A DIRECT EXTENSION OF MELLER S CALCULUS

Transcription

1 Inerna. J. Mah. & Mah. Sci Vol. 5 No. 4 (1982) A DIRECT EXTENSION OF MELLER S CALCULUS E.L. KOH Universiy of Peroleum and Minerals Dhahran, Saudl Arabia (Received June 26, 198) ABSTRACT. This paper exends he operaional calculus of Meller for he operaor -e d +i d B e d- o he case where e e (, oo). The developmen is la Mikuslnskl calculus and uses Meller s convoluion process wih a fracional derlvalve operaor. KEY WORDS AND PHRASES. Operaional Calculus, Mikusinski Calculus, Bessel Operaor, Convoluion Proems, Fracional Derivaive. 198 MATHEK4TICS SUBJECT CLASSIFICATION CODES. 44A4, 44A35, 33A4, 26A33, 93C5. 1. INTRODUCT ION. Familiariy wih inegral ransforms of disribuions is assumed This paper is accessible o readers familiar wih references [1-3]. The generalis reader ineresed in his area may sar wih hese references. Meller [4], [5] consruced an operaional calculus for he operaor B e -e e+l d-d wih -i < e < 1 by embedding i in a field of convoluion quoiens. The convoluion process was given by he formula: f()*g() P(l + e)p(l e) d e ( )-ed d x x i dx (1 x) ef (x)g[(1 x) (E n)] d d This calculus reduces o Dikin s calculus [6], [3] for when e. (1.1) Recenly, Koh [7], [8] and Conlan [9] exended Meller s calculus o he case e e (-i, o). A modified convoluion process was used which yields resuls analogous o Meller s. In he presen work, we give a direc exension of Meller s calculus 1 d by reaing he operaor F(I e) d ( )-d in (i.i) as a fracional derl- I1 va lye.

2 786 E.L. KOH Specifically, we le n be he leas ineger greaer han > O. For any n- imes differeniable funcion f(), he h-order derivaive of f() is: Df() D n I n- f() (1.2) d where D is and i is he Riemann-Liouville inegral of order > given in Ross [i] by lf () ( 6) f()d. (1.3) F() J I is easy o see ha I saisfies he semigroup propery ii= i+ bu D does no. Thus DaD in (2.1) below canno be wrien D+I. (1.4) 2. THE CONVOLUTION QUOTIENTS. Le _> be a fixed real number. Le C denoe he linear space of infiniely differeniable funcions on [,oo). For every pair of funcions () and () in C define heir convoluion by 1 DD I I I ()*() F( + N(I i) x) (xd)[(l x)( N)]dxdN (2.1) From his definiion, he following properies are clear: (1) C is closed under convoluion, (ii) convoluion is bilinear on C x C (iii) convoluion is disribuive wih respec o he usual addiion of funcions. I also follows immediaely ha equaion (2.1) specilizes o Meller s convoluion for < i and o Dikin s for. No so immediae are he following properies. PROPOSITION I. Convoluion is commuaive. V PROOF. Le x i and N in (2.1) and noing ha he Jacobian (x) i for all g (,) we have (v,o 1 [li vq v v ()*() F(a + DeDD )a( [(i )( i) o( )][ ()]dvd - 1 i F( + i) DDD J v(l )(v)[(l- )(- v)]ddv ()*(). q.e.d. PROPOSITION 2. For every complex nber %, and any #() e C %*() %(). PROF. l*() F(e + i) F(a + i) DDD (i- x)l(xn)dxdn Da+l i (i x)a(x)dx

3 EXTENSION OF MELLER S CALCULUS 787 r( + 1) r(+ l) %Dnln-ala() DaD u)(u)du Dnl n- ( u) (u)du %(). The las sep follows from (1.4) and (1.2). q.e.d. In view of Proposiion 2, here is no disincion beween consans and consan funcions in our calculus. PROPOSITION 3. Convoluion is associaive. PROOF. A direc calculaion shows ha, for nonnegaive inegers q and r, Hence on using (2.2) again, q, r q!r!r(q + + l)r(r + + i). q+r (q + r)!r( + l)r(q + r + + i) P,(q,r p!q!r! r(p+a+l)!r(q+a+l)r(r+a+l} p+q+r (p+q+r) F (i) F (p+q+r++l) F (+i) ( p, q), r (2.2) (2.3) Due o he bilineariy of our convoluion, equaion (2.3) sill holds for polynomials. Our proposiion follows from Weiersrass s Approximaion Theorem and he fac [9] ha he space of C funcions wih compac suppor is dense in C q.e.d. PROPOSITION 4. C has no zero divisors, i.e. if () and () belong o C and () * (), hen eiher () or (). PROOF. () * () implies ha ( ) -F C_l x) (xr])[(l x)( n)]dxdn n-i n-2 C1 (n i)! + C2 (n 2). - As O, some i, hen () and () have o be polynomials. Bu if hey are polynomials, + + Cn (2.4) Cn O. Now, by an argumen leading o (2.3), we see ha, if C i # for he lef side of (2.4) will be of degree a leas n. Hence, he righ side of (2.4) has o be zero. A similar argumen, ogeher wih Tichmarsh s Theorem 2], yields i ll rlc(1 x)aq(xrl)[ (I x)( ri)]dxdr] O. (2.5)

4 788 E.L. KOH To complee he proof, le x z and v in (2.5). We hen have Ivl -( r y) (y.)q[( r- y)(v-.)]dyd- Oo By a heorem of kuslnsi and Ryll-Nardzewski [ii], i follows ha z(yz) or y(yz). Thus, () or () O. q.e.d. The above properies esablish C as an inegral don der he operaions of addiion and convoluion as mulplcaon. By virue of Proposiion 2, he mulpllcave denly for C s he nber i. We may now exend C no he field F of convoluion quoiens consising of equivalence classes of ordered pairs (,) of elemens n C wih #. The equivalence relaion is gven by As usual, convoluion quoiens are called operaors 2 and are denoed by Operaors of he form.() consiue a subring of F isomorphic o C hrough he canonical maps () -+ () 3. AN OPERATIONAL CALCULUS. We now show ha he operaor B belongs o F. Firs, noe ha a righ inverse o B is given by A na(n)dnd; PROPOSITION 5 For any () g C * () A() +i PROOF. We shall assume ha a is no an ineger. Oherwise, he proof is more sraighforward, obviaing he use of fracional inegrals. i * qb() l ll n( (Ic x)(- ) e + 1 F(o + DDD i) (i- x) (x) )u )u o + 1 dxdn 1 DeDD ( n) (n- )a+ l($)ddn F(e + 2) n 2 Dc{ F(a (rl-$) + 2) ()ddn + ( ) ()d} On D F( + I) f frl a i (r- ) o ()ddn (3.1) o- o - i.e. B A, for g. C If we resric he domain of B o { g I() }, hen A is also a lef inverse; i.e., ABa

5 EXTENSION OF MELLER S CALCULUS 789 Le () In() where n leas ineger greaer han a. Then (3.1) becomes * () D + i i i -n i ) F( n + i ) ()dd I )n-a-i (n a) lu i D I n a-n D n _( u 1.- ) F(a n + i) ()ddndu F (n-e+l) F (a-n+l) () I rl-l(rl )c-n( rl)n-drld. The inner inegral reduces, via he Bea funcion, o {()-n-l}f(-n+l)f(n- ). Thus, + i * () n O. fl )(J){(-- 1}d D n n- (w)(w -n l)dw -n -i (wn- 1 fo n c -I ( [( ) -e (x() (wno(w) + nwn-llo(w))dw [( ) ()n]($)d [( ) ()hi(. (u)dud 1]d rl -c-i drld $ ()ddr A(). q.e.d. This resul implies ha operaors of he form ()_ wih #() may be idenified wih locally inegrable funcions f() such ha Af() < for every > O. Indeed,..O(). f() e Llo c[, oo) iff O() *f() (e + 1) A f < oo, V >. The nex resul follows from Proposiion 5 and Equaion (2.2) by inducion. PROPOSITION 6. Le k be a posiive ineger. Then, for any #() e C Le k n! (k + n) lk! V be he operaor * () Ak() where Ak() A(A(-.- (A))). k-imes + i and V k he k-imes applicaion of V.

6 79 E.L. KOH PROPOSITION 7. For any 2k imes differeniable funcion (), k j--1 (3.2) PROOF. () ABa() + () a + I Ba() + (). V() Ba() + ()V and (3.2) is proved for k i. Suppose now ha (3.2) is Thus rue for k m- i. Then for any 2m imes differeniable funcion (), "- Vine() m-i m-l V(Bo () + Z Bm-l-Jc (1)() l_+ j=l Vj) m-i B B m-1 a a B m-1 j=l m *() + *() I+O + V + Z Bm-l-J*().^+ V j+l B m ]--1 The proposiion follows by inducion. A number of operaional formulas such as hose in Theorems 5 and 6 of [7] may be generaed by using (3.2). The proofs are similar, muais muandis. A generalizaion of Theorem 5 of [7] is obained by parameric differeniaion. PROPOSITION 8. (V a) V P( + I) in(a)- --i m+l m! la+m (2k/) &+m V F(a + I) m(a)--t" (2 (V + a) m+l m! J&+m where l(x) and J(x) are Bessel funcions of order RHARKS. i. All he resuls of Meller are exendible o he case (,) via he meho given in his paper. 2. The operaional calculus may be applied o cerain ime-varying sysems and o Krazel s problem as done in [8]. -n-1 d n+l d 3. In [12], a convoluion for he operaor A d- d--[ where n is a naural number, is given which is associaive, commuaive, and disribuive wih respec o addiion. However, he ring under convoluion as muliplicaion conains zero divisors. ACKNOWLEDGEICENT. The auhor is on leave of absence from he Universiy of Regina, Regina, Canada. The research is parly suppored by he Naional Research Council of Canada under a gran numbered A-7184.

7 EXTENSION OF MELLER S CALCULUS 79i REFERENCES 1. ZEMANIAN, A.H..isrlbuion Theory and Transform Analysis, McGraw-Hill, New York, MIKUSINSKI, J. Operaional Calculus, Pergamon Press, Oxford, DITKIN, V.A. and PRUDNIKOV, A.P. Inegral Transforms and Operaional Calcul.us., Pergamon Press, New York, MELLER, N.A. n an operaional calculus for he operaor B-- Vichishielnaya Maemaika 6 (196), d +I d d 5. MELLER, N.A. Some applicaions of operaional calculus o problems in analysis, Zh. Vichisl. Ma. i. l. Fiz. 3(i) (1963), DITKIN, V.A. Operaional calculus heory, Dokl. Akad. Nauk. SSSR. ii 6 (1957), KOH, E.L. A Mikusinski calculus for he Bessel operaor B, Proc. Diff. Eq., Springer-Verlag Lec. Noes #564 (1976), KOH, E.L. Applicaion of an operaional calculus o cerain ime-varying sysems, In. J. Sysems Sc. 9 (1978), CONLAN, J. and KOH, E.L. On he Meijer ransformaion, In. J. Mah. and Mah. Sc. i (1978), i. ROSS, B. (ed.) Fracional Calculus and is Applicaions, Springer-Verlag, Berlin, ii. MIKUSINSKI, J. and RYLL-NARDZEWSKI, C. Un heorm sur le produc de composiion des foncions de plusieurs variables, Sudla Mah. 13(i) (1953), CHOLEWINSKI, F.M. A Hankel convolu.ion compl.e.x inversion Memoirs #58, rovidence, heory, A.M.S.

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