C.K. Li a a Department of Mathematics and

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1 This article was downloaded by: [Brandon University GST] On: 31 August 2013, At: 07:46 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Integral Transforms and Special Functions Publication details, including instructions for authors and subscription information: A note on the product C.K. Li a a Department of Mathematics and Computer Science, Augustana University College, Avenue, Camrose, Alberta, T4V 2R3, Canada Published online: 03 Apr To cite this article: C.K. Li (2001) A note on the product, Integral Transforms and Special Functions, 12:4, , DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever

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3 Integral Transfoms and Special PA (Overseas Publishers Association) N.V. 2001, V01.12, Nn 4, pp Published by license under Reprints available directly from the publisher the Gordon and Breach Science Publishers imprint, Photocopying permitted by license only a member of the Taylor & Francis Group A NOTE ON THE PRODUCT* r-" V (Ar2 ' ") C.K. LI Department of Mathematics and Computer Science, Augustana University College, Avenue, Camrose, Alberta Canada T4V 2R3 (Received July 12, 2000) The difficulties inherent in defining products, powers or nonlinear operations of generalized functions have not prevented their appearance in the literature (see e.g. [I, 2, 31). In 141, a definition for a product of distributions is given using a &sequence. However, they show that 6a does not exist, Extending definitions of products from one-dimensional space to m-dimensional by using appropriate delta-sequences has recently been an interesting topic in distribution theory. From a single variable function p(s) (see 151) deflned on R+, Li and Fisher introduce a 'workable' delta sequence in Rm to deduce a non-commutative neutrix product of rdk and A6 (A denotes the Laplacian) for any positive integer k between 1 and m - 1 inclusive. In (61, Li provides a modified &sequence and defines a 'bridge' distribution U, - ' d(x), which can be used to obtain the more general product of rwb and ~ '6. The object of this paper is to utili~e the fact that E = r2-"' is an elementary solution of Laplacian equation Au(x) = 6(x) and to apply a much simpler version of the main result in [7] to derive the product r-b. V(AT'-~) where na is any dimension greater than 2. Finally, we 1 also compute the products r-b.a In r and r-'.v(a In T) based on the equation z;; A In r = 6 for m = 2. KEY WORDS: Pisetti's formula, delta sequence, neutrix limit, neutrix product and distribution MSC (2000): 46F10 1. THE DISTRIBUTION rx Let r = (2: s;)'/~ and consider the functional rx (see [8]) defined by *This research ia supported by a UCCB research grant. 341

4 342 C.K. LI where Re X > -m and $(x) E Dm, the space of infinitely differentiable functions with compact support. Because the derivative of the variable x = (XI, x2,..., x,) exists, the functional rx is an analytic function of X for Re X > -m. For Re A 5 -m, we should use the following identity (2) to define its analytic continuation. For Re X > 0, we could deduce simply by calculating the left-hand side. By iteration we find for any integer k that ~k~x+2k rx = (X+2).--(X+2k)(X+m)...(X+m+2k-2) (2) On making substitution of spherical coordinates in (I), we come to where dw is the hypersurface element on the unit sphere. The integral appearing in the above integrand can be written in the form where R, = is the hypersurface area of the unit sphere imbedded in Euclidean space of rn dimensions, and S4 is the mean value of 4 on the sphere of radius r. It was proved in [8] that S+(r) is infinitely differentiable for r 1 0 and has bounded support, and that for any positive integer k (and ak to be determined). From (3) and (4), we obtain [r" 4) = 0, Jr'+m-1S 4 ( 4 dr 0 which indicates the application of 0,xf; with = X + m - 1 to the testing function S+(r). Using the following Laurent series for xi about X = -k we could show that the residue of (rx,t$(x)) at X = -m - 2k for nonnegative integer k is given by

5 A NOTE ON THE PRODUCT r-b. V(A~'-~) On the other hand, the residue of the function ra of (2) for the same value of A is R~A~~(x) Therefore we get 2kk!m(m+2)..-(m+2k-2) This result can be used to write out the Taylor's series for S,+(r), namely which is the well-known Pizetti's formula. 2. THE PRODUCT r-'. V6 Let p(x) be a fixed infinitely differentiable function defined on R with the following properties: 0) P(X) l 0, (ii) p(x) = 0 for 1x1 > 1, (iii) p(x) = p(-x), 1 (iv) lp(r)dx = The function dn(x) is defined by 6,(x) = np(nx) for n = 1,2,.... It follows that {6,(s)) is a regular sequence of infinitely differentiable functions converging to the Dirac delta-function d(x). Now let 2) be the space of infinitely differentiable functions of a single variable with compact support and let 2)' be the space of distributions defined on 2). Then if f is an arbitrary distribution in D', we define fn(x) = (f * fin)($) = (f(t),6n(~ - t)) for n = 1,2,.... It follows that {fn(x)) is a regular sequence of infinitely differentiable functions converging to the distribution f (a) in 2)'. The following definition for the non-commutative neutrix product f - g of two distributions f and g in V' was given by Fisher in [9]. Definition 1. Let f and g be distributions in 2)' and Let gn = g * 6,. We say that the neutrix product f g of f and g exists and is equal to h if

6 344 C.K. LI for all functions 4 in D, where N is the neutrix (see [lo]) having domain N' = {1,2,...) and range N" the real numbers, with negligible functions that are finite linear sums of the functions nx Inr-I n, Inr n (A > 0, r = 1,2,...) and all functions of n which converge to zero in the normal sense as n tends to infinity. The product of Definition 1 is not symmetric and hence f.g # g. f in general. In order to give a definition for a neutrix product f. g of two distributions in DL, the space of distributions defined Dm. We may attempt to define a &sequence in Dm by simply putting (see [12]) where 6, is defined as above. However, this definition is very difficult to use for distributions in DL which are functions of r. We therefore consider the following approach (see [7]). Let p(s) be a fixed infinitely differentiable function defined on R+ = [0, oo) having the properties: 0) p(s) 2 0; (ii) p(s) = 0 for s 2 1; (iii) / 6. (3) dx = 1, R"' where d,(x) = cmnmp(n2r2) and c, is the constant satisfying (iii). It follows that {6,(x)) is a regular &sequence of infinitely differentiable functions converging to 6(x) in DL. Definition 2. Let f and g be distributions in DL and let gn(~)=(g*bn)(x)=(g(~-t),6n(t)), where t=(tl,tz,..., t,). We say that the neutrix product f. g of f and g exists and is equal to h if where 4 E Dm and the N-limit is defined as above. With Definition 2 and the normalization procedure of p(x) x$, Li (also in [7]) shows that the non-commutative neutrix product r-' - Vb, (V is the gradient m operator, that is, V = 5) irl exists and and where k is a positive integer.,,1-2' V6 = 0,

7 3. THE DISTRIBUTION A NOTE ON THE PRODUCT T -'. V(AT'-~) 345 We note that r2-" for m 2 3 is a harmonic function in any region which does not contain the origin, i.e., that Ar2-m vanishes in the ordinary sense for all r # 0. For the case of generalized functions we have Now applying Green's theorem to this integral, we have where ds is the element of area on the sphere r = c. Now 4 $ r2-" ds, since outside the ball r < e the function r2-" is harmonic. As for the second term, by noting that 84/ar is bounded near the origin. On the other hand, / 4 $ r2-" ds = (2- m) el-' J Qds=(2 -m)nmse(4), r=e r=e where Se(4) is the mean value of 4(x) on the sphere of radius e. In the limit as e + o+, of course, Se(4) -+ 4(O), so that Hence we may write (AT'-", 4) = lim / 4 dt~ c+o+ rlc (2 - m) am and it immediately follows that rm-2 = (2 - m) Qm4(0). = a(z) is an elementary solution of Laplacian equation Au(x) = 6(x). A similar calculation for dimension m = 2 leads to the result

8 346 C.K. LI 4. SOME PRODUCTS In this section, we will provide a nicer version of the main theorem in [7] as well as the results obtained in [6] to provide the products outlined in the abstract. It obviously follows from equations (5) and (7) that Theorem 1. The non-commutative neutrix product r-k - V(A~'-~) exists for m 13. Furthermore (m- 2) 7rm/',-'k 9 2(xiAk+'6) (9) v(ar2-m) = 2k(k + l)! (m + 2). - (m + 2k) r(mj2),=, and r1-2k. v(arz-") = 0, (10) where k is a positive integer. Remark. The product of xi and ~ " in ~ our 6 theorem is well defined since (ziakt16, 4) = (6, Akt1(~i4)) (11) The following lemma will play an important role in simplifying the above theorem. Lemma 1. Proof. We use an inductive method to show the lemma. It is obviously true for k = 0. When k = 1, we have A2(xi4) = 4 & A$+X~A'# simply by calculating the left-hand side. Then, Assume, equation (12) holds for the case of k - 1, that is Hence it follows that This completes the proof of Lemma 1.

9 Lemma 2. A NOTE ON THE PRODUCT r-b. V(AT'-~) 347 m ~(siak++'6) = -2(k + 1) v(ak6), where k 2 0. (13) i=l Proof. Applying equation (12), we have Therefore, we have reached our conclusion in Lemma 2. Now, using equations (9) and (13), we can prove the following theorem. Theorem 2. The non-commutative neutrix product r-k V(Ar2-m) exists for and where k is a positive integer. In particular, we obtain (x: + x: +.:)-I. v (A (x: + x: + x:)-l") = - V(A6) 2 (XI + XI + X: + xi)-'. v (A(x: + C: + Z: +.:)-I) = -t 96 v(a2a) by noting that r(3/2) = 6 /2. The following result can be found in [6]. Theorem 3. The non-commutative neutrix product T -.6 ~ exists. Furthermore where k is a positive integer. Theorem 4. The non-commutative neutrix products r-k. V (A In r) exist for m = 2, and T - ~. A lnr and where k is a positive integer.

10 348 C.K. LI Proof. It immediately follows from equations (5), (B), (13) and (16). In particular for k = 1, we get ACKNOWLEDGEMENTS I am deeply indebted to Professor Doug Grant who made several productive suggestions, which improved the quality of this paper. REFERENCES 1. S. Gaeiorowicz, Elementary Particle Physics, J. Wiley and Sons, Inc., N.Y., C.Y. Chen, A survey of the budding nonlinear theory of distribution, J. of Math. 1 (1981), (in Chinese). 3. J.H. Bremermann, Diatributiona, Comples Variables, and Fourier Tkansforma, Addison-Wesley, Reading, Maesachusetts, P. Antosik, J. Mikusinski and R. Sikorski, Theory of Distributions, the Sequential Approach, PWN-Polish Scientific Publishers, Warsawa, C.K. Li and B. Fisher, Examples of the neutrix product of diatributions on Rm, Rad. Mat., 0 (1990), C.K. Li, A new product of distributions in m variables, to appear in the Proc, of the "International Conference on Mathematics and its Applications in the New Millemmium", July 2000, organized by Dept. of Mathem., University of Putra Malaysia. 7. C.K. Li, The product of r-' and V6 on Rm, Int. J. of Mathematics and Math. Sci., 24 (ZOOO), Nh 6, I.M. Gel'fand and G.E. Shilov, Generalized Functions, Vol. I, Academic Press, B. Fisher, On defining the convolution of distributions, Math. Nachr., 100(1982), J.G. van der Corput, Introduction to the neutrix calculus, J. Analyse Math., 7 ( ), L.Z. Cheng and C.K. Li, A commutative neutrix product of diatributiona on Rm, Math. Nachr., 111 (l991), B. Fisher and C.K. Li, On defining a non-commutative product of distributions in m variables, J. of Natural Sciences and Mathematics, 2 (l991), C.K. Li and E.L. Koh, The neutrix convolution product in Z1(m) and the exchange formula, IJMMS, 21 (1998), E.L. Koh and C.K. Li, On the diatributions 6' and (6')', Math. Nachr., 117(1992), B. Fisher, A. Kilicman and C.K. Li, An extension of a result on the non-commutative neutrix convolution product of distributions, Int. J. of Appl. Mathem., 8 (2000), Nh 1,