Photometric Objects Around Cosmic Webs (PAC) Delineated in a Spectroscopic Survey. III. Accurate Measurement of Galaxy Stellar Mass Function with the Aid of Cosmological Redshift Surveys

We present a novel method to accurately measure the galaxy stellar mass function (GSMF) based upon the Photometric objects Around Cosmic webs (PAC) method developed in our first paper (Paper I) of the series. The method allows us to measure the GSMF to a lower mass end that is not accessible to the spectroscopic sample used in the PAC. Compared with Paper I, the current measurement of GSMF is direct and model independent. We measure the GSMFs in the redshift ranges of $z_s<0.2$, $0.2<z_s<0.4$ and $0.5<z_s<0.7$ down to the stellar mass $M_{\ast }={10}^{8.2}$, ${10}^{10.6}$ and ${10}^{10.6}{M}_{\odot }$, using the data from the DESI Legacy Imaging Surveys and the spectroscopic samples of Slogan Digital Sky Survey (i.e. Main, LOWZ and CMASS samples). Our results show that there is no evolution of GSMF from $z_s=0.6$ to $z_s=0.1$ for $M_{\ast }>{10}^{10.6}{M}_{\odot }$, and that there is a clear up-turn at $M_{\ast }\approx {10}^{9.5}{M}_{\odot }$ towards smaller galaxies in the local GMSF at $z_s=0.1$. We provide an accurate double Schechter fit to the local GSMF for the entire range of $M_{\ast }$ and a table of our measurements at the three redshifts, which can used to test theories of galaxy formation. Our method can achieve an accurate measurement of GSMF to the stellar mass limit where the spectroscopic sample is already highly incomplete (e.g. $\sim {10}^{-3}$) for its target selection.