Equispaced data with gaps vs Non-equispaced data #20

Can anyone help clear up my misunderstanding?

I used the forward transform to convert a sine wave to frequency domain, then adjoint transform to go back to time domain.
If the data is equispaced, this works well (not pictured).

If the data is equispaced but with even a small gap (<1% missing) the reconstruction is smooth, but with incorrect amplitude:

If the data is non-equispaced, the reconstruction is very poor. The Fourier coefficients appear totally random - not an ideal spike, and not spectral leakage.

Is this technique not intended for just these applications? I believe it is and that I've made a mistake.

`import nfft
import numpy as np
import matplotlib.pyplot as plt

freq = 13 #Rad/s
N = 300

x = np.linspace(-0.5, +0.5, N)
f = np.sin(freq*x)

#Equally spaced data with one gap:

gapsize = 2
x=np.delete(x, slice(N//2,N//2+gapsize), axis=0)
f=np.delete(f, slice(N//2,N//2+gapsize), axis=0)

N = len(x)
k = -N//2 + np.arange(N)

f_k = nfft.ndft(x, f)

plt.figure(2)
plt.scatter(k, f_k.real, label="Real")
plt.scatter(k, f_k.imag, label="Imag")
plt.legend()

f_hat = nfft.ndft_adjoint(x, f_k, N)/N
plt.figure(1)
plt.scatter(x, f, s=20, marker='o', label="Data with gaps")
plt.scatter(x, f_hat, s=35, marker='v', label="NDFT fit")

plt.legend()

#Randomly spaced data
x = 0.5 + np.random.rand(N)
f = np.sin(freq*x)

N = len(x)
k = -N//2 + np.arange(N)

f_k = nfft.ndft(x, f)

plt.figure(4)
plt.scatter(k, f_k.real, label="Real")
plt.scatter(k, f_k.imag, label="Imag")
plt.legend()

f_hat = nfft.ndft_adjoint(x, f_k, N)/N
plt.figure(3)
plt.scatter(x, f, s=20, marker='o', label="Data non-equispaced")
plt.scatter(x, f_hat, s=35, marker='v', label="NDFT fit")

plt.legend()
plt.show()
`