$$\underset{R\rightarrow\infty}\lim \Big|{\underset{0<|m|\leq R}\sum \frac{e^{2\pi i m\cdot x}}{|m|^{\alpha}} }\Big|=\infty.$$
What is important to emphasis here is to precise that we are abandoning the mode of convergence established by S. Bochner and those who have followed since, namely: The series $$\underset{m\in \mathbb{Z}^n, m\neq 0}\sum \frac{e^{2\pi i m\cdot x}}{|m|^{\alpha}}$$ is convergent if the limit
$$\underset{\varepsilon\rightarrow 0}\lim{\underset{m\in\mathbb{Z}^{n}, m\neq 0}\sum} \Phi(\varepsilon m)\frac{e^{2\pi i m\cdot x}}{|m|^{\alpha}}$$
exists, where $\Phi$ is an appropriate continuous function with $\Phi(0)=1$ verifying the following assumptions
i) $\Phi(y)=\widehat{\varphi}(y)$ with $\int_{\mathbb{R}^{n}}\varphi(x)dx=1$.
ii)$|\Phi(y)|\leq A(1+|y|)^{-n-\delta}, |\varphi(x)|\leq A(1+|x|)^{-n-\delta}$ for some $\delta>0$.
Note that we have noted
$$\underset{m\in\mathbb{Z}^{n}, m\neq 0}\sum \Phi(\varepsilon m)\frac{e^{2\pi i m\cdot x}}{|m|^{\alpha}}=\underset{R\rightarrow\infty}\lim {\underset{0<|m|\leq R}\sum} \Phi(\varepsilon m)\frac{e^{2\pi i m\cdot x}}{|m|^{\alpha}}.$$
That is we have considered the spherical convergence of the series $$\underset{m\in\mathbb{Z}^n, m\neq 0}\sum \Phi(\varepsilon m)\frac{e^{2\pi i m\cdot x}}{|m|^{\alpha}}.$$
A classical example consists to take $\Phi(y)=e^{-2\pi|y|}$. In this case we obtain the appropriate convergence which I call the convergence for the Poisson (or Abel-Poisson) means (Note that there exists an other mode convergence called convergence for the Riesz means):
\begin{eqnarray}
\underset{m\in\mathbb{Z}^{n},m\neq 0}\sum \frac{e^{2\pi i m\cdot x}}{|m|^{\alpha}}
&\equiv & \underset{m\in\mathbb{Z}^{n},m\neq 0}\sum e^{-2\pi|m|\varepsilon} \frac{e^{2\pi i m\cdot x}}{|m|^{\alpha}}\\
&=& \underset{R\rightarrow\infty}\lim {\underset{0<|m|\leq R}\sum} e^{-2\pi|m|\varepsilon} \frac{e^{2\pi i m\cdot x}}{|m|^{\alpha}}\nonumber.
\end{eqnarray}\begin{eqnarray}
\underset{m\in\mathbb{Z}^{n},m\neq 0}\sum \frac{e^{2\pi i m.x}}{|m|^{\alpha}}
&\equiv & \underset{\varepsilon\rightarrow 0}\lim\underset{m\in\mathbb{Z}^{n},m\neq 0}\sum e^{-2\pi|m|\varepsilon} \frac{e^{2\pi i m.x}}{|m|^{\alpha}}\\
&=& \underset{\varepsilon\rightarrow 0}\lim\underset{R\rightarrow\infty}\lim {\underset{0<|m|\leq R}\sum} e^{-2\pi|m|\varepsilon} \frac{e^{2\pi i m.x}}{|m|^{\alpha}}\nonumber.
\end{eqnarray}
For this convergence mode the series $$\underset{m\in\mathbb{Z}^n,m\neq 0}\sum e^{2\pi i m\cdot x}{|m|^{\alpha}}
\equiv \underset{m\in\mathbb{Z}^{n},m\neq 0}\sum e^{-2\pi|m|\varepsilon} e^{2\pi i m\cdot x}{|m|^{\alpha}}$$$$\underset{m\in\mathbb{Z}^{n},m\neq 0}\sum e^{2\pi i m.x}{|m|^{-\alpha}}
\equiv \underset{\varepsilon\rightarrow 0}\lim\underset{m\in\mathbb{Z}^{n},m\neq 0}\sum e^{-2\pi|m|\varepsilon} e^{2\pi i m.x}{|m|^{-\alpha}}$$ is convergent for any $\alpha> 0$ and $x\in \mathbb{R}^n,$ cf
Remark in page $40$ in
"Special Trigonometric series In Dimension" by Stephen Wainger. Whereas for a spherical convergence mode the series
$$\underset{m\in\mathbb{Z}^n,m\neq 0}\sum \frac{e^{2\pi i m\cdot x}}{|m|^{\frac{n}{2}-\frac{1}{2}}}
\equiv \underset{R\rightarrow\infty}\lim {\underset{0<|m|\leq R}\sum} \frac{e^{2\pi i m\cdot x}}{|m|^{\frac{n}{2}-\frac{1}{2}}}$$
diverges almost everywhere , according to E. Stein as we have just seen.
The two modes of convergence are completely different in nature.
In the Mode convergence for the Poisson means, we are in the presence of absolutely convergent series. Whereas in spherical convergence, we are in the presence of
semi-convergence series. The technique to introduce the convergence factor $e^{-2\pi|m|\varepsilon}$ in $(1)$ is a real denaturalization of the problem and is just a headlong rush for lack of means.
I understand that in time that for lack of means of investigation of the convergence of semi convergent series like the series $\underset{m\in\mathbb{Z}^{n},m\neq 0}\sum \frac{e^{2\pi i m\cdot x}}{|m|^{\alpha}}$, one resorted to this technique of convergence of the Abel or Riesz
means. But what I do not understand at all is how S.Stein, who has the merit of studying the series
$$\underset{m\in\mathbb{Z}^n,m\neq 0}\sum \frac{e^{2\pi i m\cdot x}}{|m|^{\frac{n}{2}-\frac{1}{2}}}
\equiv \underset{R\rightarrow\infty}\lim {\underset{0<|m|\leq R}\sum} \frac{e^{2\pi i m\cdot x}}{|m|^{\frac{n}{2}-\frac{1}{2}}},$$
did not even ask himself the question for any case of $\alpha\neq \frac{n}{2}-\frac{1}{2}$. I remark the same for all of S.Stein's followers: There is only a dead silence, a very dubious and worrying silence. This worries me and I wonder, for lack of information if I am simply on the wrong track.
Allow me to confide to you that I have indeed solved this problem which I have not yet published, that my interest in this problem is not at all a simple intellectual curiosity but indeed a necessity dictated by the study of the fractional powers of the Laplacian on the torus by means of the Riesz operators on the torus and that the study of the latter require an absolute mastery of the spherical convergence and not the convergence for the mean Abel of the series $ \underset{m\in\mathbb{Z}^n,m\neq 0}\sum \frac{e^{2\pi i m\cdot x}}{|m|^{\alpha}}$. Of course in Stein and Wainger this problem of Riesz operators on the torus is well considered and studied but without finding any link with the fractional powers of the Laplacian on the torus
and this has a major interest in the end for the study of the Radon transformation on the torus and that is actually my ultimate goal and I think I have completed his study. This problem is not yet properly studied elsewhere in the world. Those are my concerns about this issue. Thank you for your help and your listening to these occupations.
