With PSTV you can play ALL PSP & PS 1 titles on it, plus VITA titles that didn't use all the rear controls of the VITA handheld console.....quite a few.
If you have Pure & Pulse on a PSP that was bought from the PStore, there is no reason to buy it again.
You can download Sony's Media GO! software to a PC, and transfer the same download onto your new PSTV unit. [some cavets, and you need to know the procedure...so post back if that's your situation and I'll show you how to do it ]
Here, see the thread I made a while back when I first got my PSTV.
http://www.wipeoutzone.com/forum/sho...highlight=PSTV
Just click on the photos to see larger versions, all those photos are FULL screen on that TV..... NO Boarders.
When I was running the same sort of setup via a PSP GO! and using the video upscaling function of my surround amp, the picture still wasn't full screen.
I suspect the same would happen to you when running your PS2, as the PSP GO! was using component out, where the PSTV uses HDMI, plus the VITA processor is a lot more powerful.
Do yourself a favour, Get a PSTV and a PS4 controller [you can use a PS3 controller] but if you want to try the 2048 hack, you'll need a PS4 controller.
PM me if you get stuck, as I don't always browse all the different sections of the forum