(* zero-th cosine fourier series coefficient *)
Subscript[c, 0] = (1/\[Pi]) Integrate[f[\[Beta]], {\[Beta], 0, \[Pi]}]
(* other cosine fourier series coefficients *)
Subscript[c, k] = (2/\[Pi]) Integrate[f[\[Beta]] Cos[k \[Beta]] , {\[Beta], 0, \[Pi]}]
(* l.h.s. approximation, very crude *)
Subscript[c, k] = (2/\[Pi]) Sum [f[Subscript[\[Beta], i]] Cos[
k Subscript[\[Beta], i]] \[CapitalDelta]\[Beta], {i, 1, N}]
\[CapitalDelta]\[Beta] = \[Pi]/N
Subscript[c, k] = (2/N) Sum[
f[Subscript[\[Beta], i]] Cos[k (i - 1) \[CapitalDelta]\[Beta]], {i,
1, N}]
Subscript[c, k] = (2/N) Sum[
f[Subscript[\[Beta], i]] Cos[\[Pi] k (i - 1)/N], {i, 1, N}]
(* k = L = 2 N *)
Subscript[c, L] = (1/N) Sum[f[Subscript[\[Beta], i]] Cos[2 \[Pi] (i - 1)], {i, 1, N}]
(* Cos[2 \[Pi] (i - 1)] === 1, so Subscript[c, L] has to have the same norm as \
Subscript[c, 0] which explains
the delta functions Subscript[\[Delta], k0] and the Subscript[\[Delta], kL] *)
I could not reproduce the
Subscript[\[Omega], k]
as seen in your inlet.
