We discuss the physical nature of quasi-PDFs, especially the reasons for the strong nonperturbative evolution pattern which they reveal in actual lattice gauge calculations. We argue that quasi-PDFs may be treated as hybrids of PDFs and the rest-frame momentum distributions of partons. The latter is also responsible for the transverse momentum dependence of TMDs. The resulting convolution structure of quasi-PDFs necessitates using large probing momenta $p_3 gtrsim 3$ GeV to get reasonably close to the PDF limit. To deconvolute the rest-frame distribution effects, we propose to use a method based directly on the coordinate representation. We treat matrix elements $M(z_3,p_3)$ as distributions ${cal M} ( u, z_3^2)$ depending on the Ioffe-time $ u = p_3 z_3$ and the distance parameter $z_3^2$. The rest-frame spatial distribution is given by ${cal M} (0, z_3^2)$. Using the reduced Ioffe function ${mathfrak M} ( u, z_3^2) equiv {cal M} ( u, z_3^2)/ {cal M} (0, z_3^2)$ we divide out the rest frame effects,including the notorious link renormalization factors. The $ u$-dependence remains intact and determines the shape of PDFs in the small $z_3$ region. The residual $z_3^2$ dependence of the ${mathfrak M} ( u, z_3^2)$ is governed by perturbative evolution. The Fourier transform of ${cal M} ( u, z_3^2)$ produces pseudo-PDFs ${cal P}(x, z_3^2)$ that generalize the light-front PDFs onto spacelike intervals. On the basis of these findings we propose a new method for extraction of PDFs from lattice calculations.
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