Wrong FFT results

[aplavin]

As I understand, fft/ifft implementations in https://github.com/mcabbott/AxisKeys.jl/blob/master/src/fft.jl are fundamentally wrong and return incorrect results. For one, the original axiskeys don't affect the resulting content at all - this cannot be right, shifting the original array should change the phase of the FFT.

These are the implementations for zero-centered axiskeys (as they @assert), seem correct --- eg ifft inverts fft, fft of delta function at zero is constant, etc:

function AbstractFFTs.fft(A::KeyedArray)
	@assert all(ks -> ks[end÷2+1] |> iszero, axiskeys(A))
	data = fft(AxisKeys.keyless(fftshift(A)))
	keys = map(axiskeys(A)) do ks
		fftfreq(length(ks), inv(step(ks)))
	end
	KeyedArray(data, keys) |> fftshift
end

function AbstractFFTs.ifft(A::KeyedArray)
	@assert all(ks -> ks[end÷2+1] |> iszero, axiskeys(A))
	data = ifft(AxisKeys.keyless(fftshift(A)))
	keys = map(axiskeys(A)) do ks
		fftfreq(length(ks), inv(step(ks)))
	end
	KeyedArray(data, keys) |> fftshift
end

Implementations for arbitrary (ranges) axiskeys should involve phase gradient multiplication, at least.