Writing a Fourier Series Equation

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.

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You need to define your function so that it is dependent on both the list a and kMax, as well as t.
– J. M.♦
Oct 6 at 3:20

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1 Answer
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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.

  

 

Thanks for responding. Regrettably, I’m not trying to do standard Fourier series. I am working on an interval that is not a direct overlap of the data range, and I have re-derived the theory slightly to modify for the mapping. In addition, I will not use a least squares fit because I need a robust set of norms.
– JimT
Oct 7 at 16:31

  

 

The result is that I am not able to use the standard capabilities built into Mathematica which give the standard derivation of the coefficients and use least squares norms when fitting. Unfortunately, I am still learning how to build functions correctly. I manually derived a version of my equations and programmed the matrix solutions into Mathematica, but the matrices are large and a little unstable, hence my desire to move to alternative norms. Now I need to learn how to program the series expansion in an arbitrary form.
– JimT
Oct 7 at 16:31

  

 

Still getting used to function definition in Mathematica, so any help on an generalized equation form would be great. Thanks.
– JimT
Oct 7 at 16:31