Trigonometric interpolation of a sampled signal

dodoPN
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Given N sampled points, using the FFT we can get the Fourier transform of those N points Xk. With N/2 the Nyquist frequency and X0 the DC value. Using the inverse we can then get back the original function we just measured. However if we would like more points then just the N we have measured but instead we would like M, how can u use the inverse FFT to find the trigonometric interpolation? We can assume the N is even and that M>N. And wat if we would drop values out of Xk, how would you find a trigonometric interpolation of the original signal.
 
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