I am told that Fourier showed that we can represent an arbitrary continuous function, $f(x)$, as a convergent series in the elementary trigonometric functions
$$f(x) = \sum_{k = 0}^\infty a_k \cos(kx) + b_k \sin(kx)$$
Also, suppose that $\{\phi_n(x)\}^\infty_{n = 0}$ is a set of orthogonal functions with respect to a weight function $w(x)$ on the interval $(a, b)$. And let $f(x)$ be an arbitrary function defined on $(a, b)$. Then the generalised Fourier series is
$$f(x) = \sum_{k = 0}^\infty c_k \phi_k (x)$$
I have the following questions relating to this:
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How does a Fourier $\sin$/$\cos$ series arise from a "normal" Fourier series $f(x) = \sum_{k = 0}^\infty a_k \cos(kx) + b_k \sin(kx)$?
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How does this relate to the generalised Fourier series $f(x) = \sum_{k = 0}^\infty c_k \phi_k (x)$?
I would greatly appreciate clarification on this.
EDIT: When I say Fourier $\sin$/$\cos$ Series, I'm referring to what is known as "Fourier sine series" and "Fourier cosine series".
Best Answer
The "normal" Fourier series is simply a specific case of the generalized Fourier series for which $$\phi_k = \{\sin(kx),\cos(kx)\},\ w(x) = 1$$ where $k$ is appropriately defined based on the domain of $f$ precisely to make each function in the family orthogonal to each other.
Consequently, the coefficients $\{a_k, b_k\}$ of the "normal" Fourier series are calculated in the same way as the generalized ones $c_k$, through an inner product with the "target" function $f$.
The Fourier sine and cosine series thus appear as individual "parts" of the "normal" Fourier series since
$$f(x) = \sum_{k = 0}^\infty a_k \cos(kx) + b_k \sin(kx) = \sum_{k = 0}^\infty a_k \cos(kx) + \sum_{k = 0}^\infty b_k \sin(kx)$$
and only one of them may be all that is required to approximate $f$ when either $a_k$ or $b_k$ are collectively $0$. This is equivalent to saying that $f(x)$ is orthogonal to either all $\sin(kx)$ or all $\cos(kx)$ on the function domain.
Such a situation can usually be qualitatively deduced before inner products are calculated; for example, a symmetric function $f(x) = f(-x)$ defined on a symmetric domain $(-a, a)$ will have $b_k = 0$ and thus can be fully approximated with a Fourier cosine series. This is described here and here for sine and cosine series respectively.