the multiplicative products between the distribution (P ± i0) λ and the operators L r {δ} and K r {δ}

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1 Revista de Matemática: Teoría y Aplicaciones 31: the multiplicative products between the distribution P ± i0 λ and the operators L r {δ} and K r {δ} Manuel A. Aguirre 1 Abstract In this note we give a sense to some multiplicative products of distributions: i P ± i0 λ. L r {δ} ii P ± i0 λ. K r {δ} where P ±i0 λ is the distribution defined by the formula, P is the quadratic form defined by the formula 1, L r is the ultrahyperbolic operator defined by 5 and K r is the Klein-Gordon operator iterated r-times defined by the formula 15. Resumen En esta nota damos sentido a algunos productos multiplicativos de distribuciones: i P ± i0 λ. L r {δ} ii P ± i0 λ. K r {δ} donde P ±i0 λ es la distribución definida en la fórmula, P es la forma cuadrática definida en la fórmula 1, L r es el operador ultrahiperbólico definido por 5 y K r es el operador de Klein-Gordon iterado r veces y definido por la fórmula Introduction Let x = x 1,...,x n be a point of the n-dimensional Euclidean space R n. Consider a non-degenerate quadratic form in n variables of the form, P = Px = x x µ x µ+1 x µ+ 1 where n = µ +. The distributions P ± i0 λ are defined by P ± i0 λ = lim ε 0 P ± iε x λ [3], page 75 1 Facultad de Ciencias Exactas, Universidad del Centro de la Provincia de Buenos Aires, Pinto 399, 3er. piso, 7000 Tandil, Argentina 1

K r {δ} where P ±i0 λ is the distribution defined by the formula, P is the quadratic form defined by the formula 1, L r is the ultrahyperbolic operator defined by 5 and K r is the Klein-Gordon

2 m. a. aguirre where ε > 0, x = x x n, 3 and λ C. A. González Domínguez proves the validity of the following formula cf.[4], 1980, page 19, formula 9.: P ± i0 k.l r {δ} = ak,n,rl k+r {δ}, 4 k = 0,1,,..., r = 0,1,,..., and k n/. Here L r is the ultrahyperbolic operator iterated r-times: { } r L r = X1 + + Xµ+1 Xµ+1 Xµ+µ 5 n n and ak,n,r = 4 k r r + k + r... + r + k 1 6 The following formulae were established [1], 1984, P ± i0 k.l{δ} = 0 if n k for n even ; 7 P ± i0 k.l r {δ} = ak,n,rl r+k {δ} if n k for n odd; 8 P ± i0 k.l r {δ} = bk,n,rl r k {δ} if r k, k = 0,1,... ; r = 0,1, k r! n where, bk,n,r = + r r k! n ; 10 + r k ak,n,r is defined by the formula 6 and α = Also, we know, cf. [], 1988 that 0 e x x α 1 dxi,1,11 11 P ± i0 n +k 1.L k {δ} = 0 1 [], page 5, formula 17 for n with r = 0,1,,.... In this paper, we intend to give a sense to certain multiplicative products: P ± i0 λ.l r {δ} 13 and where λ is a complex number, P ± i0 λ.k r {δ} 14 K r {δ} = L m r {δ} 15 Klein-Gordon operator iterated r-times and L k is defined by the formula 5.

.. + r + k 1 6 The following formulae were established [1], 1984, P ± i0 k.l{δ} = 0 if n k for n even ; 7 P ± i0 k.l r {δ} = ak,n,rl r+k {δ} if n k for n odd; 8 P ± i0 k.

3 the multiplicative products between the distribution... 3 The multiplicative product P ±i0 λ. L r {δ} and P ±i0 λ. K r {δ} theorem 1. Let λ be a complex number such that λ ±k, k = 0,1,,...; then the following formula is valid: P ± i0 λ.l r {δ} = 0 16 where the distribution P ± i0 λ is defined by the formula and L k by the formula 5. Proof: From [1], page 7, formula 17, we have, where Cα,β,n = H α.h β = C α,β,n H α+β n 17 H γ = H γ P ± i0,n = γ n P ± i0 D n γ D n γ = eγπi/ e ±qπi/ n γ γ π n/ γ/ α+β n n α n Π n/ α β From 17, 0 and taking into account the formula we have P ± i0 α n.h β = z1 z = n β By making β = r and λ = α n in we have, P ± i0 λ.h r = π 1 n β π sin zπ πi n±q e n α+n β 1 β π 1 sin β π e ±qπi/ e βπi/ P ± i0 α+β n n n + k 1 + ke ±qπi/. On the other hand, from [5], page 40, formula II,3,7, we have. sin kπ.p ± i0 λ n k 3 L r {δ} = H r, r = 0,1,,... 4 The distribution P ± i0 γ [3], page 84 have poles at the points λ = n k, k = 0,1,,...

Proof: From [1], page 7, formula 17, we have, where Cα,β,n = H α.

4 4 m. a. aguirre Therefore, substituting 4 into 3 for we have λ n k n, n 1,... P ± i0 λ.l r {δ} = 0 5 where λ ±k, k = 0,1,,... From 5 we conclude the proof of theorem 1. In particular from 16, when λ = n + r 1 and n odd, we obtain the product 1. theorem. Let the distribution P ± i0 λ and K r {δ} be the Klein-Gordon operator iterated r times defined by the formula 15; then the following formulae are valid: P ± i0 λ.k r {δ} = 0 when λ ±k, k = 0,1,,... 6 P ± i0 k.k r {δ} = 0 if k n for n even. 7 P ± i0 k.k r {δ} = =0 if k n m r ak,n,l +k {δ} 8 P ± i0 k.k r {δ} = =0 if k n m r L +k {δ} for n odd. 9 P ± i0 k.k r {δ} = =0 m r bk,n,l k {δ} if k 30 and P ± i0 n +k 1.K r {δ} = 0 if n with r = 0,1,, where ak,n, is defined by the formula 6 and bk,n, by the formula 10. Proof: From 15 we have K r {δ} = =0 where L is defined by the formula 15. m r L {δ} 3

Let the distribution P ± i0 λ and K r {δ} be the Klein-Gordon operator iterated r times defined by the formula 15; then the following formulae are valid: P ± i0 λ.k r {δ} = 0 when λ ±k, k = 0,1,,.

5 the multiplicative products between the distribution... 5 From 3 we have, P ± i0 λ.k r {δ} = =0 m r P ± i0 λ.l {δ} 33 From 33 and taking into account 16 we obtain the result 6. From 33 and taking into account the formulae 7, 8 and 9, we obtain the results 7, 9 and 30. Finally from 33 and taking into account the formulae 1 and 16 we obtain the results 31 and 6. References [1] Aguirre, M.A. Productos multiplicativos y de convolución de distribuciones. Tesis doctoral. Facultad de Ciencias Exactas y Naturales de la Universidad de Buenos Aires. [] Aguirre, M.A. Los productos multiplicativos P ±i0 n +k 1..L k {δ} y P ±i0.δ x, Preprint No. 13, Instituto Argentino de Matemática, Buenos Aires, Argentina. [3] Gelfand, I.M.; Shilov, G.E Generalized Functions, Vol. I. Academic Press, New York. [4] González Domínguez, A On Some Heterodox Distributional Multiplicative Products, Revista de la Unión Matemática Argentina, Vol. 9. [5] Trione, S.E Distributional Products, Cursos de Matemática, No. 3, Serie II, IAM-CONICET.

References [1] Aguirre, M.A. Productos multiplicativos y de convolución de distribuciones. Tesis doctoral. Facultad de Ciencias Exactas y Naturales de la Universidad de Buenos Aires. [] Aguirre, M.A. Los productos multiplicativos P ±i0 n +k 1.

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