During some calculation regarding a physics problem, I came across a function, which we will denote as $u(x)$, which is $1/2$ if $x=y$, where $y$ is a positive real number, and $0$ elsewhere. Since I then have to integrate this function $u$, I need a suitable representation in order to do the actual calculation. To give a little bit of context, the function $u$ derive from the time-average of the product of two trigonometric functions at different frequencies. In particular,
$$ \langle \cos (x t - \phi) \, \cos (y t - \phi') \rangle := \lim_{T\rightarrow+\infty} \frac{1}{T} \int_{0}^{T} dt \, \cos (x t - \phi) \, \cos (y t - \phi') $$
The time-average is $\ne 0$ only when the frequencies are the same, and so here's where $u$ comes from. Now, I know that, in the "discrete" case, i.e. when $x$ and $y$ are integers, the time average is proportional to the Kroneker delta $\delta_{x,y}$. However, in this case $x$ and $y$ are continuous variables. My "physical" guess is that the Kronecker delta becomes a Dirac delta, but, since $\delta (x-y)$ have the dimension of the inverse of a frequency and the result must be dimensionless, we have to multiply with something that have dimension of a frequency. Thus, I was expecting something proportional to $y \, \delta (x-y)$. This is a very heuristic argument; and I need something more precise. For istance, while doing some research, I learned that every distribution $v$ defined at a single point $y$ can be represented, if I understand correctly, as
$$ v(x) = \sum_{n=0}^{N} c_n (y) \, \delta^{(n)} (x-y) \, ,$$
where $N$ is called the order of the distribution $v$, $c_n$ are constants (that I guess depend on the point at which $v$ is defined) and $\delta^{(n)} (x-y)$ is the $n$-th derivative of the Dirac distribution. Unfortunately, my knowledge of distribution theory is pretty basic, and I can't quite understand how to determine the degree $N$ and the constants $c_n $ associated to the function $u$. Any help would be very, very appreciated. Also, if the problem can be solved without explicitly determining $N$ and $c_n $, this kind of solution if obviously welcome.
I thank you in advance, and I apologize if such a question already exists without me having found it on the website.
If $u$ is really defined by $u(y) = 1/2$ and $\forall x\neq y,\, u(x) = 0$, then $$\int u(x)dx = 0.$$
No "representation" is going to change the fact that the integral of $u$ is zero. In particular, this function $u$ is not the Dirac delta.
If the integral of $u$ is not zero, then $u$ is not the function defined above.