I could use some advice on the following question.
I have a vector $\mathbf{x}=\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}$ and I want to use the standard inner product to determine the Fourier expansion of $\mathbf{x}$ with respect to basis $\mathbf{U} = \{u_1,u_2,u_3\}$ where $u_1, u_2, u_3$ are the three orthonormal vectors that span $\mathbf{U}$.
I understand that I can project $\mathbf{x}$ onto the space spanned by $\mathbf{U}$ by using a projection matrix of the form $\mathbf{P} =\mathbf{U}\mathbf{U}^{\text{T}}$. The projection would be $$ \mathbf{x}_P = \mathbf{Px} =\mathbf{U}\mathbf{U}^{\text{T}}\mathbf{x} = \mathbf{U}(\mathbf{U}^{\text{T}}\mathbf{x}) $$
where the coefficients $(\mathbf{U}^{\text{T}}\mathbf{x})$ are found by multiplying the transpose of each basis vector of $\mathbf{U}$ with $\mathbf{x}$ (i.e. taking the standard inner product of the basis vectors of $\mathbf{U}$ with $\mathbf{x}$).
What I do not follow is how to tie this to Fourier series. When I think Fourier series I think of a summation of sines and cosines. When I perform these calculations I get a set of scalars from the inner product and then that gets applied to my basis vectors in $\mathbf{U}$, none of which contain any trigonometric functions.
What am I missing? Surely there is a gap in my understanding here. If I haven't been clear please let me know.