Consider a transformation $\phi: \mathbb{R}^N \to \{ [-\pi,+\pi) \to \mathbb{C} \}$ given by: $$ [\phi(v)](x) := \sum_{n=0}^{N-1} v_n e^{-inx} $$ where $v \in \mathbb{R}^N$ and $x \in [-\pi,+\pi)$. We can also define a function $\psi: \{ [-\pi,+\pi) \to \mathbb{C} \} \to \mathbb{R}^n$ going in the opposite direction: $$ [\psi(f)]_n := \frac{1}{2 \pi} \int_{-\pi}^{+\pi}dx f(x) e^{+inx} $$ This pair of functions has the property that $\psi \circ \phi$ is the identity on $\mathbb{R}^N$, though of course this does not hold in the opposite direction.
Now consider bilinear forms $A: (\mathbb{R}^N,\mathbb{R}^N) \to \mathbb{R}$. It is true in general that $$ A(v,w) = \int_{-\pi}^{+\pi}dx \int_{-\pi}^{+\pi}dy \, \left[\phi \left(v\right)\right]\left(x\right) \, \left[\phi \left(w\right)\right]\left(y\right) \, \hat{A}(x,y) \tag{*} \label{*} $$ where $\hat{A}$ is a function from $\left( \left[ -\pi, +\pi \right), \left[ -\pi, +\pi \right) \right)$ to $\mathbb{C}$ defined by $$ \hat{A}(x,y) := A(\psi(\delta_x),\psi(\delta_y)) $$ where $\delta_x(x') := \delta(x'-x)$ is the Dirac delta function. Now suppose we have a form $A$ that is translation invariant in the sense that $$ A(e_n,e_{n+m}) \equiv A_{n,n+m} = A_{0,m} \equiv a_m $$ where $[e_n]_{n'} := \delta_{nn'}$ and addition/subtraction of indices is understood to be modulo $N$. We can show that for this form we have $$ \hat{A}(x,y) = e^{i(x+y)}\hat{A}(x,y) $$ which implies $\hat{A}(x,y) = 0$ unless $y = -x$, in which case we have simply $$ \hat{A}(x,-x) = \frac{N}{(2 \pi)^2} \sum_{n=0}^N a_n e^{inx} $$ Here is where I am confused, because referring back to \ref{*} we have now in our integrand of the integral over the square $-\pi \leq x < +\pi, -\pi \leq y < +\pi$ a factor $\hat{A}(x,y)$ that is 0 everywhere except for a set of measure zero (i.e. the line $x+y = 0$) where $\hat{A}(x,y)$ is finite, so the RHS of \ref{*} must vanish while in general the LHS does not.
I do not know where lies the error that leads to this contradiction. I would have expected the calculation of $\hat{A}(x,-x)$ to include itself a Dirac delta à la $$ \hat{A}(x,y) = \delta(x+y) \frac{N}{(2 \pi)^2} \sum_{n=0}^{N-1} a_n e^{inx} $$ but that it not what my calculations indicate.
Any help in clarifying the source of this contradiction would be appreciated!
The error is in the assertion $$ \hat{A}(x,y) = e^{i(x+y)}\hat{A}(x,y) $$ which does not in general hold if $x$ or $y$ are not of the form $\frac{2 \pi n}{N}$ for some integer $n$. The assertion however is in some qualified sense true in the limit $N \to \infty$.
I was unwittingly asserting something equivalent to the statement that any two abritrary sinusoids are orthogonal on the unit interval, which is of course not true (though it is true if the domain is taken to be the real line).