The numerical method based on the fast Fourier transform (FFT) is generally applied to calculate the Fresnel diffraction field, which would suffer from sampling constraints. To break this limit, in this Letter, the semi-analytic Fresnel diffraction calculation method is proposed based on polynomial decomposition. The diffraction field is computed by using properly analytic Fresnel diffraction basis functions (FDBFs) according to the application requirements. Analytic FDBF is calculated based on Legendre or Chebyshev polynomials by using the object-domain frequency division multiplexing method. The proposed method offers arbitrary sampling, high-flexibility, and high-accuracy diffraction calculation in the full Fresnel region. The computational efficiency and accuracy of the proposed method are compared with FFT-based methods. It has potential application in light field analysis, wavefront sensing, and image processing.