Wikipedia provides the following representation of the Kronecker delta:
\begin{equation}\label{eq1}
\delta_{jk} = \frac{1}{N}\sum_{n=1}^N \mathrm{e}^{\mathrm{i}2\pi(j-k)n/N}
\end{equation}
At first glance, there doesn't seem to be any issue. When $j=k$, the summation goes to one, otherwise it's zero. Except, that is not exactly true. Consider what happens when $j = k + mN$, where $m$ is some non-zero integer: the "Kronecker delta" is equal to 1, yet $j \ne k$. Surely this representation cannot be correct?
Why is this an issue? Let us look at what happens to a $2\pi$-periodic function in $\theta$, $y(\theta)$, that consists of the sum of $N$ shifted versions of the following function $x(\theta)$ (another $2\pi$-periodic function):
\begin{equation}
y(\theta) = \sum_{n=1}^N x(\theta – 2\pi n/N)
\end{equation}
where each of the shifted functions have been shifted by equal amounts, such that $y(\theta)$ is in fact a $2\pi/N$-periodic function: we can show that $y(\theta + 2\pi/N) = y(\theta)$.
$x(\theta)$ is a periodic function, so we can express it as a Fourier series:
\begin{equation}
x(\theta) = \sum_{j = -\infty}^{\infty} x_j \, \mathrm{e}^{\mathrm{i}j\theta}
\end{equation}
Therefore, $y(\theta)$ can be expressed as
\begin{equation}
y(\theta) = \sum_{n=1}^N \sum_{j = -\infty}^{\infty} x_j \, \mathrm{e}^{\mathrm{i}j(\theta – 2\pi n/N)}
\end{equation}
Then, by changing the order of summation, we get
\begin{equation}
y(\theta) = \sum_{j = -\infty}^{\infty} x_j \, \mathrm{e}^{\mathrm{i}j\theta} \sum_{n=1}^N \mathrm{e}^{\mathrm{i} 2\pi j n/N}
= \sum_{j = -\infty}^{\infty} x_j \, \mathrm{e}^{\mathrm{i}j\theta} N \delta_{j0}
\end{equation}
But, by the property of the Kronecker delta, this implies that $y(\theta) = N x_0$ is a constant value. However, this cannot be, since we can always choose $x(\theta)$ to ensure $y(\theta)$ is non-constant. Therefore, something has gone wrong, which I assume is due to the fact that the above representation of the Kronecker delta is misleading. Am I missing something from the above discussion?
The above summation representation instead seems to represent a Kronecker comb (mentioned later in the same article). Is this true? If so, is the article calling the above representation the Kronecker delta inaccurate, or am I missing an unstated assumption?
Best Answer
The formula is to be understood as being on the discrete circle, that is for $j$ and $k$ in $\mathbb Z / N\mathbb Z$. You can rewrite this on the discrete line: $$ \sum_{p\in\mathbb Z}\delta_{j(k+pN)} = \frac{1}{N}\sum_{n=1}^N \mathrm{e}^{\mathrm{i}2\pi(j-k)n/N} $$ Technically, this is the discrete form of the Poisson summation formula.