Summation Notation: Is \sum_{u,v} Equal to \sum_u\sum_v?

[gnome]

Is

$$\sum_{u,v} H_{i-u,j-v}F_{u,v}$$

the same as

$$\sum_u\sum_v H_{i-u,j-v}F_{u,v}$$

?

(Don't worry about what H,F,i,j,u,v are. I'm only asking about the notation.)

 

[Crosson]

I only see this in the case that there could be no confusion about the range of the indicies, and so I would say yes, they are the same.

 

[gnome]

Yes, thanks, assuming that the ranges of the indices are unambiguous...

I read the first form in a textbook; the context is applying convolution to image data, and I don't see any way to interpret it other than as a double summation. I just wanted some reassurance that I'm not overlooking something.

 

[AlephZero]

It's written that way to save space. Similar to writing $$\int f \, dV$$ rather than $$\int \int \int f \, dx\, dy\, dz$$