That depends on which 15" 'Apple Studio Display' you are referring to, there are four.
The original, came in dark blue (DB15 video connector), or blueberry & white (VGA connector). Both use analog connectors, and would work with your iMac, but the original dark blue Studio Display would also require the DB15/VGA adapter to work with the video out adapter for your iMac.
Alternatively, you could get the AV (S-Video & composite) video adapter for your iMac as both original Studio Displays included AV imputs.
The newer style 15" 'Apple Studio Displays' are both digital (1st gen uses DVI, 2dn ADC). As far as I know there is no way to convert the analog VGA signal to digital (DVI or ADC).