Fourier Series of Cosh

With the given function I can plot:

CoshFou[z_, s_] := 
  Block[{a0, aN, Fun}, 
   a0 = 2/(2 b) NIntegrate[Cosh[gS[[s]] zz], {zz, -b, b}];
   aN = Table[
     2/(2 b) NIntegrate[
       Cosh[gS[[s]] zz] Cos[(2 Pi nIt zz)/(2 b)], {zz, -b, b}], {nIt, 
      nMax}];
   Fun = a0/2 + 
     Sum[aN[[nIt]] Cos[(2 Pi nIt z)/(2 b)], {nIt, nMax - 1}]];

Plot[Sum[CoshFou[z, s], {s, 1, Length@gS}], {z, -1, 1}]

This shows already the rapid divergence and figures up to $10^303$.

For a proper definition of the Fourier cosine series look up FourierCosSeries.

The function is a sum of cosine hyperbolicus at different frequencies. This can not be meant as frequencies of the coefficients of the Fourier cosine series. Mathematica allows to make use of the nMax parameter in the built-in function. There is no such periodic meaning like a cosine hyperbolicus transform.

FourierCosSeries[Sum[Cosh[gS[[s]] t], {s, 1, Length@gS}], t, 8]

(* 31.724 - (32.4347 - 3.2029810^-16 I) Cos[ t] + (14.5598 - 1.4135810^-16 I) Cos[ 2 t] - (7.65296 + 3.9007910^-16 I) Cos[ 3 t] + (4.60424 + 1.2913110^-16 I) Cos[ 4 t] - (3.04553 - 9.6337610^-16 I) Cos[ 5 t] + (2.15436 - 1.7638910^-16 I) Cos[ 6 t] - (1.60082 - 4.3831510^-16 I) Cos[ 7 t] + (1.23477 + 1.9865610^-16 I) Cos[8 t] *)

ReImPlot[%, {t, -[Pi], [Pi]}]

For the calculation of such coefficients there is not the real big interval important but the periodizity of the cosine function. So the coefficient are not calculated on the large interval but on the periodizity interval of the cosine. That keeps the the coefficients small and the integration short. To be valid on the large interval extrapolation can be hoped. In most cases this is for divergent function not the case. The Fourier transforms make the functions periodic.

If the series is dropped there is the integral Fourier transform: FourierTransform. FourierTransform[Sum[Cosh[gS[[s]] t], {s, 1, Length@gS}], t, [Omega]] (* fails *)

that makes use of the infinite real axis and thereby for approximations big intervals.

There is main value definition: $$sqrt{frac{vert bvert}{(2pi)^{1-a}}}int_{-infty}^{infty}f(t)e^{i b omega t}dt$$

Integrate[
 Sum[Cosh[gS[[s]] t], {s, 1, Length@gS}] Exp[i b [Omega] t], {t, -b, 
  b}]

(* (-(9.669354397782823*^303/(-0.00282744 + 1.*i*[Omega])) - 7.514651931406424*^269/
    (-0.00251328 + 1.*i*[Omega]) - 5.840099693021041*^235/(-0.00219912 + 1.*i*[Omega]) - 
   4.550062773378836*^201/(-0.0018849700000000001 + 1.*i*[Omega]) - 
   3.5379030434103284*^167/(-0.001570812 + 1.*i*[Omega]) + 
   (1.9733621830656194*^179 + 3.140496185412215*^182*i*[Omega] - 
     2.345487709898561*^186*i^2*[Omega]^2 - 3.732713269301929*^189*i^3*[Omega]^3 + 
     3.655743265116067*^192*i^4*[Omega]^4 + 5.817912128582447*^195*i^5*[Omega]^5 - 
     1.424544082331069*^198*i^6*[Omega]^6 - 2.2670826951605273*^201*i^7*[Omega]^7)/
    (5.46950432817499*^-26 - 3.5264830524668625*^-38*i*[Omega] - 
     7.886171420873058*^-19*i^2*[Omega]^2 - 2.9582283945787943*^-31*i^3*[Omega]^3 + 
     2.659733639289811*^-12*i^4*[Omega]^4 + 6.203854594147708*^-25*i^5*[Omega]^5 - 
     2.9610912131639997*^-6*i^6*[Omega]^6 + 2.168404344971009*^-19*i^7*[Omega]^7 + 1.*i^8*[Omega]^8) + 
   E^(500000.*i*[Omega])*(3.5379030434103284*^167/(0.001570812 + 1.*i*[Omega]) + 
     4.550062773378836*^201/(0.0018849700000000001 + 1.*i*[Omega]) + 
     5.840099693021041*^235/(0.00219912 + 1.*i*[Omega]) + 7.514651931406424*^269/
      (0.00251328 + 1.*i*[Omega]) + 9.669354397782823*^303/(0.00282744 + 1.*i*[Omega])))/
  E^(250000.*i*[Omega]) *)

This has to be multiplicated with the leading factor still.

if $infty$ if shortened to $b$ this makes sense. But for that nMax is meaningless. From this approximated integral the nMax gets the meaning back. Some common choices for {a,b} are {0,1} (default; modern physics), {1,-1} (pure mathematics; systems engineering), {-1,1} (classical physics), and {0,-2Pi} (signal processing).