Given a downscaled image, produce Gaussian blur of the original image
I don't think you can do that.
You can probably get a similar result, in that you won't be able to tell by eye that they are different.

The reason I don't think you can do what you have in mind is this:
Imagine that you downscale (downsample, basically throw away pixels) an image, apply a blur, upscale the image.  Suppose now that you apply a blur to the original image.  If those two are the same, then you could deconvolve both images and from both of them recover the original image.

This would be the best compression format ever.  Downscale by 16X, blur.  Small file, send by email.  Upscale by 16X, deblur, original image.  Hell, why stop at 16X?  You could compress all of your images to a single pixel each and be able to recover them.

Unfortunately, things don't work that way.  Downscaling will (must) throw away data, and you can't recover it.  

So, the downscaled and blurred image will be different from the blurred original.