The apple IIgs, like many computers from the 80's and early 90's, has a analog RGB output at a 15KHz horizontal frequency (same as NTSC/PAL). The majority of modern monitors and TVs will not sync to anything below 31.5KHz on their VGA input. If you have a TV with a SCART connector it is possible to wire the RGB directly to the TV. Most TVs in USA, however, use the YPbPr component instead of SCART making a direct connection not possible. GS2HD converts the IIgs RGB output to a YPbPr signal compatible with HDTVs with a component input.
The circuit is pretty straightforward. U1A, U2A and U3A buffer and amplify the RGB signals (standard non inverting configuration with a gain of 2). U1B, U2B and U3B perform the matrix transformation from RGB to YPbPr according to the standard coefficients for SDTV.
Y = 0.299 * R + 0.587 * G + 0.114 * B
Pb = -0.169 * R - 0.331 * G + 0.500 * B
Pr = 0.500 * R - 0.419 * G - 0.081 * B
U5, L1, L2 and the associated capacitors provide a cleaned up power rails for the sensitive video circuits.
The basic circuit will work with pretty much any operational amplifier. However to achieve good video quality a high speed opamp optimized for video applications is required. Suitable devices are Analog Devices AD8056ANZ and Texas Instruments LM6172IN.