This paper presents an approximate Gaussian filter which can run in one-pass with high accuracy based on spectrum sparsity. This method is a modification of the cosine integral image (CII), which decomposes a filter kernel into few cosine terms and convolves each cosine term with an input image in constant time per pixel by using integral images and look-up tables. However, they require much workspace and high access cost. The proposed method solves the problem with no decline in quality by sequentially updating sums instead of integral images and by improving look-up tables, which accomplishes a one-pass approximation with much less workspace. A specialization for tiny kernels are also discussed for faster calculation. Experiments on image filtering show that the proposed method can run nearly two times faster than CII and also than convolution even with small kernel.