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Let $\alpha ,\beta \in 𝐂$, $\alpha +\beta =n+2h$, $\alpha \ne n+2h$, $\beta \ne n+2h$, $h=0,1,\mathrm{\cdots }$. We prove under these conditions, the formula of interchange of the Fourier transformation of convolution of $Pf(H_{\alpha }(P\pm i0,n)\ast H_{\beta }(P\pm i0,n))$ into the product of their Fourier trasforms: $${\{Pf(H_{\alpha }(P\pm i0,n)\ast H_{\beta }(P\pm i0,n))\}}^{\mathrm{\Lambda }}={\{H_{\alpha }(P\pm i0,n)\}}^{\mathrm{\Lambda }}\cdot {\{H_{\beta }(P\pm i0,n)\}}^{\mathrm{\Lambda }}$$ (see, for the definitions of these notations, formulae (1), (1') and Theorem). As an immediate consequence of formula (2) we obtain $${\{Pf({(P\pm i0,n)}^{\frac{1}{2}t}\ast {(P\pm i0,n)}^{\frac{1}{2}s})\}}^{\mathrm{\Lambda }}={\{{(P\pm i0,n)}^{\frac{1}{2}t}\}}^{\mathrm{\Lambda }}\cdot {\{{(P\pm i0,n)}^{\frac{1}{2}s}\}}^{\mathrm{\Lambda }},$$ where $t+s=-n+2h$, $t\ne 2h$, $s\ne 2h$, $h=0,1,\mathrm{\cdots }$. It may be observed that, in the particular case $p=n$, $q=0$, the distributions $H_{\alpha }(P\pm i0,n)$ turn out to be the elliptic M. Riesz kernel of which they are "causal" ("anticausal") analogues; and...

We give a sense to certain divergent convolutions of the form ${(P\pm i0)}^{\lambda }\ast {(P\pm i0)}^{\mu }$ (see, for the definition of these notations, formulae (2), (4), (5)). As an application of our formulae we obtain the explicit value of a constant which appears in a formula of Fuglede (see [1], p. 7, Lemma 3.1). We observe that many of the "divergences" appearing in quantum field theory are precisely divergent convolutions of the form ${(P\pm i0)}^{\lambda }\ast {(P\pm i0)}^{\mu }$ (see [2], pp. 151-183).

We evaluate (formula (2,2)) the Hankel transform of the distribution ${\delta }_a^{(m)}(t)$. As a consequence of (2,2) we obtain $$\underset{A\to \mathrm{\infty }}{lim}\frac{{\mathrm{\Omega }}_n}{{(2\pi )}^{n/2}}{A}^{n/2}{J}_{n/2}(At)=\delta ,$$ where ${\mathrm{\Omega }}_n$ designates the area of the unit sphere in $R^n$. We also give a direct proof of this formula, which plays a role in the theory of the spherical summability of Fourier integrals.

Let $n$ and $k$ be fixed integers; $n$ even $\ge 4$; $1\le k\le \frac{n-2}{2}$. Let $f(t)$, , be differentiable and such that . We prove that, under these conditions, the integral equation (2) admits the solution (4). In the particular case $n=4$, $k=1$ (which is important in the quantum theory of fields) the reciprocal formulae (2) and (4) have already been obtained (on the basis of heuristical considerations) by Källen [1].

The distributions $G_{\alpha }\{P\pm i0,m,n\}$ (formula (1)) share many properties with the Bessel kernel, of which they are "causal" ("anticausal") analogues. In particular (Theorem 1), $G_{\alpha }\ast G_{-2k}=G_{\alpha -2k}$, $\mathrm{\Lambda }\alpha \in 𝐂$, $\mathrm{\Lambda }k=0,1,2,\mathrm{\cdots }$. The essential tool for the proof of this formula is the multiplication formula (4), namely $${\{m^2+Q(y)\mp i0\}}^{\alpha }{\{m^2+Q(y)\mp i0\}}^{\beta }={\{m^2+Q(y)\mp i0\}}^{\alpha +\beta },$$ which is valid for every $\alpha ,\beta \in 𝐂$. It follows from Theorem (1) that $G_{2k}\{P\pm i0,m,n\}$, is, for $n\ge 2$, $k=1,2,\mathrm{\cdots }$, a causal (anticausal) elementary solution of the n-dimensional Klein-Gordon operator, iterated $k$ times (Theorem 2). The particular case $n=4$, $k=1$ is important in the...

As a generalization of formula (1,3), due to Guerra (cfr. [3]), which is useful in quantum theory of fields, we prove formula (3,1).