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It seems like none of the answers I see up to now are actually producing heat maps. The difference between a heat map and a ListDensityPlot is important. In Mathematica vocabulary, the heat map is a SmoothDensityHistogram.

First of all, I tried to directly use the 'heatMap' function in my answer. I just tried it with the data in that post:

data = RandomReal[1, {100, 2}];

Show[heatMap[data, "Points" -> 300, "Radius" -> {10, .02}, 
  PlotRange -> {{0, 1}, {0, 1}}, 
  ColorFunction -> ColorData["Rainbow"]], Graphics[Point@data], 
 PlotRange -> {{0, 1}, {0, 1}}]

smearedsquare

All I did is to specify a tuple {10, .02} for the "Radius" option. Its first entry is the radius in the vertical direction, and with a choice of 10 this smears all the data out over the entire vertical image range.

This shows it works without modifying the code. But of course I have to tweak the function in order to make it look more "one-dimensional":

heatMap[data_, opts : OptionsPattern[]] := 
 Module[{n, size, xRange, pr}, 
  n = "Points" /. {opts} /. {"Points" -> 100};
  pr = PlotRange /. {opts} /. {PlotRange :> 
      Map[{Min[#], Max[#]} &, Transpose[data]]};
  xRange = -Subtract @@ pr[[1]];
  size = Floor[
    n ("Radius" /. {opts} /. {"Radius" -> xRange/6})/xRange];
  Graphics[
   {Inset[
     ArrayPlot[
      Rescale@GaussianFilter[
        ImageData@ColorNegate@ColorConvert[
           Rasterize[
            Graphics[
             Point[data],
             Background -> White,
             PlotRangePadding -> 0,
             ImagePadding -> 0,
             ImageMargins -> 0,
             PlotRange -> pr
             ],
            "Image",
            ImageSize -> n
            ],
           "GrayScale"
           ],
        {3 size, size},
        Padding -> 0
        ],
      ColorFunction -> (ColorFunction /. {opts} /. {ColorFunction -> 
           ColorData["LakeColors"]}),
      ImagePadding -> 0,
      PlotRangePadding -> 0,
      Frame -> False
      ],
     pr[[All, 1]],
     {0, 0}, xRange]},
   PlotRange -> pr,
   Frame -> True,
   FrameTicks -> {Automatic, None}
   ]
  ]

So here I removed the PlotRangePadding and the FrameTicks on the left side, as well as the PlotRangePadding. I think that's all you need to change. Having collapsed the data onto a single axis, the vertical smearing for GaussianFilter needs to be only of order 1 (in relation to the horizontal axis) - so that's what I used. Then I set the PlotRange appropriately and get this:

data = RandomReal[{0, 1}, 100];
data = {#, 0} & /@ data;

heatMap[data, "Points" -> 300, "Radius" -> {1, .02}, 
 PlotRange -> {{0, 1}, {0, .04}}, PlotRangePadding -> 0, 
 FrameLabel -> None]

heatmapThin

The meaning of the option "Points" (number of horizontal sampling points) is the same as described in the linked post.