DiscreteHankelTransform[f1_, p_, Np_, rmax_] :=
  Module[{a, aNp1, rv, uv, res, umax, T , J, F1, F2},
  a = Table[N[BesselJZero[p, n]], {n, 1, Np}];
  aNp1 = BesselJZero[p, Np + 1];
  umax = aNp1/(2 Pi rmax);
  rv = a/(2 Pi umax);
  uv = a/(2 Pi rmax);
  J = Abs[BesselJ[p + 1, a]];
  T = BesselJ[p, TensorProduct[a, a]/(2 Pi  rmax umax)]/(
  TensorProduct[J, J] Pi rmax umax) ;
  F1 = (If[MatchQ[Head[f1], List], f1, f1[rv]] rmax)/J;
  F2 = T.F1;
  Return[Transpose[{uv, J/umax F2}]];
];

InverseDiscreteHankelTransform[f2_, p_, Np_, umax_] :=
 Module[{a, aNp1, rv, uv, res, rmax, T , J, F1, F2},
 a = Table[N[BesselJZero[p, n]], {n, 1, Np}];
 aNp1 = BesselJZero[p, Np + 1];
 rmax = aNp1/(2 Pi umax);
 rv = a/(2 Pi umax);
 uv = a/(2 Pi rmax);
 J = Abs[BesselJ[p + 1, a]];
 T = BesselJ[p, TensorProduct[a, a]/(2 Pi rmax umax)]/(
  TensorProduct[J, J] Pi rmax umax) ;
 F2 = (If[MatchQ[Head[f2], List], f2, f2[uv]] umax)/J;
 F1 = T.F2;
 Return[Transpose[{rv, J/rmax F1}]];
];

f = Sin[2 Pi g #]/(2 Pi g #) &;
F0[u_, p_, g_] =
 Piecewise[{
   {HankelTransform[f, u, p, {p >= 0, 0 < u < g}], 0 <= u < g},
   {Sin[p ArcSin[g/u]]/(2 Pi g Sqrt[u^2 - g^2]), u > g},
   {Infinity, u == g}
  }, Null]
SetAttributes[F0, Listable];

dF01 = DiscreteHankelTransform[f /. g -> 5, 1, 256, 3.0];
dF04 = DiscreteHankelTransform[f /. g -> 5, 4, 256, 3.0];
dF01max = Max[Abs[dF01]];
dF04max = Max[Abs[dF04]];
dynerr1 = Transpose[{dF01[[;; , 1]], 20 Log10[Abs[F0[#[[1]], 1, 5] - #[[2]]]/dF01max] & /@dF01}];
dynerr4 = Transpose[{dF04[[;; , 1]], 20 Log10[Abs[F0[#[[1]], 4, 5] - #[[2]]]/dF04max] & /@dF04}];
GraphicsGrid[{
 {Show[
  Plot[F0[u, 1, 5], {u, 0, 20},
    PlotRange -> {-0.005, 0.05},
    Exclusions -> None,
    Frame -> True,
    Axes -> False,
    FrameLabel -> {"u", "!(*SubscriptBox[(f), (2)])[[Nu]]"},
    PlotLabel -> "HT, p[Equal]1"],
  ListPlot[dF01]],
  ListLinePlot[dynerr1,
   Frame -> True,
   Axes -> False,
   FrameLabel -> {"[Nu]", "dB"},
   PlotLabel -> "e([Nu]), p[Equal]1",
   PlotRange -> {{0, 20}, {-180, 0}}]},
{Show[
  Plot[F0[u, 4, 5], {u, 0, 20},
   PlotRange -> {-0.03, 0.03},
   Exclusions -> None,
   Frame -> True,
   Axes -> False,
   FrameLabel -> {"[Nu]", "!(*SubscriptBox[(f), (2)])[[Nu]]"},
   PlotLabel -> "HT, p[Equal]4"], ListPlot[dF04]],
  ListLinePlot[dynerr4,
   Frame -> True,
   Axes -> False,
   FrameLabel -> {"[Nu]", "dB"},
   PlotLabel -> "e([Nu]), p[Equal]4",
   PlotRange -> {{0, 20}, {-180, 0}}]}},
   ImageSize -> Large]