one might dispute that faking it can be as difficult as making it 'for real', but here we go.
How to do it depends on your constraints.
a) Level playing field or not ? i.e. are there platforms on different levels, or just one long flat like Moon Patrol?
b) Fixed FrameRate or not ? this decides whether you can force the position without using a Timer.
Examples: (numbers are just off the top of my head )
take Moon Patrol example, long flat ground, fixed frame rate:
one can just say, buggy will be at +3, +8, +10, +8, +3 heights along the jump.
1) decide the time a jump will take in frames, say 24 frames
2) make the array have values for each frame until the jump is complete.
3) modify the height the sprite is drawn at according to how many frames have passed since starting the jump.
take mario example, platforms with no fixed frame rate.
one can say, jump upwards power is + 10 pixels.
mark the starting time, and on every update modify the upvector by a gravity value, say -10 pixels per second.
cap the value to -30 pixels or something so you dont max out the falling speed over your tilesize.
again, with feeling:
1) start the jump
2) update, add the jump power to the sprite, then subtract the (gravity * time) from the jump power.
3) update, repeat. ( and let user continue to move left or right as normal )
4) dont forget to check for landing.
This is not really very far from 'real' jumping...
I might even venture to say for many games faked jumping is better than the 'real gravity' simulation...
Even a lot of modern FPS have jumping that feels like you are popping and floating, quite clumsy.
Maybe watch Osamu Tezuka's animation 'Jumping' for more inspiration
[EDIT] lol totally forgot that main point there...