# Nonparametric method - Parzen windows

I don't understand the main function of the Parzen Window

Let $u=[u_1, u_2,..., u_d]$ and define a window function

$φ(u)=\left\{ \begin{array}{l l} 1 & \quad \text{$|u_j|<\frac{1}{2}$,$j=1,2,...,d$}\\ 0 & \quad \text{otherwise} \end{array} \right.$

What exactly it means?

I found lots of presentation on the internet, so please do not direct me to one.

-

It means that $\varphi(u)$ is $1$ in a unit hypercube centred on the origin and $0$ outside (and on the boundary).
$u$ is a $d$ dimensional sample, we choose a "window" which is: a line with length h when $d=1$, a square with edge $h=2$ ($h^2$ space), an hypercube with edge $h=2$. The "window" is centered at the point $u$ and we normalize is so $h=1$.
Now our function assign 1 to each other sample that is close to $u$ and 0 to samples that are far.
Close means that the sample $x$ is in the range of $u-\frac{1}{2}h\leq{x}\leq{u+\frac{1}{2}h}$