h={0,15, 2, 8, 2, 7, 37, 20, 56, 126, 75}

x={0., 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.}

tempf = Table[(i - j), {i, 0, 1, 0.1}, {j, 0, 1, 0.1}]
 tempf2=tempf  /. x_ /; x < 0 -> 0;
 tempf3=tempf2  /. x_ /; x > 1/2 -> 0;
 F=tempf/. {0.1 -> f1, 0.2 -> f2, 0.3 -> f3, 0.4 -> f4, 0.5 -> f5}

tempg = Round[Table[j, {j, 0, 1, 0.1}], 0.01];
 g = tempg /. {0. -> g1, 0.1 -> g2, 0.2 -> g3, 0.3 -> g4, 0.4 -> g5,
 0.5 -> g6, 0.6 -> g7, 0.7 -> g8, 0.8 -> g9, 0.9 -> g10, 1. -> g11}

Solve[ h == F.g
&&g1>=0&& g2>=0&& g3>=0&& g4>=0&& g5>=0&& g6>=0&& g7>=0&& g8>=0&&g9>=0&& g10>=0&& g11>=0
  &&f1>=0&& f2>=f1&& f3>=f2&& f4>=f3&& f5>=f4&& 1>=f5,
 {f1, f2, f3, f4, f5, g1, ge2, g3, g4, g5, g6, g7, g8, g9, g10, g11}]

FindInstance[
h - F.g < 0.001 && g1 >= 0 && g2 >= 0 && g3 >= 0 && g4 >= 0 &&
g5 >= 0 && g6 >= 0 && g7 >= 0 && g8 >= 0 && g9 >= 0 && g10 >= 0 &&
g11 >= 0 && f1 >= 0 && f2 >= f1 && f3 >= f2 && f4 >= f3 &&
f5 >= f4 && 1 >= f5
, {f1, f2, f3, f4, f5, g1, g2, g3, g4, g5, g6, g7, g8, g9, g10,
g11}]

{f1 -> 1., f2 -> 1., f3 -> 1., f4 -> 1., f5 -> 1., g1 -> 16.,
g2 -> 0, g3 -> 0, g4 -> 0, g5 -> 0, g6 -> 128., g7 -> 0, g8 -> 0,
g9 -> 0, g10 -> 0, g11 -> 1.}

F .g /. {f1 -> 1., f2 -> 1., f3 -> 1., f4 -> 1., f5 -> 1.,
g1 -> 16., g2 -> 0, g3 -> 0, g4 -> 0, g5 -> 0, g6 -> 128.,
g7 -> 0, g8 -> 0, g9 -> 0, g10 -> 0, g11 -> 1.}

{16., 16., 16., 16., 16., 128., 128., 128., 128., 128.}