 x 3.1) π +1) fo =,1,,... =

6 33 ADEM KILIÇMAN Poof. We put [S + x)] ν = S + x)τ ν x). Then the convoluton poduct [S + x)] ν x exsts and [ S+ x) ] ν ν+ν ν ν x = St)x t) dt + τ ν t)st)x t) dt. 3.13) ν We have π ν St)x t) dt ν = t x t) snu dudt ) ν +1 = 1 +1 = x [ ν) +1 u) +1] snu du, 3.14) and t follows that ν ) +1 N-lm St)x t) dt = 1) +1 L +1 x. 3.15) ν π +1) = Futhe, t s easly seen that fo each fxed x, ν+ν ν lm ν ν τ ν t)st)x t) dt =, 3.16) and 3.1) now follows mmedately fom 3.14), 3.15), and 3.16). Coollay 3.5. fo =,1,,... The neutx convoluton S x) x exsts and ) +1 S x) x = 1) L +1 x 3.17) π +1) = Poof. n 3.1). Equaton 3.17) follows on eplacng x by x and L by 1) +1 L Coollay 3.6. The neutx convoluton Sx) x exsts and Sx) x = 3.18) fo =,1,,... Poof. Equaton 3.18) follows fom 3.1)and3.17) on notng that Sx) = S + x)+s x). Acknowledgments. The autho s vey gateful to Pofesso H. M. Svastava fo hs valuable comments and help n the mpovement of ths pape. The pesent eseach has been patally suppoted by Unvest Puta Malaysa unde Gant IRPA EA1.

7 ON THE FRESNEL SINE INTEGRAL AND THE CONVOLUTION 333 Refeences [1] B. Fshe, Neutces and the convoluton of dstbutons, Unv. u Novom Sadu Zb. Rad. Pod.-Mat. Fak. Se. Mat ), no. 1, [] B. Fshe and A. Kılıçman, Commutatve neutx convoluton poducts of functons, Comment. Math. Unv. Caoln ), no. 1, [3], Some commutatve neutx convoluton poducts of functons, Comment. Math. Unv. Caoln ), no. 4, [4] I. M. Gel fand and G. E. Shlov, Genealzed Functons. Vol. 1, Academc Pess, New Yok, [5] I. S. Gadshteyn and I. M. Ryzhk, Table of Integals, Sees, and Poducts, Academc Pess, Calfona,. [6] F. W. J. Olve, Asymptotcs and Specal Functons, Compute Scence and Appled Mathematcs, Academc Pess, New Yok, [7] J. G. van de Coput, Intoducton to the neutx calculus, J. Analyse Math /196), Adem Kılıçman: Depatment of Mathematcs and Insttute of Advanced Technology ITMA) and Mathematcal Reseach Insttute INSPEM), Unvesty Puta Malaysa, 434 UPM Sedang, Selango, Malaysa E-mal addess: aklc@fsas.upm.edu.my

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