On Distributional Marchi-Zgrablich Transformation

Transcription

1 IMA Journal of Applied Mathematics (1985) 34, On Distributional Marchi-Zgrablich Transformation S. D. BHOSALE Department of Mathematics and Statistics, Marathwada University, Aurangabad , India AND S. V. MORE Institute of Science, 15 Madam Cama Road, Bombay-32, India [Received 20 June 1984 and in revised form 8 January 1985] The purpose of this paper is to extend the classical Marchi-Zgrablich transformation to a class of distributions, in the spirit of L. Schwartz's extension of the Fourier transformation to distributions. The inversion and uniqueness theorems have been proved. An operational transform formula is also established. A differential equation of the form P(A, J*u = g has been solved by using Marchi-Zgrablich transform of distribution. 1. Introduction THE FINITE HANKEL TRANSFORMATION has been extended to a class of generalized functions by Zemanian (1968), by using orthonormal series expansion. In our earlier paper (Bhosale & More, 1984) we extended Marchi-Zgrablich transformation to distributions by using orthononnal series expansion. In this work we extend the classical Marchi-Zgrablich transformation to a class of distribution by quite a different method from the two mentioned above. The method involved in this work is related to Hilbert-space techniques and is in the spirit of L. Schwartz's extension of the Fourier transformation to distributions. Further, the classical inversion theorem for the Marchi-Zgrablich transformation (Marchi & Zgrablich, 1964) is extended to a class of distributions which gives rise to the Fourier-Bessel series expansion of generalized functions. The convergence of the series is interpreted in the weak distributional sense. An operation transform formula is also established. A differential equation of the form P(A,J*u = g has been solved by using Marchi-Zgrablich transform of distribution. The classical Marchi-Zgrablich transform of a function / defined on (a, b), is defined (Marchi-Zgrablich, 1964) as whose inversion theorem is given by Rn)=[ b xfix)sjlk 1,k 2, l i n x)dx (1.1) ^) = I-7tn)S^i,*2,A*.*X (1-2) 0272^*960/85/ $03.00/ Academic Press Inc. (London) Limited

2 214 S. D. BHOSALE AND S. V. MORE where SJLk lt k 2, ft,*) = JJ^x)lYJ,k u ^a) + YJLk 2, and = r JJp m x) and y^i,x) are the Bessel functions of the first and second kind respectively of order v and // are the positive roots of the equation 2. Notation and Terminology 0..(1.3) In notation and terminology we follow Zemanian (1968). n m n= 1,2,... will denote the positive roots of the equation (1.3) with (i t < \x 2 < n 3 <... The open set (a, b) will be denoted by the letter /. The letters x and t will be real and onedimensional variables restricted to the open interval /. We will use the following Bessel operator, and the expression ^ ^ J fc = 0,l,2,..., v^-i, > x = ^ (2.1) GJt, x) = - SJLk u k 2, /V)S.(*i, k 2, n n x). (2.2) 3. The Testing Function Space MZ^ 1) and Its Dual Let x e I = (a,b) and m, v be fixed real numbers satisfying v 3= ^ and m > \. Let (x) be an infinitely differentiable function defined over /, satisfying (x) > 0 for all x > 0 and such that (»( x ) = 2t-i» 0 < a < x < b and for large values of /v Now we define MZ mi,(/) to be the collection of all infinitely differentiable complex valued functions #(x) on / such that pr? (#c))= sup IftxK^x-^x)]! (3.1) a<ki exists (i.e. finite) for each k = 0, 1, 2, MZ^^i) is a linear space over the field of complex numbers. We assign the topology to it by taking {pi 1 *'(< )} as its seminorms and in addition Po'"(4>) is a norm on MZ K,^/). We equip MZ M-y (J) with the topology that is generated by the countable multinorm {pr*(^)}» -o aq d this makes MZ,,(/) a complete countable multinormed space. The dual space MZ^/) consists of all continuous linear functional on MZ M,^7). By Zemanian [1968, th. (1.8.3)], MZ' m, J(I) is also complete. It is clear that MZ M, J(I) is a Frechet space.

3 MARCM-ZGRABLICH TRANSFORMATION 215 LEMMA 3.1 MZ^XO is a testing Junction space. Proof. Clearly MZ^ T) satisfies the first two conditions (Zemanian, 1968, p. 39) of a testing function space. Now we shall prove the third. We have the result for all k Ik n-0 ), for (3.2) Let {< a (x)} -i converge in MZ m,(/) to zero. In view of (3.2) and the seminorms defined in (3.1) it follows by induction on k that for each k, [D m (j> B (x)} converges uniformly to zero on every compact subset of /. This proves the third condition. Hence MZ,, XO is a testing function space. We now list some further properties of the space MZ m _XO and its dual. (i) D{I) is a subspace of MZ m>,(/) and convergence in D(I) implies convergence in MZ M X0- The topology of D{T) is stronger than the induced topology on it by MZ n,(/). Consequently the restriction of any member of MZJ.,,(/) to D(I) is (ii) in iy(i). Moreover convergence in MZ^ JJ) imphes convergence in I>(I). MZ m,,(/) c (/). Moreover it is dense in (/) because D(I) cz MZ M>,(/) and D{I) is dense in (/). The topology of MZ m> JJ) is stronger than that induced on it by (/). Hence '(/) can be identified with subspace of MZ^,,(/). (iii) For each / e MZ^, XO there exists a non-negative integer r and a positive constant C such that </,0> <C max pr'(#*)) for every ^»(x) e MZ_,(/). Here r and C depend on/but not on <f>(x). The proof of this follows by the boundedness property of generalized functions. (iv) Let f[x) be a locally integrable function defined on / such that f xf(x) dx exists. Then_/fa) generates a regular generalized function in MZ^,(/) defined by if, <t>> = \ b Ax)4>(x) dx, Indeed, let <j>{x) e MZ M,XO- Then 4>(x) e MZ M,X/). (3.3)

4 216 S. D. BHOSALE AND S. V. MORE X/{X) o- y (<t>(x)) r dx < oo for all 4>{x) e MZ W 1) and hence our assertion. (v) For each n = 1,2,3,... and v^ \ and a < x < b, the function xsj^k u k 2, u B x) is a member of MZ my (/). Indeed, an easy computation leads to ~D X - x"v j SJk lt k 2, u n x) = -uls^ky, k 2, u n x). (3.4) Since a < x < b, \x n are the positive zeros of (1.3) and S^k u k 2, u n x) = O(l//xJ (Wankhede, 1972) uniformly on (a, b). Therefore p"' y [_xs,(k u k 2, /x,x)] < oo for allfc = 0, 1, 2,... (vi) The operation <j> - xa* -x [x" 1 </>(x)] is a linear continuous mapping of MZ M,(/) onto itself since 4. The Generalized Marchi-Zgrablich Transformation for all </>emz M>v (J). Let m, v satisfy v > ^ and m > ^. We shall call a generalized function / Marchi- Zgrablich transformable if it belongs to MZ^i). We are now in a position to define our generalized Marchi-Zgrablich transformation, which we denote by /?(/). For a given Marchi-Zgrablich transformable generalized function /, the Marchi- Zgrablich transform F(n) of / is defined as the application of / to the kernel xs^fci.fcj, n n x), that is, P(f)(n) = Fin) = <J[x), xsjk u k 2, ^x)>, (4.1) where fi n are.the positive roots of (1.3). (4.1) is well defined in view of the fact that xs,(k lt k 2, n n x) e MZ., JH) for each n = 1, 2, 3 The boundedness property of the generalized Marchi-Zgrablich transform has been established by proving the following theorem. THEOREM 4.1 Let f be a member o/mz^i), m ^ i v > -^, a < x < b and Then F(n) satisfies the inequality F(n) = Cflx), xsjik lt k 2, ft.x)> for n = 1, 2, 3,... \F{nlH ^ (A/J-POO, (4.2) where the polynomial Pfjtf'J will depend upon the choices of a,b and n. Proof. Since / e MZ'^J), then by Section 3, property (iiix there exists a nonnegative integer r and positive constant C such that max

5 MARCJfl-ZGRABLICH TRANSFORMATION 217 = C max p Since m 2* }, v ^(x)5,(fc!, fe 2> /x,,x) = C max sup \i(x)sjlk u k 2, xml -\ and S^k v k 2, n,x) = 0(l//O (Wankhede, 1972), then.4, for all x e /, where i4. is a constant Hence [F(n)\ max This completes the proof of the theorem. 5. Inversion and Uniqueness We shall now derive an inversion formula for Marchi-Zgrablich transformation of generalized functions. The proof of the inversion formula requires the following lemmas. LEMMA 5.1 For any positive integer N, tg^t, x) e MZ m,(i). Proof. To prove this we shall show that for all k = 0, 1,2,... sup (x)a* i,[r 1 {rg*(r,x)}] is finite. Therefore it/ since X ^ 1 1,2k- 1 - SJLk lt k 2, fi m t)s,(k u k 2, nn mr fr-fl t, k 2, (Wankhede, 1972). Therefore since sup MM' oo, is bounded, where M and M' are constants. Hence tg^t, x) e MZ m

6 218 S. D. BHOSALE AND S. V. MORE Now it is clear that rtgjt,x)x<mx)dx is a member of MZ n,(/), for an arbitrary < {x) e >(/). LEMMA 5.2 Let f e MZ^,,(/), then for any positive integer N and for an arbitrary 4>{x) e T <yt ). tg.it, x)>#x)x dx = </U), T dx>. (5.1) Proof. In view of Lemma (5. IX rthe function tg^t, x) and t, x)4>(x)x dx are the members of the space MZ m,(/) with t as the variable of testing functions. Expressions on both the sides of (5.1) have a sense. Using the technique of Riemann sums (Pandey & Zemanian, 1968, th. 2\ (5.1) can be easily established. LEMMA 5.3 Let 4>(x) be an arbitrary member of D(I) and I = (a, b), then for a ^ $, f* G^t J x)[^i(x)-il/{t)']xdx O asn-*oo (5.2) uniformly for all t e (a, b) where the support of ip(x) is contained in the interval (a, b) and 0 < a < a' < b' < b. Proof. Let us divide the interval (a, b) into two mutually disjoint sets (a, a') u {b', b) and [a', b']. For t e (a, d) u(fc, b\ 4>{t) = 0 as support of \f/(t) is contained in [a', i/]. Therefore P G^t, x)mx)- mix dx = f * Cfr, dx. Ja' Jo' In view of Lemma (5.1) and of the analogue of the Riemann Lebesgue Lemma (Watson, 1958), for a given \: e > 0 there exists a positive integer N Q such that where M, M' are the constants. Therefore for all N > N o and for all t e (a, a 7 ) u (b\ b), f\{ b G N {t,x)il>(x)xdx since a > \ and a < t < b. Hence as e is arbitrary we have J>, f >0 as N -* oo uniformly for all t e (a, d) u (b', b).

7 Next we want to show that MARCHI-ZGRABLICH TRANSFORMATION 219 t, x)o(x)- ij/(t)]x dx -> 0 uniformly for all t e [a', #]. Let F(t, x)(x 2 -t 2 ) = x~'[i/r(x)-^(t)] for a < x < b and a < t < b. Now define the function H(t, x) in the square domain {a < t < b; a < x < b} as { F(t, x\ t =x, ->'(*) t_ x Obviously H(t, x) is a continuous function of t and x in the domain {a < t < b, a < x < b). Now Jo' Ja' = rx' +1 if(f,x)(x 2 -t 2 Ja' as the value of the integral remains unchanged by replacing the expression F(t, x)(x 2 t 2 ) by H(t, x)(x ). Let us now divide the interval d < x ^ b' into P equal parts by the points d = XQ, x lt x 2,..., x p = b' and after choosing a positive number e, p so large that m-l where u M and L^, are respectively the upper and lower bounds of H(t, x) taken over {x m _, < x < x B a' ^ ^ b'}, m = 1, 2,..., p. Using uniform continuity of the function H(t, x) over the region {a 1 < t ^ ', a 7 < x < b'} and following the line in the proof of the analogue of the Riemann Lebesgue Lemma (Watson, 1958) for an arbitrary e > 0, we get ir x^h {t,x)[x 2 -t 2 ) Gt At,x)dx ^MM>(fc ;- ar where M and M' are constants. Hence for t e [d,b'] and as e is arbitrary, AfM'(b'-oOVe as N -* oo uniformly for all t e [d, b"~\. r Thus uniformly for all r e (a, b) as N - co when the support of tp(x) is contained in the interval \d, f].

8 220 S. D. BHOSALE AND S. V. MORE LEMMA 5.4 Let <f>(x) e D(I) with the support contained in (a, b) then t\ G^t,x)x4>{x)dx converges in MZ.,,(/) to t<f>(t) as N -* oo for all t e (a, b). Proof. In view of the operation relation (3.4) we can easily see that Since the integrand in t, x)x<p(x) dx is a smooth function and <p{x) is of bounded support, we may differentiate under the integral sign and by integration by parts we obtain v>, j*, x)x<kx) dx = J A^G^t, x)]xflx) dx 1 Operating the operator A,, successively and using the integration by parts, it can be shown that Hence as N -» oo we have At., f" G^t, x)x<ftx) dx = T GA x)at. x [0(x)]x dx. A*. t P G^t, x)x[0(x)-*(t)] dx = T G^t, JC)[<^W-A:,,#))]X dx = f*g(t,x)[^(x)-^(t)]xdx, where ip(x) = At x 0(x) which is obviously a member of D(/) with support contained in (a, b). Hence it suffices to show that ^G^t, converges to zero as N -> oo uniformly for all t e (a, b) which is true in view of Lemma (5.3). This completes the proof. THEOREM 5.1 (Inversion). Letf(x) be a Marchi-Zgrablich transformable generalized function and F(n) be the generalized Marchi-Zgrablich transform offe MZ^/) as

9 MARCK-ZGRABLJCH TRANSFORMATION 221 defined by (4.1). Then in the sense of convergence in D'(I) M= lira t -F(n)SAk u k 2,H.x)- W-co -1 C n Proof. Let < {x) e D(I). We wish to show that - F(n)S,(k u k 2, ^x), <Kx)) = <M <Kt)> asat - oo. (5.3) We know that #c) e D(-0 iff x<j>(x) e >(/). (5.3) is equivalent to showing as N -> oo. (5.4) Suppose that the support of <f>(x) is contained in (a, b). We prove (5.4) by justifying the following steps. As _ 1 \ p{fj\ ^ I Is If II v\ is locally integrable and since x< (x) e D{I) then without limit notation (5.4) can be written as f»_ 1 y F(n)S^k 1> k 2,n, Since <p(x) has a compact support and the integrand is continuous in (a, b), therefore By Lemma (5.2) we have JO \ C B " Cn I u k 2,» n x)x4>(x) dx Ja i C n Then by Lemma (5.4) ( - F(n)SAk lt k 2, n n x)x<p(x) dx) ^ <J{t), t<j>(t)y as N -» oo. \Ja " C n I This completes the proof.

10 222 S. D. BHOSALE AND S. V. MORE THEOREM 5.2 (Uniqueness). Let F(n) = fhj) and G{n) = P(g) be the Marchi- Zgrablich transform of f and g. lff[n) = G(n)for each n = 1, 2,..., thenf= g in the sense of equality in Proof. By Theorem 5.1 (f-g)= lim -{.F(n)-G as F(n) = G(n) for each n = 1, 2, Hence/ = g in the sense of equality in 6. Operational Transform Formula and Operational Calculus For v ^ i and m $s ^, we define a generalized operator [A,. J* on MZ^/) as the adjoint of the operator [xa, Jix'^x)] on MZ m ^/). More specifically for arbitrary ftx) 6 MZ m>,(/) and / e MZ^,.(/) ^), <W^)> = <^), xajx-'««x))). (6.1) The right-hand side of (6.1) makes sense, because xa, JC (x" 1^)(x)) e MZ^XO when < (x) e MZ my (/). Since the mapping <^(x)-»-xa T ^"^(x)) is linear continuous on MZ,, t,(/) into itself, then [A, J* is also a continuous linear mapping on MZ^,,(/) into itself. It can also be seen inductively that for any integer k we have <(AJ, x)*flx), #x)> = y[x\ xaj,.(x "»0(x))>. (6.2) which leads to the following operation transform formula /*[AU*/=(-l)W(A «= 1,2,... (6.3) Indeed, for/e MZ^_,(/) and xs v (ki, k 2, fi,x) e MZ Hi i/) we have ; t, k 2, p B x)> = </(x), xa^jx-ixsakt, k 2, n n x))> where /?(/) = (J[x\ xs,(/c 1( k 2, rt.x)>. The formula (6.3) represents the property of Marchi-Zgrablich transformation which makes it so useful as an operational tool for solving differential equations involving generalized functions. The Marchi-Zgrablich transformation generates an operational calculus by means of which certain differential equations involving generalized functions can be solved. Now the differential equations that can be solved by using the generalized Marchi-Zgrablich transformation are of the form P(A,, J*u = g, (6.4) where P is a polynomial and given g and unknown u are required to be in MZ',, XO- Here our object is to find a generalized function u e MZ^, XO satisfying the operational formula (6.3). Applying generalized Marchi-Zgrablich transform to the equation (6.4) and using (6.3) we get = G{ n ), (6.5)

11 MARCHI-ZGRABLICH TRANSFORMATION 223 where U and G are the generalized Marchi-Zgrablich transform of u and g respectively. If P( /^) ± 0 for every x we can divide by P( 1 ) to obtain (6.5) and applying the inversion theorem (5.1) to (6.5) we get Therefore u(x) = lim Gvv ""'7' r "" / (6.6) is a solution of (6.4) where the equality is considered in the sense of D'(f). This solution is unique in view of the uniqueness Theorem 5.2. The first author is grateful to CSIR for providing financial assistance. The authors also wish to thank the referee for the valuable suggestions for the improvement of the paper. REFERENCES BHOSALE, S. D. & MORE, S. V On Marchi-Zgrablich transformation of generalized function. IMA J. appl. Math. 33, MARCHI, E. & ZGRABUCH, G Heat conduction in hollow cylinders with radiation. Proc. Edinb. Math. Soc. 14(2), PANDEY, J. N. & ZEMANIAN, A. H Complex inversion for the generalized convolution transformation. Pacif. J. Math. IS (1), SCHWARTZ, L Theoriedes Distributions, Vol. II. Paris: Hermann. WATSON, G. N Theory of Bessel Functions. Cambridge: CUP. WANKHEDE, P. C A study of some aspects of integral transforms with applications to problems of physics and engineering. Ph.D. Thesis. Jabalpur University (M.P.), India. ZEMANIAN, A. H Generalized Integral Transformations. Interscience.