What is the impulse response used in an interpolation filter when upsampling? I need to downsample a single period waveform from $M$ samples to $L$ samples. If that matters, in my particular case, $L=M-1$.

I read about downsampling and came to the conclusion I need:


expand the original waveform by a factor $L$ (some other source call the step “zero stuffing”), 
interpolate the missing samples, 
decimate by a factor of $M$:


If I understood it correctly, expansion only requires adding $L-1$ zeros between each original sample. Decimation seems simple too since it only requires keeping every $M^{th}$ samples.

But I'm stuck with the interpolation process. The naive solution of using linear interpolation gave me surprisingly good results at low frequencies but quickly degraded when the frequency increases (i.e., .when the number of samples in the waveform decrease).

I found several mentions of using an interpolation filter that looks like a dot product1. For example, on the Upsampling Wikipedia page, I see:


  $$y[j+nL] = \sum_{k=0}^K x[n-k]\cdot h[j+kL],\ \ j = 0,1,\ldots,L-1,$$
  
  where the h[•] sequence is the impulse response, and K is the largest value of k for which h[j + kL] is non-zero


In my understanding, $x[•]$ is the sequence after the expansion step. $y[•]$ is the upsampled waveform. But I don't understand what are $h[•]$ and $K$.  I read they are related to the impulse response and I understand the concept. But I can't see where it comes from and how to obtain it in practice.

Could someone explain to me how to obtain $h[•]$ and $K$? 



1It also looks like a convolution to me, but I'm not quite sure of that. And apparently this would be an instance of FIR filter. We already talked about that in other questions, but this is still not clear to me.
The dot in that summation is just scalar multiplication.  And yes, it's a convolution -- you're convolving the input signal by the filter.