For a function \(f(x), x\in\Omega\), and a basis function sequence \(\{\rho_k\}_{k\in\kappa}\), basis expansion is to compute \(\int_\Omega f(t)\rho_k(t) dt\). Here we do basis expansion for all \(f_i(t), t\in\Omega = [t_0,t_0+T]\) in functional variable data, \(i=1,\dots,n\). We compute a matrix \((b_{ik})_{n\times p}\), where \(b_{ik} = \int_\Omega f(t)\rho_k(t) dt\). The basis used here is the Fourier basis, $$\frac{1}{2},\ \cos(\frac{2\pi}{T}k[x-t_0]),\ \sin (\frac{2\pi}{T}k[x-t_0])$$ where \(x\in[t_0,t_0+T]\) and \(k = 1,\dots,p_f\).
fourier_basis_expansion(object, order_fourier_basis)# S4 method for functional_variable,integer
fourier_basis_expansion(object, order_fourier_basis)
Returns a numeric matrix, \((b_{ik})_{n\times p}\), where \(b_{ik} = \int_\Omega f(t)\rho_k(t) dt\).
a functional_variable class object.
the order of Fourier basis, \(p_f\).
Heyang Ji