Writing a Fourier Series Equation

Question

I am trying to program a Fourier Series equation that I will use for a normed fit that is not a least squares fit. If I write

f[t_]:= a[[1]]/2 + Sum[a[[2 k]] Sin[2 k π t] + a[[2 k + 1]] Cos[2 k π t], {k, 1, kMax}]

the equation compiles. However, I cannot use it later. I believe the problem is that I have not defined the coefficients correctly.

Can anyone tell me how to define this equation? I want to create a list of the function at points on an interval for which I have other data to be fitted, so I can create an error function and minimize for the best fit coefficients. I am trying to create the function with unknown coefficients, so I can use a range of norm functions.

Answer 1

You could define a function for a[n] and b[n] and apply the definition of Fourier series. Here is a quick example

Suppose you want to find F.S. of this function

ClearAll[t, n, f];
T0 = 1;(*period*)
f[t_] := Piecewise[{{t, 0 < t < 1/2}, {1/2, 1/2 < t < T0}, {0, True}}];
Plot[f[t], {t, 0, T0}, PlotLabel -> "f(t)"]

Then

an[n_] = If[n == 0, 1/(T0) Integrate[f[t], {t, 0, T0}], 
   2/T0 Integrate[f[t] Cos[n 2 Pi/T0 t], {t, 0, T0}]];
bn[n_] = 2/T0 Integrate[f[t] Sin[n 2 Pi/T0 t], {t, 0, T0}];
f[t_, max_] := 
  an[0] + Sum[ 
    an[n] Cos[n 2 Pi/T0 t] + bn[n] Sin[n 2 Pi/T0 t], {n, 1, max}];

Now plot it for 20 terms

nTerms = 20;
Show[Plot[f[t], {t, 0, T0}, PlotStyle -> {Red}], 
 Plot[Evaluate@f[t, nTerms], {t, -T0, T0}], PlotRange -> All]

or 50 terms for better approximation

nTerms = 40;
Show[Plot[f[t], {t, 0, T0}, PlotStyle -> {Red}], 
 Plot[Evaluate@f[t, nTerms], {t, -T0, T0}], PlotRange -> All]

btw, Fourier series support is already build-in Mathematica, so you do not have to implement it, and it has better performance than the above, which becomes slow for large number of terms.