series expansion - Chebyshev Approximation

Is there functionality in Mathematica to expand a function into a series with Chebyshev polynomials?

The Series function only approximates with Taylor series.

Answer

You can just take Bob Hanlon's answer from 2006 directly, and modify the plot just a bit to update it.

ChebyshevApprox[n_Integer?Positive, f_Function, x_] := 
  Module[{c, xk}, xk = Pi (Range[n] - 1/2)/n;
   c[j_] = 2*Total[Cos[j*xk]*(f /@ Cos[xk])]/n;
   Total[Table[c[k]*ChebyshevT[k, x], {k, 0, n - 1}]] - c[0]/2];

f = 3*#^2*Exp[-2*#]*Sin[2 #*Pi] &;

ChebyshevApprox[3, f, x] // Simplify


((-(3/4))*((-E^(2*Sqrt[3]))*(Sqrt[3] - 2*x) - 2*x - Sqrt[3])*x*
   Sin[Sqrt[3]*Pi])/E^Sqrt[3]

GraphicsGrid[
 Partition[
  Table[Plot[{f[x], ChebyshevApprox[n, f, x]}, {x, -1, 1}, 
    Frame -> True, Axes -> False, PlotStyle -> {Blue, Red}, 
    PlotRange -> {-2, 10}, 
    Epilog -> Text["n = " <> ToString[n], {0.25, 5}]], {n, 9}], 3], 
 ImageSize -> 500]

Comments

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