cheb2ultra seems unreasonably slow

[jishnub]

julia> x = rand(1000);

julia> y1 = @btime FastTransforms.cheb2ultra($x, 2);
  24.224 ms (2 allocations: 7.98 KiB)

julia> y2 = @btime ApproxFunBase.mul_coefficients(Conversion(Chebyshev(), Ultraspherical(2)), $x);
  45.568 μs (13 allocations: 63.27 KiB)

julia> y1 ≈ y2
true

[MikaelSlevinsky]

Almost all the time is spent on the plans. Good planning makes better execution.

julia> @btime cheb2ultra($x, 2);
  15.481 ms (2 allocations: 7.98 KiB)

julia> p = plan_cheb2ultra(x, 2);

julia> @btime $p*$x;
  204.918 μs (1 allocation: 7.94 KiB)

julia> p = plan_cheb2ultra(x, 1.9);

julia> @btime $p*$x;
  197.862 μs (1 allocation: 7.94 KiB)

BUT, the cheb2ultra plan doesn't specialize on integer lambda, which is a banded conversion.

[jishnub]

Would it make sense to cache the plans, so that one only needs to compute then once for a specific combination of parameters? Something like an LRUCache could ensure that there's no explosion in the number of plans cached