sph methods slow?

Maybe this is totally expected, but:

• A lot slower than FFT. It has been a while, but I seem to recall (going back to 2018) that the first version was much more comparable to FFT
• Allocations that don't go away with in-place (hard to make an efficient time evolution algorithm in this case)
• Which method is the bottleneck depends on size

using FastTransforms
using FFTW
using LinearAlgebra

n,m = 1024,2047

x = randn(ComplexF64,n,m)
# FFTW
@time fft(x);
  0.033259 seconds (259 allocations: 32.001 MiB)
P! = plan_fft!(x);
@time P!*x; 
  0.035154 seconds (232 allocations: 14.969 KiB)

# Spherical 
P = plan_sph2fourier(x)
PS = plan_sph_synthesis(x)

@time PS*(P*x);
  0.235185 seconds (549 allocations: 127.972 MiB, 1.57% gc time)
@time P*x; # slow
  0.145820 seconds (8 allocations: 63.969 MiB, 1.85% gc time)
@time PS*x;
  0.051036 seconds (545 allocations: 64.003 MiB)

@time lmul!(P,x);  # slow
  0.139856 seconds (4 allocations: 31.984 MiB, 0.91% gc time)
@time lmul!(PS,x);
  0.047418 seconds (539 allocations: 32.018 MiB)

## small array
n,m = 60,121
x = randn(ComplexF64,n,m);

# FFTW
@time fft(x);
  0.000560 seconds (242 allocations: 129.836 KiB)
P! = plan_fft!(x);
@time P!*x; 
  0.000411 seconds (218 allocations: 14.492 KiB)

# Spherical 
P = plan_sph2fourier(x);
PS = plan_sph_synthesis(x);

@time PS*(P*x);
  0.101827 seconds (19.31 k allocations: 1.626 MiB)

@time P*x; 
  0.000266 seconds (8 allocations: 227.406 KiB)
@time PS*x;  # slow
  0.031964 seconds (20.48 k allocations: 1.440 MiB)

@time lmul!(P,x);  
  0.000364 seconds (4 allocations: 113.594 KiB)
@time lmul!(PS,x);  # slow
  0.037324 seconds (20.38 k allocations: 1.326 MiB)

Reproduce the reported timings using plan_sph2fourier, plan_sph_synthesis, lmul!, and the FFTW comparisons in the issue. Start by locating those spherical-transform entry points and profiling both the large and small array cases. Done would require identifying and addressing the reported slowdown or allocations, but the issue does not define a specific target or test.