Intuition behind using non-hypercubic kernels in density estimation Suppose that we perform density estimation in m-dimensional space: we estimate the value $p(a)$ for some point $a$ given observations $\{x_1, \dots, x_n \}$.

It is known that if region $A \subset \mathbb{R}^m$ is "small" enough to consider density being constant on points from $A$ then we can make the following estimate:
$$ p(a) \approx \frac{k / n}{|A|} $$
where $k$ is the number of observations that lie in $A$ and $|A|$ is Lebesgue measure of $A$. 

Let parameter $h$ be small enough to consider density as constant inside hypercube centered at $a$ with side length equal to $h$. The volume of this hypercube is equal to $h^m$ and point $x$ lies inside this hypercube iff $K(\frac{x-a}{h}) = 1$ where
$$K(u) =\cases {
      1\text{, if $|\frac{u^k - a^k}{h}| \leq \frac{1}{2}, k = 1,\dots,m$}\cr
      0\text{, otherwise}
}$$
It's easy to see that the number of observations inside this hypercube equals to
$$k = \sum_{i = 1}^{n} K(\frac{x-a}{h})$$ 
and so the estimation described above gets the following form:
$$p(a) \approx \frac{1}{n h^m} \sum_{i = 1}^{n} K(\frac{x-a}{h}) $$

We can interpret $K$ as "weight" given to particular observations and one of the drawbacks of hypercubic approach is that all observations lying inside hypercube have equal weights despite having different distances from $a$. Yet another drawback is that the resulting estimate is not continuous. That's what i understand to be the main reason of using non-hypercubic kernels such as gaussian kernel which give more weight to points close to $a$ and yields continuous estimate.

But i have troubles with interpreting the usage of such kernels. The sum $\sum_{i = 1}^{n} K(\frac{x-a}{h})$ is no longer equal to $k$ so we can't justify the usage of these kernels by formula $p(a) \approx \frac{k / n}{|A|} $. Finally here are my questions: how do we justify the usage of smooth kernels? how can one interpret this usage?

Thank you for any ideas.
Histograms and methods based on binning have a number of well-known problems. Different anchor points etc. can introduce artificial patterns that make interpretation unreliable. Smooth kernels don't use a grid and thus smooth out the noise. 

This also has the advantage that it makes it easier to get a single overall picture of the data because it takes into account neighboring points and smooths the data into areas where no data is observed.

Smooth kernels can also be justified by their favorable statistical properties. Popular methods like fastKDE use the fact that one can find "an empirical kernel that is optimal in the sense that the integrated, squared difference between the resulting KDE and the true PDF is minimized."