Gaussian smoothing and its first and second differentials are very important for image processing and computer vision. We already have methods based on the sliding Fourier transform (SFT) in order to calculate Gaussian smoothing efficiently. However, errors in floating point calculation accumulate quickly when we use the single-precision floating-point format. Also, the resultant discontinuities at the edges of SFT intervals generate false extreme points easily. To resolve the abovementioned problems, we propose the attenuated SFT (ASFT) by introducing a decay factor to the original SFT together with a new criterion for determining coefficients to effectively approximate Gaussian function and its differentials. Extensive experiments demonstrate a decisive superiority of the proposed ASFT over the original SFT.