Fourier transformation is an extensively studied problem in many research fields. It has many applications in machine learning, signal processing, compressed sensing, and so on. In many real-world applications, approximated Fourier transformation is sufficient and we only need to do the Fourier transform on a subset of coordinates. Given a vector $x \in \mathbb{C}^{n}$, an approximation parameter $ε$ and a query set $S \subset [n]$ of size $k$, we propose an algorithm to compute an approximate Fourier transform result $x'$ which uses $O(ε^{-1} k \log(n/δ))$ Fourier measurements, runs in $O(ε^{-1} k \log(n/δ))$ time and outputs a vector $x'$ such that $\| ( x' - \widehat{x} )_S \|_2^2 \leq ε\| \widehat{x}_{\bar{S}} \|_2^2 + δ\| \widehat{x} \|_1^2 $ holds with probability of at least $9/10$.