Sopra un teorema d'intercambio

Trione, Susana Elena

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti, Tome 59 (1975), p. 357-361 / Harvested from Biblioteca Digitale Italiana di Matematica

Résumé

Let $\alpha,\beta \in \mathbf{C}$, $\alpha+\beta=n+2h$, $\alpha \ne n+2h$, $\beta \ne n+2h$, $h=0,1,\cdots$. We prove under these conditions, the formula of interchange of the Fourier transformation of convolution of $Pf (H_{\alpha}(P \pm i 0,n) \ast H_{\beta} (P \pm i 0,n))$ into the product of their Fourier trasforms: $$\{ Pf (H_{\alpha}(P \pm i 0,n) \ast H_{\beta} (P \pm i 0, n)) \}^{\Lambda} = \{ H_{\alpha}(P \pm i 0,n) \}^{\Lambda} \cdot \{ H_{\beta}(P \pm i 0,n) \}^{\Lambda}$$ (see, for the definitions of these notations, formulae (1), (1') and Theorem). As an immediate consequence of formula (2) we obtain $$\{ Pf ((P \pm i 0,n)^{\frac{1}{2}t} \ast (P \pm i 0,n)^{\frac{1}{2}s}) \}^{\Lambda} = \{ (P \pm i 0,n)^{\frac{1}{2}t} \}^{\Lambda} \cdot \{ (P \pm i 0,n)^{\frac{1}{2}s} \}^{\Lambda},$$ where $t+s =-n+2h$, $t \ne 2h$, $s \ne 2h$, $h = 0,1,\cdots$. It may be observed that, in the particular case $p=n$, $q = 0$, the distributions $H_{\alpha} (P \pm i 0, n)$ turn out to be the elliptic M. Riesz kernel of which they are "causal" ("anticausal") analogues; and from formula (18), we arrive at $\{ Pf (r^{t} \ast r^{s}) \}^{\Lambda} = \{ r^{t}\}^{\Lambda} \{ r^{s} \}^{\Lambda}$, which is valid for $t+s = - n + 2h$, $t \ne 2h$, $s \ne 2h$, $h = 0,1,\cdots$. The last formula is an extension of formula (VII, 8; 8), p. 271, obtained by L. Schwartz (cf. [6]).

Détail

Publié le : 1975-11-01

MR 0487443 | Zbl 0354.42018

Bibliographie

[1] Riesz, M. (1949) - L'intégrale de Riemann-Liouville et le problème de Cauchy, «Acta Mathematica», 81, 1-223. | MR 30102 | Zbl 0033.27601

[2] Trione, S. E. (1974) - Sopra alcune convoluzioni divergenti, «Rend. Accad. Naz. Lincei», ser. VIII, 57 (3-4), 143-146. | MR 420264

[3] Gelfand, I. M. e Shilov, G. E. (1964) - Generalized Functions, Vol. I, Academic Press. New York. | MR 166596

[4] Trione, S. E. (1973) - Sobre una fórmula de L. Schwartz, «Revista de la Union Mat. Argentina», 26. | MR 373501

[5] Trione, S. E. (1972) - Tesis doctoral, «Sobre soluciones elementales causales de ecuaciones diferenciales en derivadas parciales con coeficientes constantes», F.C.E. y N., Universidad de Buenos Aires.

[6] Schwartz, L. (1966) - Théorie des Distributions, Paris, Hermann. | MR 209834