Int. J. Pure Appl. Sci. Technol., 16(1) (2013), pp. 20-25 International Journal of Pure and Applied Sciences and Technology ISSN 2229-6107 Available online at www.ijopaasat.in Research Paper Generalized Laplace-Fractional Mellin Transform and Operators V.D. Sharma 1,* and M.M. Thakare 2 1 Department of Mathematics, Arts, Commerce and Science College, Amravati, 444606 (M.S.), India 2 Department of Mathematics, P.R. Patil Inst. of Polytechnic and Technology, Amravati, 444606 (M.S.), India * Corresponding author, e-mail: (vdsharma@hotmail.co.in) (Received: 16-2-13; Accepted: 23-3-13) Abstract: The theory of integral transforms is presented a direct and systematic technique for the presentation of classical and distribution theory. The main view of our work is the generalization of Laplace-Fractional Mellin transform to distribution of compact support using kernel method. Definition of Distributional Generalized Laplace fractional Mellin Transform (LMrT) is given. Some operators and properties for the Generalized Laplace fractional Mellin transform are proved. Keywords: Fractional Mellin Transform, Generalized Function, Laplace-Fractional Mellin Transform, Laplace Transform, Pattern Recognition. 1. Introduction In the theory of integral transforms of generalized functions the monograph of Zemanian [1] takes a remarkable place. The investigation of transform is the most effective procedure for solving many problems concerning PDE and functional equations. In recent years many authors pointed out that derivatives and integrals of fractional ordered are suitable for description of properties of various real materials. It has provided to be very useful tool for modeling of many phenomenons in Physics, Chemistry, Engineering, Bioscience and other areas. The Mellin transform is regarded as the multiplicative version of the two-sided Laplace transform. It is closely connected to Laplace transform. The Mellin transform is widely used in fractional calculus because of its scale invariance property [3]. Sazbon [4] has given the fractional Mellin transform in the conventional sense and used this tool for solving and supporting the visual navigation problem. Also Fractional Mellin based correlator can be used to control the range of rotation and scaling.
Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 20-25 21 Number of extensions of Laplace, Mellin and fractional Mellin transform were studied by different mathematicians. In our previous work we have extended Generalized Two-dimensional fractional Mellin transform [5]. Khairnar S. M. [6, 7] had extended fractional Mellin transform in the range 0, and had given Applications of the Laplace-Mellin integral transform to Differential equations. Sarkar [8] had applying the Fourier-modified Mellin transform to Dopller distorted wave form processing. Motivated by the above we aim to study Laplace-fractional Mellin transform. In this paper we extend this concept to generalized Laplace-fractional Mellin transform. The paper is organized in the following manner. In section 2 we define Laplace-fractional Mellin transform and testing function space. Section 3 deals with the Definition of distributional generalized Laplace-fractional Mellin transform. Some operators are given in section 4. In section 5 properties for Laplace-fractional Mellin transform are derived. Some results are given in section 6. Section 7 concludes the result. 2. Laplace-Fractional Mellin Transform 2.1 Definition The Laplace-fractional Mellin transform with parameter of (,) denoted by {(,)} performs a linear operation, given by the integral transform {(,)}= {(,)}(,)= (,)= (,) (,,,) Where the kernel, (,,,)= ( ) (2.1) =, 0< 2 (2.2) 2.2 Test Function Space,, : An infinitely differentiable complex valued smooth function on belongs to ( ), if for each compact set, where ={:,,>0},. And be the open sets in such that,,, ()= (,) (2.3 <,, =0,1,2. The space,, are equipped with their natural Hausdoff locally convex topology,,. This toplogy is respectively generates by the total families of seminorms,,, given by (2.3).
Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 20-25 22 3. Distributional Generalized Laplace-Fractional Mellin Transform(LMrT) Let,, is the dual space of,,. This space,, consist of continuous linear functional on,,. The distributional Laplace-fractional Mellin transform of (,) ( ) is defined as {(,)}= {(,)}(,) where the kernel = (,), (,,,) (,,,)= (3.1) (3.2) For each fixed (0<< ), >0 and 0<, the right hand side of (3.1) has sense as the application of (,),, to (,,,), 4. Operators on LMrT 4.1 Differential Operator L-Type,,, If (,), then differential operator of L type also belongs to, i.e. (,),, where (,)= (,).,., Proof: For,,,,, (,)= = = (,) (,) (,) <, (,),,.. (,),,, Thus (,),, If (,),. 4.2 Proposition: The differential operator of L type, :(,) (,) is topological, automorphism on,., 4.3: Theorem: For and (,),, ()=(,)=, (,),., Proof: Let (,), Consider,,, (,)= (,)
Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 20-25 23 = = (,) () (,) < Then (,),,, if (,),,., 4.4 Proposition: The Exponential multiplier operator :, isomorphism.,, is topological 5. Properties for Generalized Laplace-Fractional Mellin Transform 5.1 Shifting Property If (,),, and is any fixed real number then (+,),,,+>0. Proof: Consider,,, (+,)= = = (+,) (+,,,) (,,,) = where =+ (,,,) <,for any ixed, and any fixed and, 0<. Thus (+,),,,for +=0 5.2 Proposition: The translation (Shifting) operator :(,) (+,) is a topological, automorphism on,, for +>0. 5.3 Scaling Property:,, If (,), and >0, strictly positive number then (,),. Proof: consider,,, (,)= = (,) (,,,)
Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 20-25 24 = < (,,,) where = for any fixed, and any fixed,, 0<. Thus (,),,, >0. 5.4 Proposition: If >0 and (,),, then the scaling operator :,,,, defined by = where (,)=(,) is a topological automorphism. 5.5 Proposition: If and such that +>0 and >0, then the shifting scaling operator :,,,, defined by ()=, where (,) (+,) is a topological automorphism. 6. Theorem For =(, ), where, =0,1,2, if (,),,, then (,),,, where (,)= (,). Further the mapping = : is one-one, linear and continuous. Proof: For (,),,,,,, (,)= = = (,) (,) (,) < (6.1) Thus,, (,),, if (,),. It is obviously linear. It is injective for, if =0 then =, c is a constant. If =0 then =0 and D is injective. But if 0 then, = for ==0. As the right hand side is not bounded we conclude that,,, which is a contradiction. Hence c must be zero and therefore =0. For continuity we observe from equation (6.1) that,,, ( ),,, (), where M is some constant. Thus the theorem is proved.
Int. J. Pure Appl. Sci. Technol., 16(1) (2013), 20-25 25 7. Conclusion This paper presents extension of generalized Laplace-fractional Mellin transform. Some operators and properties for the generalized Laplace-fractional Mellin transform are derived. References [1] A.H. Zemanian, Generalized Integral Transformation, Interscience Publisher, New York, 1968. [2] Y. Singh and H.K. Mandia, Relationship between double Laplace transform and double Mellin transform in terms of generalized hypergeometric function with applications, Int. Jou. of Sci. & Eng. Research, 3(5) (2012), 1-5. [3] B.K. Dutta and L.K. Arora, Multiple Mellin and Laplace transforms of I-functions or r variables, Journal of Fractional Calculus and Applications, 1(10) (2011), 1-8. [4] D.D. Sazbon, Fourier Mellin based correlators and their fractional version in navigational tasks, Pattern Recognition, 35(2002), 2993-2999. [5] V.D. Sharma and P.B. Deshmukh, Generalized two-dimensional fractional Mellin transform, Proc of II nd Int. Conf. on Engineering Trends in Engineering and Technology, IEEE, (2009), 900-903. [6] S.M. Khairnar, Application of the Laplace-Mellin et al integral transform to differential equations, Int. Journal of Scientific & Research Publications, 2(5) (2012), 1-8. [7] S.M. Khainar et al, Fractional Mellin integral transform in (0, ), Int. Journal of Scientific and Research Publications, 2(5) (2012), 1-9. [8] Y.J. Sarkar et al, Applying the Fourier-modified Mellin transform to Dopller distorted waveform, Digital Signal Processing, 17(2007), 1030-1039.