  We reinterpret the shear estimator developed by Zhang & Komatsu (2011) within
the framework of Shapelets and propose the Fourier Power Function Shapelets
(FPFS) shear estimator. Four shapelet modes are calculated from the power
function of every galaxy's Fourier transform after deconvolving the Point
Spread Function (PSF) in Fourier space. We propose a novel normalization scheme
to construct dimensionless ellipticity and its corresponding shear responsivity
using these shapelet modes. Shear is measured in a conventional way by
averaging the ellipticities and responsivities over a large ensemble of
galaxies. With the introduction and tuning of a weighting parameter, noise bias
is reduced below one percent of the shear signal. We also provide an iterative
method to reduce selection bias. The FPFS estimator is developed without any
assumption on galaxy morphology, nor any approximation for PSF correction.
Moreover, our method does not rely on heavy image manipulations nor complicated
statistical procedures. We test the FPFS shear estimator using several HSC-like
image simulations and the main results are listed as follows. (i) For
simulations which only contain isolated galaxies, the amplitude of
multiplicative bias is below 1 % . (ii) For more realistic simulations which
also contain blended galaxies, the blended galaxies are deblended by the first
generation HSC deblender before shear measurement. Multiplicative bias of
(-5.71 +- 0.31) % is found. The blending bias is calibrated by image
simulations. Finally, we test the consistency and stability of this
calibration.
