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I would like to interpolate the following function

 g[z_, e_] := 
     PDF[TruncatedDistribution[{e - 2, e + 2}, 
       NormalDistribution[e, 0.3]], z]

    a1 = N[g[-2, 0]];
    a2 = N[g[-1, 0]];
    a3 = N[g[0, 0]];
    a4 = N[g[1, 0]];
    a5 = N[g[2, 0]];

With your help I am now using:

 k[x_, e_] = 
      Interpolation[{{-2 + e, a1}, {-1 + e, a2},  {0 + e, 
     a3}, {1 + e, a4}, {2 + e, a5}}, 
   Method -> "Spline"] [x];

enter image description here I need e as an variable, which I can easily access. Can i somehow get rid of the error shown in the picture? Or do I have to stick to InterpolatingPolynomial? Can i somehow abuse InterpolatingFunction?

Why do i get negative values in the approximation function? Can i get rif of them with other, better methods?

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    $\begingroup$ Interpolation[data, Method -> "Spline"]? $\endgroup$ – vapor Mar 13 '17 at 11:47
  • $\begingroup$ have a look at 15879 $\endgroup$ – vapor Mar 13 '17 at 11:52
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    $\begingroup$ You can take derivatives with that function, and you can get a BSplineFunction with explicit parameters, what else do you want? $\endgroup$ – vapor Mar 13 '17 at 12:24
  • $\begingroup$ The suggestion of @happyfish works for me: i.stack.imgur.com/3horZ.png $\endgroup$ – Michael E2 Mar 13 '17 at 12:33
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    $\begingroup$ as to why you get negative values, its because you are doing a polynomial interpolation of something that's not a polynomial, with far too few points. You might have a look at FunctionInterpolation, or set InterpolationOrder->1 $\endgroup$ – george2079 Mar 13 '17 at 15:18
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This is where a monotonic interpolation method like Steffen's interpolation or Fritsch-Carlson interpolation is useful. Using the routine from this answer:

k[x_, e_] := SteffenInterpolation[{{-2 + e, a1}, {-1 + e, a2}, {0 + e, a3},
                                   {1 + e, a4}, {2 + e, a5}}][x]

Plot[k[x, 0], {x, -2, 2}]

monotonic interpolant

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