(* 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.