Positive Integral Operators with Analytic Kernels

Transcription

1 Int. Journl of Mth. Anlysis, Vol. 5, 2, no. 34, Positive Integrl Opertors with Anlytic Kernels Cn Murt Dikmen onguldk Krelms University Art nd Science Fculty Deprtment of Mthemtics 67, onguldk, Turkey Abstrct. In this work, s in [2] nd [3], we construct exmples of positive definite integrl kernels which re lso nlytic using Fourier nd Lplce Trnsforms Mthemtics Subject Clssifiction: 45P5, 45H5 Keywords: Lplce Trnsform, Fourier Trnsform, Positive Integrl Opertors, Anlytic Kernels Introduction Let I,J be intervls nd suppose k L 2 (I J),i.e. k (s, I J u) 2 duds <. Then the formul Sf (s) k (s, u) f(u)du J where s I,f L 2 (J) defines compct liner opertor S mpping L 2 (J) into L 2 (I). ThedjointS : L 2 (I) L 2 (J) is given by S g (u) g(t)k (t, u)du. J Whenever k L 2 (I J),T SS will be positive integrl opertor on L 2 (I) with kernel K(s, t) k(s, u)k (t, u)du. J This gives us method of constructing exmples of positive integrl opertors on L 2 (I).

2 684 Cn Murt Dikmen We will use this theorem to give exmples of positive definite kernels K using kernels k which rise in nturl wy in mthemticl nlysis. (See []) 2 Exmples Suggested by the Fourier Trnsform Here we will find positive definite kernels using Fourier trnsforms. We will give two exmples of this method.now we define the Fourier trnsform. Definition Fourier trnsform of function f(u) is defined by ˆf(s) Exmple 2 Here (2.) exists if f L (), becuse ˆf f(u) du kfk. e isu f(u)du. (2.) Suppose β > nd tht ω is continuous function on stisfying ω(u) O(e β u ) s u ;thenω L 2 (). So if f L 2 (),fω L (), sothefourier trnsform of fω : F (s) f(u)ω(u)e isu du is continuous on ; infctf will be nlytic on the strip Im s < β. Now let I be ny bounded closed intervl nd define S : L 2 () L 2 (I) by Sf(s) f(u)ω(u)e isu du. Here k(s, u) ω(u)e isu.nowk(s, u) L 2 (I ), becuse k(s, u) 2 du ω 2 duds I I (b ) kωk 2 2 <. In this cse K(s, t) ω(u) 2 e i(s t)u du L 2 (I I).

3 Positive integrl opertors with nlytic kernels 685 For n explicit exmple we let ω e α u /2.Then K(s, t) e α u e iu(s t) du e αu e iu(s t) + e iu(s t) du e αu ((cos(s t)u + i sin(s t)u)+(cos(s t)u i sin(s t)u)) du ½ ¾ 2e αu cos(s t)udu 2e e (α+i(s t))u du (α > ) ½ ¾ ½ ¾ 2e (α + i(s t)) e (α+i(s t))u 2e α + i(s t) ½ ¾ α i(s t) 2α 2e α 2 +(s t) 2 α 2 +(s t). 2 Since K(s, t) is the kernel of SS it is positive definite on L 2 (I). For the next exmple we will let ω(u) Exmple 3 We let ω(u) (cosh λu) /2 (λ > ) so tht K(s, t) becomes (cosh λu) /2 K(s, t) e iu(s t) cosh λu du. To solve this we consider the following integrl: I(x) e ixu du I( x) cosh u (x>) 3πi 2 +πi γ 3 π i + πi γ 4 πi 2 γ 2 γ Figure : Integrtion Contour

4 686 Cn Murt Dikmen γ f(u)du 2πi.es(f(u), πi/2) eisu f(u) cosh u e πs f(u + πi) f(u)du 2π s. γ 2 /γ 4 cosh ( + e πs )I (s) 2πie πs/2 sinh πi/2 2πe πs/2 e isu cosh u du π cosh πs/2. Then we get the following integrl kernel e iu(s t) K(s, t) cosh u du π cosh π(s t)/2. More generlly, for λ > we hve e iu(s t) e iu(s t)/λ du /λ cosh λu cosh u du π λ cosh π(s t)/2λ. Since K is the kernel of SS, K is positive definite on L 2 (I). 3 Exmples Suggested by the Lplce Trnsform Here we shll use Lplce trnsform to find some exmples of positive definite kernels. As in the previous section we will give two exmples of such positive definite kernels. First, we define the Lplce trnsform which is similr to Fourier trnsform. Definition 4 For g belonging to L 2 ((, )) we write G(s) (Lg)(s) for the Lplce trnsform of g G(s) (Lg)(s) g(t)e st dt.

5 Positive integrl opertors with nlytic kernels 687 Exmple 5 In this exmple we hve <<bnd I [, b],j [, ). We define the opertor S : L 2 ([, )) L 2 (I) by so tht This is becuse b Sf(s) f(u)e su du k(s, u) e su L 2 (J I). e 2su duds b b µ e 2u du ds 2 ds b 2 <. Then K(s, t) e u(s+t) du s + t. since it is the kernel of SS we know tht it is positive definite on L 2 (I). Now we give our lst exmple. Exmple 6 Let I [, b] where > nd J [, ). Nowweconsiderthe following integrl Similrly, o u α e su du s α+ Γ(α +) sα+ u α e u du α > (Γ(α +)α!). In this exmple we tke which is in L 2 (J I), becuse o u α e su du u α e u du α > s α s Γ(α). α k(s, u) u α 2 e us, b u α e 2us duds b (b )C < u α e 2u duds

6 688 Cn Murt Dikmen where C is constnt. Then we hve, K(s, t) b u α e u(s+t) duds Γ(α) (s + t) α α >. Since K(s, t) is the kernel of SS,itispositivedefinite on L 2 (I). EFEENCES [] Dikmen,C.M.,(997), Positive İntegrl Opertors with Anlytic kernels, M.Sc. Thesis, The University of Mnchester. [2] Dikmen,C.M.,(26), Positive Integrl Opertors with Anlytic Kernels, Çnky Üniversitesi Journl of Arts nd Sciences, (6), [3] Dikmen,C.M.,(26), Anlitik Çekirdekli Pozitif İntegrl Opertörler., XIX. Ulusl Mtemtik Sempozyumu Bildiri kitbı, Küthy eceived: Mrch, 2