Does such filter exist? Assume the following operation on input $x$ with filter $w$ of size $2L+1$:

$y[i] = \frac{\sum_{k=-L}^{L} x[i+k] w[k]}{\sum_{k=-L}^{L} x[i+k]}$

In other words, my desired filter does a convolution and divides by the sum. 
Does this operation has a name (or implemented in conventional software such as MATLAB or python's Scipy)?



In case you are wondering why I need that, this is my problem: I'm having a 1-D temporal signal in which some textures (say edges) are to be discovered. The noise has attenuated locally (in time) some of the textures by some scaling factor which is unknown to me. Now I want to discover the potential time locations where a certain texture happens by normalizing the output of usual filter to the sum of absolute values of my signal (I've dropped the abs() since my signal is positive always).
It is not really a convolution, more a weighting. Your expression is $0$ homogeneous in $x$, as you divide $x$s values by $x$s.  Specifically:


it is not defined if $\sum_{-K,K} x[i+k] = 0$
if $x[k]$ is locally a constant, the output does not depend on it (because it will be equal to $\sum h[k]$)
if $h[k]$ is an averaging filter ($h[k] = \frac{1}{2K+1}$), the output will be $\frac{1}{2K+1}$


So globally I would not consider it as a  classical filter for $x$. But it could be seen as an adaptive weighting for $h$, based on a unit-normalized window $\chi$ with $\chi[n] = \frac{x[n]}{\sum_{|k|\le K} x[n]}$. As such, it is easily implemented in Matlab or Python. So far, I don't have a use for it. What was your purpose?