Quantum fourier transform

How would the circuit look like for only 1 qubit?

For only 1 qubit the QFT algorithm contains only one Hadamard gate.

In the 2 qubits system, what would be the result of the following expression?

QFT₂( |0> + |2> )

Actually,

QFT₂ ( |0> +  |2> ) = ( |0> +  |2> )

as a middle step, you can show that

QFT2|0> = 1/2 ( |0> +  |1>  + |2> +  |3> )

and

QFT2|2> = 1/2 ( |0> -  |1>  + |2> -  |3> )