On the solutions of the causal and anticausal n-dimensional diamond operator
In this paper, we consider the solution of the equation ◇k(p±i0)=∑mi=0Cr ◇rδ, where ◇k is introduced and named as the Diamond operator iterated k-times and is defined by ◇=[(∂2/∂x21+... +∂2/∂x2p)2-(∂2/∂x2p+1+... +∂2/∂x2p+q)2]k Let x = (x1, x2, ..., xn) be a point of the n-dimensional Euclidean space...
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| Formato: | Capítulo de libro |
| Lenguaje: | Inglés |
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2002
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| Acceso en línea: | Registro en Scopus DOI Handle Registro en la Biblioteca Digital |
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| LEADER | 04131caa a22004097a 4500 | ||
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| 001 | PAPER-5445 | ||
| 003 | AR-BaUEN | ||
| 005 | 20230518203505.0 | ||
| 008 | 190411s2002 xx ||||fo|||| 00| 0 eng|d | ||
| 024 | 7 | |2 scopus |a 2-s2.0-0036486853 | |
| 040 | |a Scopus |b spa |c AR-BaUEN |d AR-BaUEN | ||
| 100 | 1 | |a Trione, S.E. | |
| 245 | 1 | 3 | |a On the solutions of the causal and anticausal n-dimensional diamond operator |
| 260 | |c 2002 | ||
| 270 | 1 | 0 | |m Trione, S.E.; Departamento de Matemática, Fac. de Ciencias Exactas y Naturales, Instituto Argentino de Matematica, Saavedra 15-3er. Piso-(C1083 ACA), Buenos Aires, Argentina; email: trione@iamba.edu.ar |
| 506 | |2 openaire |e Política editorial | ||
| 504 | |a Gelfand, I.M., Shilov, G.E., (1964) Generalized Functions, 1. , Academic Press, New York | ||
| 504 | |a Kananthai, A., On the solutions of the n-dimensional diamond operator (1997) Applied Mathematics and Computation, 88, pp. 27-37 | ||
| 504 | |a Trione, S.E., On the elementary (P ± i0)λ-ultrahyperbolic solution of the Klein-Gordon operator iterated k-times (2000) Integral Transforms and Special Functions, 9 (2), pp. 149-162 | ||
| 504 | |a Nozaki, Y., On the Riemann-Liouville integral of ultra-hyperbolic type (1964) Kodai Mathematical Seminar Reports, 6 (2), pp. 69-87 | ||
| 504 | |a Riesz, M., L'intégrale de Riemann-Liouville et le probléme de Cauchy (1949) Acta. Mathematica, 81, pp. 1-223 | ||
| 504 | |a Trione, S.E., Distributional products (1980) Cursos de Matemática, 3. , IAM-CONICET, Buenos Aires | ||
| 504 | |a Donoghue, W.F., (1969) Distributions and Fourier Transforms, , Academic Press | ||
| 520 | 3 | |a In this paper, we consider the solution of the equation ◇k(p±i0)=∑mi=0Cr ◇rδ, where ◇k is introduced and named as the Diamond operator iterated k-times and is defined by ◇=[(∂2/∂x21+... +∂2/∂x2p)2-(∂2/∂x2p+1+... +∂2/∂x2p+q)2]k Let x = (x1, x2, ..., xn) be a point of the n-dimensional Euclidean space. Consider a non-degenerate quadratic form in n variables of the form P = P(x) = x12+...+xp2- xp+12 - ... - xp+q2, where p + q = n, Cr is a constant, δ is the delta distribution ◇0δ = δ and k = 0, 1, .... The distributions (P ± i0)λ are defined by (P±i0)λ = limε→0{P±iε|x|2}λ where ε > 0, |x|2 = x12 + ... + xn2, λ εC. The distributions (P ± i0)λ are an important contribution of Gelfand (cf. [1], p. 274). The distributions (P ± i0)λ are analytic in λ everywhere except at λ = -n/2 - k, k = 0, 1, ..., where they have simple poles (cf. [1], p. 275). By causal (anticausal) distributions, we mean distributions where P = P(x) = x12 + ... + xn-12 - xn2. The causal distributions are particularly important when n = 4 because they appear frequently in the quantum theory of field. In this note we obtain the solutions of the causal and anticausal n-dimensional Diamond operator by following, line by line, the paper entitled "On the solutions of the n-dimensional Diamond operator" by Amnuay Kananthai (cf. [2]). |l eng | |
| 593 | |a Departamento de Matemática, Fac. de Ciencias Exactas y Naturales, Instituto Argentino de Matematica, Saavedra 15-3er. Piso-(C1083 ACA), Buenos Aires, Argentina | ||
| 690 | 1 | 0 | |a CAUSAL (ANTICAUSAL) SOLUTIONS |
| 690 | 1 | 0 | |a DIAMOND OPERATOR |
| 690 | 1 | 0 | |a HOMOGENEOUS |
| 690 | 1 | 0 | |a TEMPERED DISTRIBUTIONS |
| 773 | 0 | |d 2002 |g v. 13 |h pp. 49-57 |k n. 1 |p Integr. Transforms Spec. Funct. |x 10652469 |w (AR-BaUEN)CENRE-5176 |t Integral Transforms and Special Functions | |
| 856 | 4 | 1 | |u https://www.scopus.com/inward/record.uri?eid=2-s2.0-0036486853&doi=10.1080%2f10652460212894&partnerID=40&md5=e3c1885e834cc938cb1dbdedd941979e |y Registro en Scopus |
| 856 | 4 | 0 | |u https://doi.org/10.1080/10652460212894 |y DOI |
| 856 | 4 | 0 | |u https://hdl.handle.net/20.500.12110/paper_10652469_v13_n1_p49_Trione |y Handle |
| 856 | 4 | 0 | |u https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10652469_v13_n1_p49_Trione |y Registro en la Biblioteca Digital |
| 961 | |a paper_10652469_v13_n1_p49_Trione |b paper |c PE | ||
| 962 | |a info:eu-repo/semantics/article |a info:ar-repo/semantics/artículo |b info:eu-repo/semantics/publishedVersion | ||
| 999 | |c 66398 | ||