Sopra un teorema d'intercambio

Trione, Susana Elena. "Sopra un teorema d'intercambio." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti 59.5 (1975): 357-361. <http://eudml.org/doc/290891>.

@article{Trione1975,
abstract = {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]).},
author = {Trione, Susana Elena},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti},
language = {eng},
month = {11},
number = {5},
pages = {357-361},
publisher = {Accademia Nazionale dei Lincei},
title = {Sopra un teorema d'intercambio},
url = {http://eudml.org/doc/290891},
volume = {59},
year = {1975},
}

TY - JOUR
AU - Trione, Susana Elena
TI - Sopra un teorema d'intercambio
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
DA - 1975/11//
PB - Accademia Nazionale dei Lincei
VL - 59
IS - 5
SP - 357
EP - 361
AB - 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]).
LA - eng
UR - http://eudml.org/doc/290891
ER -