innerProduct {mrct}

Pairwise inner product for L^2 functions

Description

Calculate all pairwise inner products between elements from L^2 supplied to this function. The integral is approximated by the Trapezoidal rule for uniform grids:

\int_a^b f(x) dx \approx \Delta x \left( \sum_{i=1}^{N-1} f(x_i) + \frac{f(x_N) - f(x_0)}{2} \right)

whereas \{x_i \} is an uniform grid on [a,b] such that a = x_0 < x_1 < \ldots < x_N = b and \Delta x the step size, i.e. \Delta x := x_2 - x_1. Therefore, it is assumed that the functions are evaluated at the same, equidistant grid.

Usage

innerProduct(grid, data)

Arguments

grid

A numeric vector of the uniform grid on which the functions are evaluated.

data

A numeric matrix. Each function has to be a vector stored in a column of data and evaluated at the points of grid. Thus, the number of rows and columns of data correspond to length(grid) and the number of functions, respectively.

Value

Numeric symmetric matrix containing the approximated pairwise inner products between the functions supplied by data. The entry (i,j) of the result is the inner product between the i-th and j-th column of data.

Examples

# Create orthogonal fourier basis via `fdapace` package
library(fdapace)
basis <- fdapace::CreateBasis(K = 10,
                              type = "fourier")
iP <- innerProduct(grid = seq(0, 1, length.out = 50), # default grid in CreateBasis()
                   data = basis)
round(iP,3)
# Since the basis is orthogonal, the resulting matrix will be the identity matrix.