Fourier series, the rate of convergence is not the
best possible for the application of approximating the mod function
in small intervals around multiples of the modulus.
We show a different sine series, such that the
sine series of order n has error O(epsilon^(2n+1)) for approximating
the mod function in epsilon-sized intervals around multiples of the modulus.
Moreover, the resulting polynomial, after Taylor series approximation of the
sine series, has small coefficients, and the whole polynomial can be computed
at a precision that is only slightly larger than
-(2n+1)log epsilon, the precision of approximation being sought. This polynomial can then be used to approximate the mod function to almost arbitrary precision,
and hence allows practical CKKS-HE bootstrapping with arbitrary precision.