Transcription
UNIVERSITI PUTRA MALAYSIATHE MELLIN TRANSFORM OF GENERALIZED FUNCTIONSAND SOME APPLICATIONSMUHAMMAD REZAL BIN KAMEL ARIFFINFSAS 2002 63
THE MELLIN TRANSFORM OF GENERALIZED FUNCTIONSAND SOME APPLICATIONSByMUHAMMAD REZAL BIN KAMEL ARIFFINThesis Submitted to the School of Graduate Studies, Universiti Putra Malaysia,in Fulfilment of the Requirement for the Degree of Master of ScienceFebruary 2002
Abstract of the thesis presented to the Senate ofUniversiti Putra Malaysia infulfillment of the requirements for the degree of Master of ScienceTHE MELLIN TRANSFROM OF GENERALIZED FUNCTIONS ANDSOME APPLICATIONSByMUHAMMAD REZAL KAMEL ARIFFINFebruary 2002Chairman:Adem Kilicman, Ph.D.Faculty:Faculty of Science and Environmental StudiesFrom the space of testing functions Ep,q and the semi-norms induced onthis space and from the fact that the testing function {x} {XS-J;x 0ois a member of the space Ep,q for;x O)p Re{s q, made it possible for us to choosefrom the space E ,q generalized functions that are Mellin transformable.Extending the convolution in the Mellin sense to the generalized functions givesrise to the definition of the convolution in the Mellin sense of generalizedfunctions in E .qby(f{w), (g{u 1 (uw)))Thus, this made it essential for us to discuss the characteristics to beimposed onEp.q (uw)so thate{w) (g{u1 {uw)) would also be a member ofhence giving meaning to the definition of the convolution in the Mellin sense11
Ep.qhence giving meaning to the definition of the convolution in the Mellin senseof generalized functions in E .q. Accomplishing the above gave us the result thatthe exchange formula also holds for the Mellin transform of the convolution in theMellin sense for generalized functions in Ep' q From the research due to Butzer and lansche it had discussed the Mellintransform in an independent method and most importantly the Mellin translationoperator f{x) h f{hx) which is of different characteristic from the classical translation operator T,, f{x) h f{x h), or in the particular case for c 0,J;,f{x) f{x h), it gave rise to another scope of research regarding thedifferential equations in the Mellin sense (Le. differentiating functions using theMellin translation operator) with polynomial coefficients. By considering thedifferential equations in the Mellin sense with polynomial coefficients, andsolving it using the Mellin transform, we are able to put into order and solvevarious differential equations which could be shown to belong to the space X; ,thus generalizing the work by Sasiela.III
Abstrak tesis yang dikemukakan kepada SenatUniversiti Putra Malaysiasebagai memenuhi keperluan untuk Ijazah Master Sains.JELMAAN MELLIN FUNGSI TERITLAKDAN BEBERAPA APLIKASIOlehMUHAMMAD REZAL KAMEL ARIFFINFebruari 2002Pengerusi :Adem Kilicman, Ph.D.Fakulti :Fakulti Sains dan Pengajian Alam SekitarEp.qBerasaskan ruang fungsi ujianserta semi-norma yang ditetapkan ke atasruang tersebut dan bersandarkan fakta bahawa fungsi ujian (X) {XS-1o;x 0;x 0adalah suatu unsur dalam ruang EP.q untuk()p Re s q .memungkinkan kami untukmemilih dari ruang E .q fungsi teritlak yang boleh jelma secara Mellin. Memperluaskankonvolusi secara Mellin kepada fungsi teritlak memberi takrif konvolusi secara Mellinbagifungsi teritlak dalam E .q yang diberi oleh(f(w1(g(u 1 (uw))).Oleh yang demikian. adalah perlu bagi kita untuk membincangkan ciri-ciri yangdikenakan ke atas (uw)supayae(w) (g(u 1 (uw)) juga menjadi ahli bagi EP.q' danjusteru itu memberi makna kepada takrif konvolusi secara Mellin bagi fungsi teritlakIV
dalam E'p,q , Bersandarkan di atas, ia memberi kita keputusan di mana rumus penukaranadalah benar untuk jelmaan Mellin bagi konvolusi secara Mellin untuk fungsi teritiakdalam E ,q.Berasaskan kajian oleh Butzer dan Jansche yang telah membincangkan jelmaanMellin dalam suatu kaedah bebas dan paling penting pengoperasi translasi Mellin'l" f{x) hef{hx)yangberbezaemT;, f{x) hef{x h), atau untuk kes khususdaric pengoperasl0,translasiklasikT;,f{x) f{x h), ianya telah membuka satu lagi ruang kajian, mengenai persamaan pembezaan secara Mellin (i.e.membezakan fungsi menggunakan pengoperasi translasi Mellin) dengan pekalipolinomial. Berasaskan persamaan pembezaan secara Mellin dengan pekali polinomial,dan menyelesaikannya menggunakan jelmaan Mellin, kami boleh menyusun sertamenyelesaikan beberapa persamaan pembezaan dalam ruang X:, yang akhimya telahmengitlakkan hasil yang diperolehi oleh Sasiela.v
ACKNOWLEDGEMENTSIn the name Of Allah the most Gracious the most Merciful. This thesis would nothave been possible without the will of Allah s.w.t, for it is He who gave me strength andwisdom to carry on even at times of great uncertainty.In completion of this thesis, I would like to thank my supervisors, Prof. MadyaDr. Adem Kilicman, Prof. Madya Dr. Malik Abu Hassan and Prof. Madya Dr.Suncheleyev Rustam for their comments, helpful comments, valuable guidance andencouragement throughout my research.To the Department of Mathematics, Faculty of Science and EnvironmentalStudies, Universiti Putra Malaysia, I would like to thank Prof. Madya Dr. Isa Daud forgranting me study leave form November 1999 to April 2001 and Prof. Dr. Kamel AriffinMohd Atan for granting me study leave from May 2001 to November 2001 to carry outand complete my research.Finally I would like to dedicate this thesis to my supportive parents Kamel Ariffinand Zaini, beloved wife Siti Suhailah and son Muhammad Haziq Sirhan, for theirencouragement and unfailing support during my course of study.VJ
I certify that an Examination Committee met on 6th February 2002 to conduct the finalexamination of Muhammad Rezal bin Kamel Ariffin on his Master of Science thesisentitled «Mellin Transform of Generalized Functions and Some Applications" inaccordance Universiti Pertanian Malaysia (Higher Degree) Act 1980 and UniversitiPertanian Malaysia (Higher Degree) Regulations 1981. The committee recommends thatcandidate be awarded the relevant degree. Members of the Examination Committee are asfollows:HJ. HARUN BUDIN, Ph.D.Associate ProfessorFaculty of Science and Environmental Studies,Universiti Putra Malaysia(Chairman)ADEM KILICMAN, Ph.D.Associate ProfessorFaculty of Science and Environmental Studies,Universiti Putra Malaysia(Member)HJ. MALIK ABU HASSAN, Ph.D.Associate ProfessorFaculty of Science and Environmental Studies,Universiti Putra Malaysia(Member)RUSTEM SUNCHELEYEV, Ph.D.Associate ProfessorFaculty of Science and Environmental Studies,Universiti Putra Malaysia(Member) - ----r SHAMSHER MOHAMAD RAMADILI, Ph.D.ProfessorlDeputy DeanSchool of Graduate StudiesUniversiti Putra MalaysiaDate: 14VllMAR 2002
This thesis submitted to the Senate ofUniversiti Putra Malaysia has been accepted asfulfilment of the requirements for the degree of Master of Science.AlNI IDERIS, Ph. D.ProfessorlDean,School of Graduate Studies,Universiti Putra MalaysiaDate:Vlll09 MAY 2002
hereby declare that the thesis is based on my original work except for quotations andcitations which have been duly acknowledged. I also declare that it has not beenpreviously or concurrently submitted for any other degree at UPM or other institutions.IDate:l?IXMARC \-\ ;l.OO:l-
TABLE OF CONTENTSPageABSTRACTABSTRAK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ACKNOWLEDGEMENTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . .APPROVAL. . . . . .DECLARATION . . . . . . . . . . .LIST OF TABLES. . . .LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11ivviVlJIXXlJXllICHAPTER1INTRODUCTION.Notations and Definitions . . . . . . . . . . . . . . . . . . . . . .Special Functions. . . . . . . . . . . .L'Hospital's Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Organisation of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111719202123MELLIN TRANSFORMS.252527. . . .2. . .Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . .Basic Properties of Mellin Transfonn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The Mellin Translation . .The Mellin Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Mellin Operators of Differentiation and Integration . . . . . . . . . . .3. . . .GENERALIZED FIUNCTIONS (DISTRIBUTIONS) . . . . . . .Introduction . . . . . . . . . . . . .Countably Nonned Space and their Duals . . . .The Dual Space of Countably Nonned Space . . . . . . . . . . . . . . . . . . . .Test Function Spaces . . . .The Dual Spaces of Testing Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . .Calculus of Generalized Functions . . . . . . . .The Space Ep.q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .X. . . . . . . . . .3739414949505354586470
The Space E .q (The Dual Space of Ep.q ) . . . . . . . . . . . . . . . . . . . . . . . . . .The Space E1 .,.,. . .The Space E;.rIJ " . ". ,.,. " . ,',. ".,.,.,.,.,.,. ". , , ' , , , . , . , , ,. .7579.80. .4CONVOLUTON IN MELLIN SENSE"". . ,.,.,. . . .Introduction . ,.,. .,, . . . ,.,"., . . ,.,. . ,. .,.,.,.The Mellin Direct Product of Generalized Functions in E ,q . . . . . . .838384. . .The Support of the Mellin Direct Product of Generalized Functions'in Ep,q.89The Convolution of Generalized Functions in E ,q in the Mellin. . . . . . . . . .Sense . . . . . . . . . . . . .90Some Operations on the Convolution in the Mellin Sense. . . . . . . 93. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5MELLIN TRANSFORMS&GENERALIZED FUNCTIONS,Introduction . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . .Mellin Transform of Generalized Functions in E ,q . . . . . . .Conditions for F()s'FunctIon In EP.q '.The Mellin Transformation of the Convolution of GeneralizedFunctions in E ,q in the Mellin Sense . . . .6THE MELLIN DIFFERENTIAL EQUATIONS . . . ,.Introduction . . . . . . . . . . .Laplace Transform and the Differential equations ., . . . . . . . . . . .Mellin Transform and the Mellin Differential Equations . . ,. . . . . . . . . . . . . . .Laplace and Mellin Transformable Differential Equations withPolynomial Equations ., . . . . . . . . ,, . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . .7CONCLUSIONSAND SUGGESTIONS.Major findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Conclusions . . . . . . . . . .Suggestions for further research . . . ., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .BIBILIOGRAPHYVITA. . . . . . . . . . . . . . . . . . . . . . . . ,.,. . ,. ,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Xl1011011031081 141 191 191 191 231 281311311 341 341371 40
LIST OF TABLESTableAPageTable of the Mellin transform of functions widely usedXIl1 39
LIST OF FIGURESFigure4.14.2PageMellin regularization of the function f{x) !xThe function f{x) !x Xlll9898
CHAPTERlINTRODUCTION1.1Notations and DefinitionsInthis work we will use the usual notation,N willdenote the set ofnatural numbers, Z the set of integers, Qthe set of rational numbers, R the set ofreal numbers and C the set of complex numbers. The null set will be representedby0.The interior point of a set E is a point peE if there exists E 0 suchthat Nc (P) c E, whereNcneighborhood of the point pneighborhoodNc (P)(P). { eRxA point:Ix pi E} and is called the epsilon -peR is a limit point of E if every epsilon of p contains a point q e E with q :I; p. The subset 0 of Rwill be open if every point of 0 is an interior point of O. The subsetM of R isc10sed if and only if it contains all its limit points.Supremum of a set E denoted by sup E, defines an element which is theleast upper bound of E . Infimum of a set E denoted by inf E defines an element,which is the greatest lower bound of E .Definition J. J. JThe space X is called a linear (vector) space if for anyx,y,ze X anda,p e C it is non-empty, closed under 'addition', closed under 'multiplication'by scalars and the following axioms hold.
21.x y y x11.x (y z) (x y) Zlll.X O XIV.X (-X) 0V.a(,8x) (a,B)xVI.(a p)x ax f3xVll.a{x y) ax ayVll.l·x xDejillltion 1.1.2Alinear tr ansfonnation fwhich maps a vector spacespace Y, i.e. f:X intoa vectorX Y, is called an isomorphism if the linear transfonnation isone to one and onto.Dejillltion 1.1.3LetX bea vector space onR. Afunction1111: X R satisfyingfollowing conditions1.IIxil 0 for all x e X11.Ilxll O if and only if x 0m.Ilaxll lalllxli for aU a e C and x e XIV.Ilx YII : llxll IIYIIfor aUx,y E X the
3is called a norm on X , The pair(X,IIII) is called a normed space.Example 1.1.1a.For any x E R, the norm induced on R is given by Ilxll I xl .b.Let X era, b]be the space of continuous functions on the closedinterval [a,b]. For I E X, the norm induced on this space is givenby IIIII !!1 /(x Definition 1.1.4Let X be a vector space of R A function p: X R satisfying L, iii. and.iv. in definition1 . 1 .2is called a semi-nonn on X. The pair (X, p) is called asemi-nonned space.Example 1.1.2Let D be the space of infinitely differentiable functions (Gelfand, Shilov,1964).The semi-norms induced on this space is given byk 0,1,2, . .If k 0 ,ro(;) is a norm.sup 1;(k )(X forrk (;) xe[aA
4Definition 1.1.5a.Asequence {xn } in a nonned linear space(X,I ID is aCauchy sequence if for every E 0 , there exists a positiveinteger no such tha Ilxn - xm II E for all integersA nonned linear spaceb.n,m no.(x, II II) is complete if every Cauchysequence in X converges in nonn to an element inX.Definition 1.1. 6Let Un t.] be a sequence of real-valued functions defined on a set E . Thesequence Vn } converges pointwise on E to f if for any E 0 and for everyx E E there exists a positive integer no n o(x E) such that I f(x)- fn{x E for,Definition 1.1. 7Let {fn }:.] be a sequence of real-valued functions defined on a set E . Thesequence {fn } converges unifonnly on E to f if for any E 0 there exists apositive integerwhereno such that I!(x)- /,,(x efor allno no(E) (i.e. n (E) depends only on E). on no for every x E E,
5Example 1. 1.3a. Let E [0,1] and for each x eE, neN let I,,(x) x". Each I" iscontinuous on E. Since In (1) 1 for all n , limIn (1) 1. Ifn- '"o X 1 then lim /" (x) o. Thus we may say that the sequence ofII . '"functions{ f{x} {t" t l converges pointwise to the limit function,·x 1;0 x 1. That is, for any E 0 there exists a positiveinteger nodepending on both E and x such that ! f(x) - x"! E for alln no(x,E). If x l , 1/"(x)-f(x ll- ll O E. Now letxe[O,l).Inorder for {tn } to converge to f,IXIIOI Emust betrue for allX E [0,1) for a positive integer no. Let E 1 and n no(x,E) logE .logxThennlogx logEx" Eb. We shall now show that the sequence in a. converges uniformly tooon the interval [0, a] for 0 a 1. For any E 0 there exists a. mteger.pOSItIveno logE sueh that fior every n no we haveloga--logEn - loga
6Thus, Ix" 1 a" Efor every x E [O,a]. This shows that the sequenceof functions f" (x) x" converges to unifonnly.Definition 1.1.8LetthenXXbe a nonned linear space. IfXis a complete nonned linear spaceis said to be a Banach space.Example 1.1.4Let us consider the spaceX e[a,b]of functions that are continuous onthe closed interval [a,b]. For any I E X, let the nonn be defined as1 /1 Xis amaxl/ {x . According to definition 1.1.5, a sequence of functions {f"} onxe!a.br Cauchy sequence, if for anyAny Cauchy sequence inE X 0there exists an integerN suchthat fore[a,b], with the above norm, convergesuniformly to some function in X e[a,b] (Stoll, 1997). Since the convergenceis uniform, then the limitfunction I must also be continuous. Thus, X e[a,b]is a Banach space with the above mentioned nonn.
7Definition 1.1.9LetX bea Banach space, which admits scalar multiplication, vectoraddition and vector multiplication. If for anyx,YE X,Ixyl ;5; Ilxl 'lIylthen X is called Banach Algebra.Let [a,b], a b, be a closed and bounded interval on R. By a partitionPof [a,b]we mean a finite set of points P {xo,XI".,xJa Xo XI . , Xn b, There is no requirement that the pointsspaced, For each i 1,2, ,n, set./j,x, x,-such thatx, be equallyX, I'Now let,m, inf{/(t): X, I S t;5; x,}M, sup{/(t) : X, I ;5; t;5; x,}and let us define the upper sum U(P,/) for the partition Pand function /bynnU(p,/) LMj/j,x, and the lower sum L(P,/) by L(P,/) Lml/j,xj ,Definition 1.1. J 0Let /be a bounded real-valued function on the closed interval [a,b], Theupper and lower integral of /is defined by-[/ inf{U(P,/)}
8f! sup {L(P,!)}Definition 1.1.11A functionthe limitlimmaxtu Othe partitionsf is said to be Riemann integrable (i.e. integrable) on [a,b] ifI f(x; rf{x)dx exists, and does not depend on choice ofk 1[Xi I,XJ c [a,b] for i 1,2,3, . . .or on the pointsx;in the partitions.Theorem 1.1.1 (Stoll, 1 997)A bounded real-valued function is Riemann integrable onif for everyc o there exists a partition[a,b] if and onlyP of [a,b]such thatU(P,f)-L(P,f) cDefinition 1 . 1. 12A subsetcollection of { IE c Rhas measure zero, if for any c 0 there exists a countableJnEA of open intervals such that EcU 1n and :Z(IJ c, wherenEAnEA/(1,,) denotes the length of the interval In.Theorem 1. 1.2 (Lebesgue) (Stoll, 1997)A bounded real-valued function f on[a,b] is Riemannonly if the set of discontinuities has measure zero.integrable if and
9Thus, when a function is said to beRiemannintegrable, the integral isgiven by rf(x)dx which is evaluated in the usual way.Definition 1.1.13If J is an interval we define the measure of J , denoted by m{J) to be thelength of J . If J is (a,b), [a,b), (a,b] or [a,b] thenm (J ) b - a . If J is (a, oo ) ,[a, 00) , (-oo,b) or (- oo,b] then m(J) 00.Definition 1.1.14Let E be a subset of R . A collection {OoteA of open subsets in R is anopen cover if E c UOa .aeADefinition 1.1.15Asubset K of R is compact if {Oa }aeA is an open cover of K, and therenexists al'a2,a3,.,an e A such that K c UOaj .j-IDefinition 1.1.16Let E be a subset of R . The Lebesgue outer measure of E is defined by.t inf {m{U) : E c U} where U is open. The Lebesgue inner measure of E isdefined by A. sup {m{K) : K c E} where Kis compact.
10Definition 1.1.17a.measurable) ifdefined byb.E of R is said to be measurable (LebesgueA bounded subsett(E) A.(E).Inthis case the measure ofEisl(E) t(E) A.(E).An unbounded set subsetE of R isE n [a,b] isformeasurableinterval [a,b]. By choosingthat E c [a,b]wecansaid to be measurable ifeveryclosedanda and b sufficientlydefinetheboundedlarge suchmeasureofE asl(E) ic- «Jliml (En[- k,k D.The two separate definitions are required due to the existence ofunbounded non measurable setsE for which t(E) A. (E) 00.Example 1.1.5a.Let E [a, be a bounded subset ofR Since t (E) A. (E) b -a ,.thenl (E) b -a.b.LetE (c,d)beanunboundedsubsetofR.Sincelimt (En[-k,kD A. (E) limA. (En[-k,kD d -c ,t(E) k- «Jk- «Jthen l (E) d -c.
11Definition 1.1.18f be a real-valued functionLetmeasurable if for every s e R , the setgenerally, ifthe seton[a, b].The function f is said to be{x e [a,b] : f{x) slE is a measurable set of R, a function f : E R is measurable if{x e E : f{x} sExample 1.1.6Letf : [O,l] R be defined by f(x} {o;x e Q [0,1]then we canx ; x e [O,l ] \Q11have the following set,[0,1] ;s{x e [0,1] : [(x) .) Q' (.,1) ;s;s{ Since the sets[0,1], QC ( s11measurable.Definition 1.1.19Let {In} bedisjoint if In 11 1macollection of open intervals. The family {In} is pairwise 0 whenever n m.
Table Page A Table of the Mellin transform of functions widely used 139 XIl . LIST OF FIGURES Figure 4.1 Mellin regularization of the function f{x) ! 4.2 The function f{x) ! x x Xlll Page 98
