International Journal of Pure and Applied Sciences and Technology

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1 Int J Pure Appl Sci Technol, 15(2) (2013), pp International Journal of Pure and Applied Sciences and Technology ISSN Available online at wwwijopaasatin Research Paper Operation Transform Formulae of Fourier-Laplace Transform VD Sharma 1 and AN Rangari 2, * 1 Department of Mathematics, Arts, Commerce and Science College, Kiran Nagar, Amravati (MS), India 2 Department of Mathematics, Adarsh College, Dhamangaon Rly (MS), India * Corresponding author, (ashwinarangari@gmailcom) (Received: ; Accepted: ) Abstract: The concept of Fourier Transform is the most fundamental tool in harmonic analysis and also of major importance in functional analysis, complex analysis, partial differential equation, number theory, additive combinatory, representation theory, signal processing, etc Fourier Transform is also useful to select significant frequencies of an observed noisy signal, which can be applied as a model selection tool of (weighed) Fourier series analysis of medical images The Laplace Transformation is a very powerful technique, that it replaces operations of calculus by operations of algebra Laplace transform is applied in various areas of physics, Electric circuit Theory, Power systems, Load Frequency control etc This paper will discuss Operation Transform formulae of Fourier-Laplace Transform, we will discuss about shifting and scaling properties of Fourier Transform as well as Laplace Transform, Some results on differential operator are also discussed Keywords: Fourier Transform, Laplace Transform, Fourier-Laplace Transform, Operators, Differential Operator 1 Introduction: Mathematically speaking, the Fourier transform is a linear operator that maps a functional space to another functions space and decomposes a function into another function of its frequency components[4],[7] [9] The Fourier Transform can be used for Imaging Analysis[12] The Fractional Fourier Transform can also be used for Time Frequency Representation as quoted by Luis B Almeida [6] Laplace Transform is applicable not only to continuous functions but also to piece-wise continuous functions, complicated periodic functions, step functions, Impulse functions[8] Jason

2 Int J Pure Appl Sci Technol, 15(2) (2013), Creighton found that the Laplace Transform can also be used for circuit Analysis EASudicky [2] stated that the Laplace Transform is useful for mass transport in groundwater Fourier Transforms have various properties like Shifting, Scaling, Conjugate, Translation,Duality, Composition etc Laplace Transforms also have Shifting, Scaling, Linearity, Differentiation, Time delay etc[5] The time-shifting property of Fourier Transform identifies the fact that a linear displacement in time corresponds to a linear phase factor in the frequency domain This becomes useful and important when we discuss filtering and the effects of the phase characteristics of a filter in the time domain The differentiation property for Fourier Transform is very useful In the time domain we recognize that differentiation will emphasize these abrupt changes, and the differentiation property states that, consistent with this result, the high frequencies are amplified in relation to the low frequencies [13] By all the above properties of Laplace Transform we can solve various problems The properties of Fourier and Laplace Transform can be used to find the solution of heat equations, ordinary differential equations etc with initial values[8] [11] In the present paper, Operators on the space,, is defined and Shifting and Scaling property for Fourier Transform are proved in the section 2, Shifting property for Laplace transform is proved in section 3, Some theorems on Differential operator are proved in section 4, some results on differential operator are proved in section 5, Lastly Conclusions are given in section 6 Notations and terminologies as per [1] [3] [10] 2 Operators on the Space,, 21 Proposition: (Shifting Property for Fourier Transform) If,,, and is any fixed real number then +,,,, + <0 + >0 And +,,,, Proof: - Consider,,,,, +,=, +, =,,, Where =+ = Thus, +,,,, for + > 0 Similarly it can be show that +,,,, for + <0 22 Proposition: The translation (shifting) operator, +, is a topological automorphism on,,, for + > 0, and it is a topological isomorphism from,, onto,,, for + <0 23 Proposition: (Scaling Property for Fourier Transform) If,,, and >0, strictly positive number then,,,

3 Int J Pure Appl Sci Technol, 15(2) (2013), Proof: Consider,,,,,,=,, =,,, Where M is a constant depending on p Where = = = M,,,, Where = Thus,,,,, for > 0 24 Proposition: If >0 and,,, then the scaling operator,, =, where,=, is a topological automorphism,, defined by 3 Shifting Property for Laplace Transforms 31 Proposition: If,,,, +,, and is any fixed real number then, + >0 and, +,,, + <0 Proof: Consider,,,,,,+ =,,+ =,,, Where, =+ = Thus,, +,,, for + >0 Similarly it can be shown that, +,,, for + <0 4 Differential Operator F-Type 41 Theorem: The operator,, is defined on the space,, this space into itself and transforms Proof: Let,,, If, =, we have

4 Int J Pure Appl Sci Technol, 15(2) (2013), ,,,, =,, =,, =,, +1,,, ie,,, 42 Theorem: Or,,,,, =,,,,,,,, The operator,, is defined on the space,, and transform this space into itself Proof: Let,,, If, =, We have,,,,, =,, =,, =,, +1,,, = 0,1,2, Therefore,,,, ie,,, ie,,,,, =,,,,, 43 Proposition: The differential operator of F-type,, is a topological,, automorphism on 44 Proposition: The differential operator of L-type,, is a topological automorphism on,, 5 Results on Differential Operator 51 Theorem: For =, where, =0,1,2, if,,,, then,,, where, =, Further the mapping Δ= is one-one, linear and continuous

5 Int J Pure Appl Sci Technol, 15(2) (2013), Proof: For,,,,,,,,=, =, =, (511) Thus,,, if,,, It is obviously linear It is injective for, if =0 then =, is a constant If =0 then =0 and 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 (511) that,,,,,,,,,, where M is some constant Thus the theorem is proved 52 Theorem: For and,,, Proof: Let,,,, =, =,,, Consider,,,,,, =, =, Thus,,, if,,, In view of this proposition we have, the exponential multiplier operator,,, a topological isomorphism,, is 6 Conclusions In this paper, The Shifting and Scaling Property of Fourier and Laplace transform are described, some results on differential operator are proved References [1] AH Zemanian, Generalized Integral Transform, Inter Science Publisher, New York, 1968 [2] EA Sudicky, The Laplace transform Galerkin technique: A time-continuous finite element theory and application to mass transport in groundwater, Water Resources Research, 25(8) (1989), 1833

6 Int J Pure Appl Sci Technol, 15(2) (2013), [3] IM Gelfand and GE Shilov, Generalized Function (Vol II), Academic Press, New York, 1968 [4] JF Alssm and JS Walker, Time-frequency analysis of musical instruments, Society for Industrial and Applied Mathematics (SIAM) Review, 44(3) (2002), [5] L Debnath and D Bhatta, Integral Transforms and their Applications, Chapman and Hall/CRC Taylor and Francis Group Boca Raton London, New York, 2007 [6] LB Almeida, Member, IEEE, The fractional Fourier transform and time: Frequency representations, IEEE Transactions on Signal Processing, 42(11) (November) (1994), [7] MC Anumaka, Analysis and applications of Laplace/Fourier transformations in electric circuit, IJRRAS, 12(2) (2012), [8] SM Khairnar, RM Pise and JN Salunke, Applications of the Laplace-Mellin integral transform to differential equations, International Journal of Scientific and Research Publications, 2(5) (2012), 1-8 [9] SM Khairnar and RM Pise et al, Relation of finite Mellin integral transform with Laplace and Fourier transforms, Contemporary Engineering Sciences, 4(6) (2011), [10] VD Sharma and AN Rangari, Analyticity of distributional Fourier-Laplace transform, International J of Math Sci and Engg Appls (IJMSEA), 5(V) (2011), [11] VD Sharma and AN Rangari, Generalization of Fourier-Laplace transforms, Archives of Applied Science Research, 4(6) (2012), [12] VD Sharma and AN Rangari, Generalized Fourier-finite Mellin transform, International Journal of Mathematical Archive, 3(3) (2012), [13] WT Thomson, Laplace Transformation (2nd Edition), Prentice-Hall, 1960