We present a calculation of the pion quark momentum fraction, $langle x rangle$, and its third Mellin moment $langle x^2 rangle$. We also obtain directly, for the first time, $langle x rangle$ and $langle x^2 rangle$ for the kaon using local operators. We use an ensemble of two degenerate light, a strange and a charm quark ($N_f=2+1+1$) of maximally twisted mass fermions with clover improvement. The quark masses are chosen so that they reproduce a pion mass of about 260 MeV, and a kaon mass of 530 MeV. The lattice spacing of the ensemble is 0.093 fm and the lattice has a spatial extent of 3 fm. We analyze several values of the source-sink time separation within the range of $1.12-2.23$ fm to study and eliminate excited-states contributions. The necessary renormalization functions are calculated non-perturbatively in the RI$$ scheme, and are converted to the $overline{rm MS}$ scheme at a scale of 2 GeV. The final values for the momentum fraction are $langle x rangle^pi_{u^+}=0.261(3)_{rm stat}(6)_{rm syst}$, $langle x rangle^K_{u^+}=0.246(2)_{rm stat}(2)_{rm syst}$, and $langle x rangle^K_{s^+}=0.317(2)_{rm stat}(1)_{rm syst}$. For the third Mellin moments we find $langle x^2 rangle^pi_{u^+}=0.082(21)_{rm stat}(17)_{rm syst}$, $langle x^2 rangle^K_{u^+}=0.093(5)_{rm stat}(3)_{rm syst}$, and $langle x^2 rangle^K_{s^+}=0.134(5)_{rm stat}(2)_{rm syst}$. The reported systematic uncertainties are due to excited-state contamination. We also give the ratio $langle x^2 rangle/langle x rangle$ which is an indication of how quickly the PDFs lose support at large $x$.
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