Summing up Fourier coefficients $\hat{f}({\bf m})$ of $f\in L^1({\bf T}^n)$ for ${\bf m}=(m,\ldots,m), m \in {\bf Z}$

Question

Say $f\in L^1({\bf T}^n)$ a function on the real n-torus ${\bf T}^n$ with $\sum_{{\bf m}\in {\bf Z}^n} \mid \hat{f}({\bf m})\mid<\infty$ where $\hat{f}({\bf m})=\int_{{\bf T}^n} f({\bf x}) e^{-2\pi i {\bf m}\cdot {\bf x}} dx$ is the m-th Fourier coefficient. Then by Fourier inversion

$f({\bf x})= \sum_{{\bf m}\in {\bf Z}^n}\hat{f}({\bf m})e^{2\pi i {\bf m}\cdot {\bf x}}$

almost everywhere. Now set $\Delta: {\bf Z}\rightarrow {\bf Z}^n, m \mapsto (m,\ldots,m)^t$. In this setting what (if anything non-trivial) can one say about

$\sum_{{\bf m}\in \Delta({\bf Z})}\hat{f}({\bf m})e^{2\pi i {\bf m}\cdot {\bf x}}$

or

$\sum_{{\bf m}\in \Delta({\bf Z})} \mid \hat{f}({\bf m})\mid$?

What I mean is, are there theorems (with perhaps additional assumptions) making any interesting statements about these sums?

Answer 1

For simplicity, take $n=2$ and $f \in C^\infty$.

As usual $$\sum_{m=-M}^M e^{2i \pi m(x+y)} = \frac{\sin( 2\pi (m+1/2) (x+y))}{\sin(\pi(x+y))}$$ so that $$\sum_{m=-\infty}^\infty c_m e^{2i \pi m(x+y)} = \lim_{M \to \infty} \int_0^1 \int_0^1 f(x-u,y-v) \frac{\sin( 2\pi (m+1/2) (u+v))}{\sin(\pi(u+v))}dudv \\ = \int_0^1 \int_0^1 f(x-u,y-v) \delta(u+v) du dv = \int_0^1 f(x+v,y-v)dv$$

Answer 2

There's a lot to say if you care about asking in a certain direction. I don't know what level of math you're asking from: this answer is going to be about current research, so there's a chance this flies over your head if you were asking from a less advanced motivation.

This question could be classified under the category of discrete Fourier restriction, which (to simplify things a bit) studies what happens when a Fourier series is restricted to a specific subset $E$ of frequency space. Researchers in this field are concerned with questions like finding the best constant $A_{p,N}$ in the inequality $$ \left(\sum_{|\mathbf{m}|\leq N, \mathbf{m}\in E} |\hat{f}(\mathbf{m})|^q\right)^{1/q} \leq A_{p,q,N}\left( \int_{\mathbf{T}^n} |f(\mathbf{x})|^p dx\right)^{1/p}. $$ Specific cases of $E$ are often of particular interest: for example, $E = \{(\mathbf{m},|m|^2): \mathbf{m}\in\mathbf{Z}^n\} \subset \mathbf{Z}^{n+1}$ is associated to counting the number of solutions to certain systems of quadratic Diophantine equation (see Vinogradov's mean-value theorem), which has further consequences for PDEs (in this case, the Schrodinger equation). There has been much recent research progress in this area: for instance, many reasonably general polynomial relations defining $E$ have been studied.

That said, $E$ is usually taken to be a hypersurface of one dimension lower than the ambient space, as cutting away more than one dimension's worth of Fourier data tends to lead to noninteresting conclusions. In your case, $E$ is a straight line through the origin in the direction $(1,1,\ldots,1)^t$, hence one-dimensional: therefore the nontrivial case in Fourier restriction arises in the case $n=2$. In this case you have the discrete Fourier restriction problem for a 1-dimensional hyperplane in a 2-dimensional ambient space. Unfortunately, this is also not a particularly interesting problem: it is well-known that curvature of the hypersurface is required for interesting norm bounds, while on a flat hypersurface there are no restriction inequalities because you can construct functions that are well-behaved in the physical domain while blowing up on any subset of a hyperplane in Fourier space.

However, you would arrive at a very interesting problem if you instead considered $E = \{(\mathbf{m},|\mathbf{m}|)\in\mathbf{Z}^n\times\mathbf{Z}\}$. This set is the cone in $n+1$-dimensional Fourier space, and corresponds to the discrete Fourier restriction problem for the cone, which is also related to the wave equation. For $n=1$ this matches your original problem, except that the anti-diagonal $(-m,m)$ is also included. Although the problem is still not interesting in this dimension (two straight lines through the origin are not any better than just one), in higher dimensions the cone is a genuine curved surface, albeit with one direction of zero principal curvature. This direction of zero principal curvature makes the cone problem one of the more difficult restriction problems, and indeed it is connected to some of the most notable problems in harmonic analysis and PDEs today, including the Kakeya conjecture, Bochner-Riesz conjecture, and the local smoothing conjecture. Again, there has been very recent progress on this problem as well.