Is there a periodic function (seen as a function on the torus $T^k$) such that the associated Fourier family $(c_n e_n)_{n\in \mathbb{Z^k}}$ (where the $(c_n)$ are the usual Fourier coefficients, and $(e_n)$ is the Fourier basis)
is
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summable in $C(T^k)$ (with the uniform norm) (this is equivalent to : the Fourier series $\sum_n c_n e_n$ is unconditionnally uniformly convergent)
-
not absolutely summable in $C(T^k)$ (this is equivalent to $\sum_n |c_n|=\infty$)
Note : the example given here of a function with uniformly convergent but non absolutely convergent Fourier series does not seem to me (but I could be wrong) such that its Fourier series converges unconditionally. (If that's the case, that answers my question.)
Best Answer
It turns out that unconditional convergence of a Fourier series is equivalent to its absolute convergence.
To see this, note that if $T : X \to Y$ is a bounded linear operator and if $\sum_i x_i$ converges unconditionally in $X$, then $\sum_i T x_i$ converges unconditionally in $Y$ (why?!).
Now apply this to $T : C(T^k) \to \Bbb{C}, f \mapsto f(0)$ to see that if $\sum_k c_k e_k$ converges unconditionally, then the series $$ \sum_k c_k = \sum_k c_k T e_k $$ converges unconditionally in $\Bbb{C}$. It is well-known that in $\Bbb{C}$, unconditional convergence is equivalent to absolute convergence.