Artículo
Abstract:
In this note, we study the causal (anticausal) generalized Riesz potential of order α: Rα Cf (Rα Af) of the function f ∈ S (cf. (1.8) and (1.9), respectively). The distributional functions Rα Cf (Rα Af) are causal (anticausal) analogues of the α-dimensional potentials in the ultrahyperbolic space defined by Nozaki (cf. [1, p. 85]). Therefore, we define the generalized causal (anticausal) Riesz derivative of order α of a function α by the formula (Dα Cf)(x) = (1/dn,ℓ(α))(Tα ℓf)C(x), α ε ℂ, ℓ is a nonnegative integer, ℓ > α > 0 and α ≠ 1, 2, 3, . . . , where dn,ℓ(α) and (Tα ℓf)C(x) are given by (2.4) and (2.1), respectively. Theorem 2 expresses that Dα CRβ A = MU-α+β C + NU-α+β A, where Uα C,A = Φα C,A*f, Φα C,A = rα-n ±/Cn(α); Theorem 3 says that Rα C R-2k Af = Rα-2kCf, α ≠ n + 2r, r = 0, 1, . . . . Similarly, we have Theorem 4: Rα AR-2k Cf = Rα-2k Cf, a ≠ n + 2r, r = 0, 1, . . . . Theorem 5 expresses that (cf. (3.5)) Rα C(Rβ Af) + Rα A(Rβ Cf) = K1Rα+βCf + K2Rα+β Af, f ∈ S. Finally, Theorem 6 expresses that the following formula is valid: Dα C(Dα Af) + Dα A(Dβ Cf) = C1Dα+β Cf + C2Dα+β Af, where C1and C2 appear in (3.13). © 2000 Elsevier Science Ltd. All rights reserved.
Registro:
| Documento: | Artículo |
| Título: | Some properties of the generalized causal and anticausal Riesz potentials |
| Autor: | Cerutti, R.A.; Trione, S.E. |
| Filiación: | Universidad Nacional del Nordeste, Corrientes, Argentina Fac. de Ciencias Exactas y Naturales, UBA and IAM, CONICET, Saavedra 15, 3er. Piso, 1083 Buenos Aires, Argentina |
| Palabras clave: | Generalized Riesz potentials; Riesz derivatives of order α |
| Año: | 2000 |
| Volumen: | 13 |
| Número: | 4 |
| Página de inicio: | 129 |
| Página de fin: | 136 |
| DOI: | http://dx.doi.org/10.1016/S0893-9659(99)00222-0 |
| Título revista: | Applied Mathematics Letters |
| Título revista abreviado: | Appl Math Lett |
| ISSN: | 08939659 |
| CODEN: | AMLEE |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_08939659_v13_n4_p129_Cerutti |
Referencias:
- Nozaki, Y., On Riemann-Liouville integral of ultrahyperbolic type (1964) Kodai Mathematical Seminar Reports, 6 (2), pp. 69-87
- Trione, S.E., Distributional products (1980) Cursos de Matemática, 3. , Instituto Argentine de Matemática, IAM-CONICET, Buenos Aires, Argentina
- Gelfand, I.M., Shilov, G.E., (1964) Generalized Functions, 1. , Academic Press, New York
- Riesz, M., L'integral de Riemann-Liouville et le problème de Cauchy (1949) Acta Mathematica, 81, pp. 1-223
- Cerutti, R.A., Sobre los operadores causales de Riesz The Actas del Tercer Encuentro Centroamericano de Investigadores Matemáticos, , Managua, Nicaragua, (to appear)
- Trione, S.E., On Marcel Riesz's ultrahyperbolic kernel (1988) Studies in Applied Mathematics, 79, pp. 185-191. , Massachusetts Institute of Technology, Cambridge, MA
- Trione, S.E., La integral de Riemann-Liouville (1981) Cursos y Seminarios de Matemática, , Fascículo 29, Facultad de Ciencias Exactas y Naturales, UBA
- Samko, S.G., On spaces of Riesz potentials (1976) Math. USSR, Izvestiya, 10 (5), pp. 1089-1117
- Cerutti, R.A., On the inversion of causal Riesz potentials Trabajos de Matemática, 248. , Serie I, Publicaciones previas, Instituto Argentino de Matemática, IAM - CONICET, Buenos Aires, Argentina
- Cerutti, R.A., On the inversion of Bessel potentials (1994) Revista de la Unión Matemática Argentina, 39
- Trione, S.E., Distributional convolution products (1983) Trabajos de Matemática, 42. , Instituto Argentino de Matemática, IAM-CONICET, Buenos Aires, Argentina
- Aguirre, M., Multiplicative and convolution products of causal distributions (1990) Trabajos de Matemática, 166. , Instituto Argentino de Matemática, CONICET
Citas:
---------- APA ----------
Cerutti, R.A. & Trione, S.E. (2000) . Some properties of the generalized causal and anticausal Riesz potentials. Applied Mathematics Letters, 13(4), 129-136.http://dx.doi.org/10.1016/S0893-9659(99)00222-0
---------- CHICAGO ----------
Cerutti, R.A., Trione, S.E. "Some properties of the generalized causal and anticausal Riesz potentials" . Applied Mathematics Letters 13, no. 4 (2000) : 129-136.http://dx.doi.org/10.1016/S0893-9659(99)00222-0
---------- MLA ----------
Cerutti, R.A., Trione, S.E. "Some properties of the generalized causal and anticausal Riesz potentials" . Applied Mathematics Letters, vol. 13, no. 4, 2000, pp. 129-136.http://dx.doi.org/10.1016/S0893-9659(99)00222-0
---------- VANCOUVER ----------
Cerutti, R.A., Trione, S.E. Some properties of the generalized causal and anticausal Riesz potentials. Appl Math Lett. 2000;13(4):129-136.http://dx.doi.org/10.1016/S0893-9659(99)00222-0
