TAYLOR-TYPE EXPANSION OF THE k-th DERIVATIVE OF THE DIRAC DELTA IN u(x 1,...x n ) t.

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1 Novi Sad J Math Vol 32, No, 2002, TAYLOR-TYPE EXPANSION OF THE k-th DERIVATIVE OF THE DIRAC DELTA IN u(x, x n ) t Manuel A Aguirre Abstract We obtain an expansion of Taylor style of the distribution δ (k) (u(x, x n ) t) u(x, x n ) C (R n ) without critical points and t is a real number In particular, we obtain the expansion of the distribution δ (k) (P + m 2 )(see ([3), ([4) and ([5)), m is a positive real number and P = P (x) = x 2 + x x 2 p x 2 p+ x 2 p+q, p + q = n dimension of the space AMS Mathematics Subject Classification (99): 46F0, 46F2 Key words and phrases: Theory of distributions Introduction Let φ t denote a distribution of one variable t Let u C (R n ) be such that (n-)-dimensional manifold u(x, x n ) = 0 has no critical point By φ u(x) Leray(cf [2, p 02) designates the distribution defined on R n by () φu(x), ϕ(x) = φ t, ψ(t) ([2, page 02) (2) ψ(t) = u(x)=t ϕ(x)w u (x,dx) and ϕ C o (R n ) is the set of infinitely differentiable functions with compact support and w u is a (n-)-dimensional exterior differential form on u defined as (3) du w u = dx dx 2 dx n By assumption, in the neighborhood of any point of the surface we can introduce a local coordinate system u, u 2, u n such that one of the coordinates, say u j is u(x, x n ) and such that the transformation from x i to u i, i =, 2, is given by infinitely differentiable function with the positive Jacobian D ( x u) ([,p 220) If, in particular, in the neighborhood of the given point (4) u x > 0, Núcleo Consolidado de Matemática Pura y Aplicada, NUCOMPA, Facultad de Ciencias Exactas, UNCentro, Pinto 399, CP 7000,ARGENTINA, maguire@exaeduar

2 86 M A Aguirre we may take the u coordinate to be (5) u = u(x, x 2, x n ) t u 2 = x 2 u n = x n then (6) (7) Therefore, from(6) and(7) we have ( ) x w u = D du 2 du n u ( ) [ ( ) x u D = D = u x u x (8) w u = w u (u, du) = du 2du n u x From (5) and taking into account(4), it follws that there exists such a function α = α(u 2, u n ) C (R n ) such that (9) x = α(u 2, u n ) Therefore, from (2) and considering the formulae (5),(6) and(7) we have, (0) ψ(t) = ϕ (u, u 2, u n )w u (u, du) () u =o ϕ (u, u 2, u n ) = ϕ(x, x 2, x n ) and w u (u,du) is defined by(8) On the other hand from [, p 230, formula 6,we have, (2) δ (k) (G(x, x 2, x n ), ϕ(x, x 2, x n ) = ( ) k w k (ϕ) G(x)=0 k = 0,, 2,, x = (x, x 2, x n ), G(x, x 2, x n ) is such an infinite differentiable function that ( G grad G =, G, G ) (3) 0, x x 2 x n ( ) w k (ϕ) = k x (4) u k D ϕ (u, u 2, u n ) du 2 du n, u

3 Taylor-type expansion of the k th derivative of 87 (5) (6) w o = ϕw, u = G(x, x 2, x n ) u 2 = x 2 u n = x n, ϕ is defined by equation(), w = w u is the differential form defined by (6) and (7) with (8) ( ) [ ( ) x u D = D = u x G x > 0 Otherwise, from [, p 2, formula 8, δ (k) (G(x, x 2, x n ) can be written as, δ (k) (G(x), ϕ = ( ) k G=0 f u (k) (0, u 2, u n )du 2 du n = (9) ( ) [ k k f (k) G=0 u k u (0, u 2, u n ) du 2 du n u =0 (20) G x ( ) x f(u, u 2, u n ) = ϕ (u, u 2, u n )D u ϕ is defined by equation () and D ( x u) by (7) From(2) and(4), taking into account(5), we have (2) ψ(0) = ϕw = δ(u(x), ϕ(x) u(x)=0 In this paper we obtain an expansion of Taylor style of the distribution δ (k) (u(x, x n ) t) u(x, x n ) C (R n ) whithout critical point, t is a real number and δ (k) (u(x, x n ) t) is defined by (9) 2 The expansion of δ (k) (u(x, x n ) t) We begin by showing a lemma of expansion of ψ(t) defined by (2) Lemma Let ψ(t) be the function defined by (2) Then the following expansion for ψ(t) in powers of t is valid: (22) ψ(t) = a (ϕ)t =o

4 88 M A Aguirre for every ϕ C o (R n ),, (23) (24) w (ϕ) is defined by (4) a (ϕ) = G=0 w (ϕ), G = G(x, x n ) = u(x, x n ), Proof From(0) and (4) we have (25) ψ(t) = ϕ (u(x) t, u 2, u n )w u (u, du) (26) u =o u = u(x) t On the other hand, ϕ (u(x) t, u 2, u n ) has the following Taylor series expansion in the neighborhood of t = 0 ϕ (u(x) t, u 2, u n ) = ( ) [ d (27) dt ϕ (u(x) t, u 2, u n ) t t=o Considering that ϕ Co (R n ) and the convergence uniform of the series (27), from (25) and (27) we have ψ(t) = ( ) [ d (28) u =o dt ϕ (u(x) t, u 2, u n ) w u (u, du) t t=o On the other hand, taking into account (26), we obtain [ [ d (29) dt ϕ (u(x) t, u 2, u n ) = t=o u ϕ (u, u 2, u n ) ( ), u =u(x) therefore, from (29) we have [ d u =o dt ϕ (u(x) t, u 2, u n ) t=o w u(u, du) = (30) [ ( ) u ϕ (u, u 2, u n ) w u (u, du) u =o u =u(x) Considering that the form w u does not depend on the choice of u, u 2, u n coordinate system (see[, p 222), then using (26), if t = o is u = u(x, x 2, x n ), thus, from (30) we obtain [ u =o u ϕ (u, u 2, u n ) w u (u, du) = u =u(x) (3) [ u ϕ (u, u 2, u n ) w u (u, du) u(x)=o u =o

5 Taylor-type expansion of the k th derivative of 89 Substituting (30) in(28) and taking into account(3) we arrive at ψ(t) = ( ) [ (32) u(x)=o u ϕ (u, u 2, u n ) w u (u, du) t u =o On the other hand, considering the invariance from w(u, du) on the hypersuface S given by the equation G(x, x n ) = 0 ([, p 222),the following property is valid: [ u ϕ (u, u 2, u n ) w u (u, du) = u =u(x)=o (33) [ u ϕ (u, u 2, u n )w u (u, du) In fact, from (8) we have u =u(x)=o (34) u w u (u, du) = ( ) ( u u ( ) u u x ) x du 2 du n = ( ) 2 u x du2 du n On the other hand, considering (9) we obtain (35) α From (9), (36) ( ) u u x = u ( ( u ) ( α α u + u u 2 α α u(α, u 2, u n ) ) = ) ( u 2 u + + u u n α α u = 0, thus, from (35) and (36) we have ( ) (37) u(α, u 2, u n ) = 0 u x From (34) and considering (37) we arrive at ) u n u = 2 u α 2 α u (38) u w u (u, du) = 0 Therefore, from(38) we conclude that the property (33) is valid Now, using the property (33), we have, [ u ϕ (u, u 2, u n ) w u (u, du) = u =u(x)=o (39) [ u ϕ (u, u 2, u n )w u (u, du) u =u(x)=o

6 90 M A Aguirre = 0,, 2, Substituting (39) in (32) we obtain ψ(t) = (40) u(x)=o u [ϕ (u, u 2, u n )w u (u, du) t u =o From (40) and considering (8), (),(9) and (20), we have ψ(t) = ( ) (4) δ () (u(x)), ϕ(x) t Otherwise, from (4) and (2) we have ψ(t) = ( ) (42) w (ϕ) t u(x)=o w (ϕ) is defined by(4) Finally, from (42) we conclude the Lemma, formula (22) (43) We observe from (4) that we have obtained a series expansion of δ(u(x) t) In fact, from (2) and (2) we obtain ψ(t) = u(x)=t ϕw = u(x) t=o ϕ(x)w(xdx) = δ(u(x) t), ϕ(x) and from(4) we have ( ) (44) ψ(t) = δ () (u(x))t, ϕ(x) Therefore, from (43) and (44) we obtain the following formula: (45) δ(u(x) t) = ( ) δ () (u(x))t On the other hand, a series expansion of δ (k) (u(x) t) can be considered as a generalization from the formula (45) which we will study in the following theorem: Theorem 2 Let u(x, x n ) C (R n ) be such that (n-)-dimensional manifold u(x, x n ) t = 0 has no critical point, then the following formula is valid, δ (k) (u(x) t) = ( ) q (46) δ (k+q) (u(x))t q q! q o t is a real number and δ () (G(x, x n )) is defined by (2) or(9)

7 Taylor-type expansion of the k th derivative of 9 Proof From (40) we have, (47) ψ(t) = L t, (48) L = u(x)=o u ϕ (u, u 2, u n )w u (u, du) Considering that the series (27) exibits uniform convergence, from (28) and (48) we have, (49) d k ψ(t) dt k = k L t k ( k)! = q o L q+k tq [ q o u(x)=o [ q o u(x)=o q! = q+k ϕ u q+k (u, u 2, u n )w u (u, du) q+k u q+k ϕ (u, u 2, u n )D ( x u t q q! = ) du 2 du n t q q! From (49) and taking into account (4) and (5) we obtain (50) d k ψ(t) dt k = q o ( )q+k δ (k+q) (u(x)), ϕ t q q! Now, from (2) and (4) we have (5), (52) ( ) k δ (k) (u(x) t), ϕ = ( ) q o u(x)=o w t q+k(ϕ) q q! = u =o w k (ϕ), ( w k (ϕ) = k x u k ϕ (u, u 2, u n )D du 2 du n u) From (5), (52) and considering (29) we obtain ( ) k δ (k) (u(x) t), ϕ = (53) u =o u =o k ϕ (u u k, u 2, u n )D ( x u) du2 du n = k u k ϕ (u(x) t), u 2, u n )w u (u, du) = dk ( )k u ϕ =o dt k (u(x) t), u 2, u n )w u (u, du) = d k dt k u =o ( )k ϕ (u(x) t), u 2, u n )w u (u, du) d k dt k u =t ϕ(x)w(x, dx) = dk ψ(t) dt k =

8 92 M A Aguirre By (50) and (53) we conclude the proof of the theorem In particular, putting (54) u(x) t = P (x) + m 2 in (46), m is a positive real number, (55) P = P (x) = x 2 + x x 2 p x 2 p+ x 2 p+q, p + q = n (dimension of the space), we obtain the following (56) δ (k) (m 2 + P ) = (m 2 ) q δ (k+q) (P ) q! q o The formula (56) appears in [3 (formula (77)) under conditions n being odd, in [4, (formulae (38) and (39)) for two cases: a) p and q being even and b) p and qbeing odd Finally, the formula (56) appears in [5 independently of p, q and n, p + q = n is the dimension of the space References [ Gelfand, I M, Shilov, G E, Generalized Functions, Vol I, Academic Press, New York 964 [2 Leray, J, Hyperbolic Differential Equations, The Institute for Advanced Study, Princeton, New jersey, 957 [3 Manuel Aguirre T, The expansion and Fourier s transform of δ (k ) (m 2 + P ), Integral Transforma and Special Functions, Vol 3, No 2, pp 3-34, 995 [4 Manuel Aguirre T, The expansion of δ (k ) (m 2 + P ), Integral Transform and Special Functions, Vol 8, No -2, pp 39-48, 999 [5 Manuel Aguirre T, The expansion in series (of Taylor types) of (k ) derivative of Dirac s delta in m 2 + P, to appear Received by the editors September 20, 2000