import numpy
import numpy.fft
import matplotlib.pyplot as plt

def f(x):
    return 1./(1.+numpy.cos(x)**2)

x = numpy.linspace(0, 2.0*numpy.pi, 2048)

coeff = numpy.fft.rfft(f(x))

rec5 = numpy.fft.irfft(coeff[:5],2048)
rec10 = numpy.fft.irfft(coeff[:10],2048)
rec50 = numpy.fft.irfft(coeff[:50],2048)

plt.figure()
plt.plot(x, f(x),
         x, rec5,
         x, rec10,
         x, rec50,
         )
plt.legend(('f(x)', '5', '10', '50'))

def g(x):
    return x**-12 - x**-6

x = numpy.linspace(1, 5, 2048)

coeff = numpy.fft.rfft(g(x))

rec5 = numpy.fft.irfft(coeff[:5],2048)
rec10 = numpy.fft.irfft(coeff[:10],2048)
rec50 = numpy.fft.irfft(coeff[:50],2048)

plt.figure()
plt.plot(x, g(x),
         x, rec5,
         x, rec10,
         x, rec50,
         )
plt.legend(('g(x)', '5', '10', '50'))

plt.show()