You can't recover information that isn't present in the data you have. Sometimes there's useful information hidden by noise, and filtering can help. However, esp. in the case of downscaling/upscaling, information is just gone.
These wiki links are way too general and mathy to be specifically helpful, but this is the theory behind why you can't recover quality once it's gone:
Think of it this way: if you could somehow increase the quality of a video, we'd just use that procedure on 100x100 pixel videos to upscale them to beautiful 4k displays, and fit HD movies on floppy disks.
Same deal if you compress the hell out of a JPEG image: it'll be all blurry and blocky, and the details of the source are gone. If they were recoverable from a tiny JPEG, then we could compress anything as much as we wanted and still have great-looking images.
These things are as impossible as perpetual motion machines. Sorry to dash your hopes. :/
All that said, it's possible you could do some smoothing or filtering on an upscaled version to produce something that looks to a human eye more like what you saw. With the aid of words, you might have some luck getting your point across.