Unreasonable number of Chebyshev coefficients required to approximate certain Gaussians

[johnbcoughlin]

The following function requires north of 2 million Chebyshev coefficients to represent: Fun(x -> exp(-12(x + 4.9)^2), Chebyshev(-12..12))

This is in contrast to the function's "neighbors", see as follows:

for x0 in 4.7:0.05:5.1
    fun = Fun(x -> exp(-(x + x0)^2 / (1 / 12.0)), Chebyshev(-12..12))
    @show x0, length(fun.coefficients)
end

(x0, length(fun.coefficients)) = (4.7, 144)
(x0, length(fun.coefficients)) = (4.75, 144)
(x0, length(fun.coefficients)) = (4.8, 142)
(x0, length(fun.coefficients)) = (4.85, 143)
(x0, length(fun.coefficients)) = (4.9, 2097152)
(x0, length(fun.coefficients)) = (4.95, 141)
(x0, length(fun.coefficients)) = (5.0, 142)
(x0, length(fun.coefficients)) = (5.05, 142)
(x0, length(fun.coefficients)) = (5.1, 142)

[jishnub]

julia> 2^21
2097152

This indicates that the approximation didn't converge (not entirely sure why). I'll try to look into this.

[macd]

I have noticed similar behavior with abs if you try to use the constructor by passing in a function. If , instead you create a Fun for x on your domain and then write your desired Fun in terms of x, all is well. This should help to narrow down the problem? Here is an example:

julia @v1.11> x = Fun()
Fun(Chebyshev(), [0.0, 1.0])

julia @v1.11> x = Fun(Chebyshev(-12..12))
Fun(Chebyshev(-12 .. 12), [0.0, 12.0])

julia @v1.11> for x0 in 4.7:0.05:5.1
                  fun = exp(-(x + x0)^2 / (1 / 12.0))
                  @show x0, length(fun.coefficients)
              end

(x0, length(fun.coefficients)) = (4.7, 487)
(x0, length(fun.coefficients)) = (4.75, 487)
(x0, length(fun.coefficients)) = (4.8, 486)
(x0, length(fun.coefficients)) = (4.85, 485)
(x0, length(fun.coefficients)) = (4.9, 484)
(x0, length(fun.coefficients)) = (4.95, 483)
(x0, length(fun.coefficients)) = (5.0, 481)
(x0, length(fun.coefficients)) = (5.05, 480)
(x0, length(fun.coefficients)) = (5.1, 479)

julia @v1.11>