Lebesgue Constant

Estimates the Lebesgue constant.

lebesgue(x, refine = 4, plotting = FALSE)

Arguments

x

numeric vector of grid points

refine

refine the grid with 2^refine grid points; can only be an integer between 2 and 10, default 4.

plotting

shall the Lebesgue function be plotted.

Details

The Lebesgue constant gives an estimation \(||P_n f|| \le L ||f||\) (in minimax norm) where \(P_n f\) is the interpolating polynomial of order \(n\) for \(f\) on an interval \([a, b]\).

Value

Lebesgue constant for the given grid points.

References

Berrut, J.-P., and L. Nick Trefethen (2004). “Barycentric Lagrange Interpolation”. SIAM Review, Vol. 46(3), pp.501–517.

Note

The Lebesgue constant plays an important role when estimating the distance of interpolating polynomials from the minimax solution (see the Remez algorithm).

See also

barylag

Examples

lebesgue(seq(0, 1, length.out = 6))  #=> 3.100425
#> [1] 3.100425