Convergence of the periodization series in $L^2(\mathbb{T})$

Question

I'm reading a paper using periodization of function, I know that for a given function $f\in L^1(\mathbb{R})$ we can define the periodization $\tilde{f}$ by $$\tilde{f}(x)=\sum_{k\in\mathbb{Z}}f(x+k)$$ which is obviously a periodic function and belongs to $L^1(\mathbb{T})$. However, in the paper the author took a function $f\in L^2(\mathbb{R})$ and just said that he can define by the same way the periodization function which is convergent in $L^2(\mathbb{T})$. I don't even understand why the periodization is well defined in this case and how the serie might converge in $L^2(\mathbb{T})$.

Answer 1

If $f\in L^1(\Bbb{R})$ then $$\sum_n f(x+n) = g\in L^1[0,1]$$ $\hat{f}$ is continuous and $$c_k(g) = \hat{f}(k)$$ Iff $\sum_k |\hat{f}(k)|^2<\infty$ then $$ g=\sum_k \hat{f}(k) e^{2i\pi k x} \in L^2[0,1]\cap L^1[0,1]$$ This is what your paper is using

Answer 2

People have said in comments that periodization doesn't work for an arbitrary $f\in L^2(\Bbb R)$ but I don't see a counterexample. Let $f(t)=\frac1{|t|+1}$; then $f\in L^2(\Bbb R)$ but the sum defining $\tilde f$ converges pointwise to $+\infty$ at every point. (Hence Fatou's Lemma shows that the $L^2(\Bbb T)$ norm of the partial sums tends to $+\infty$.)

(I'm told that a similar counterexample has already been posted; in fact, as stated above, I don't see it here...)