MECfda (version 0.2.1)

numeric_basis-class: Numeric representation of a function basis

Description

A s4 class that numerically represents a basis of linear space of function.
\(\{\rho_k\}_{k=1}^\infty\) denotes a basis of function linear space. Some times the basis cannot be expressed analytically. But we can numerically store the space by the value of a finite subset of the basis functions at some certain points in the domain, \(\rho_k(t_j), k = 1,\dots,p, j = 1,\dots,m\). The s4 class is to represent a finite sequence of functions by their values at a finite sequence of points within their domain, in which all the functions have the same domain and the domain is an interval.

Slots

basis_function

matrix of the value of the functions, \((\zeta_{jk})_{m\times p}\), where \(\zeta_{ik} = \rho_k(t_j), j = 1,\dots,m, k = 1,\dots,p\). Each row of the matrix is corresponding to a point of \(t\). Each column of the matrix is corresponding to a basis function.

t_points

a numeric atomic vector, represents the points in the domains of the function where the function values are taken. The \(j\)th element is corresponding to \(j\)th row of slot basis_function.

t_0

left end of the domain interval.

period

length of the domain interval.

Author

Heyang Ji

Details

The units of a basis of a linear space should be linearly independent. But the program doesn't check the linear dependency of the basis function when a numeric_basis object is initialized.

Examples

t_0 = 0
period = 1
t_points = seq(0.05,0.95,length.out = 19)
numeric_basis(
  basis_function = cbind(1/2,cos(t_points),sin(t_points)),
  t_points       = t_points,
  t_0            = t_0,
  period         = period
)