We introduce a new (amortized) functional bootstrapping framework over the CKKS homomorphic encryption (HE) scheme based on Fourier extension. While approximating the modular reduction function in CKKS bootstrapping through Fourier series is a well-known technique, how such method can be efficiently generalized to functional bootstrapping is less understood. In this work, we show that, by constructing proper Fourier extensions, any function with a bounded domain in the smoothness class \(C^{\kappa }\) can be approximated by a degree-n Fourier series with errors of order \(\mathcal {O}(n^{-\kappa -2})\) (except at the singularities), improving on previous results on a global error bound of \(\mathcal {O}(n^{-1})\) (Alexandru et al. Crypto’25). To achieve such bound, we propose a new way of constructing Fourier extensions, such that the extended functions appear as smooth as possible in the sense of a Fourier approximation. By implementing our functional bootstrapping over OpenFHE, we demonstrate that we can improve the data precision by 10–27 bits and reduce the amortized FBS latency by \(1.1\times \) – \(2\times \) over a variety of benchmarking functions.