I know that if we have some data representing some wave, for example image line values, we can use fourier transform to get frequency function of that wave. But we have N values at points x=0...N-1 And we get only N frequencies at the output. So I want to analyze the wave everywhere in the range [0, N-1] For example at point u = 1.5. How can I do it?



Calculating fourier transform at any frequency
You need to have some a priori knowledge about your wave between and beyond the sampling points to get a meaningful guess about the full Fourier transform. The $N$ values that you get doing the discrete Fourier transform have anything to do with the continuous Fourier transform only for indices much less than $N$. Note that different assumptions will lead to different answers for large frequences. If you need only relatively low frequences, your signal is compactly supported in time, and your sampling points cover the entire support, you can safely interpolate the values of the discrete FT to guess the continuous one but there is no way to get reliable high precision values for the continuous Fourier transform at frequences comparable to $N$. The $N$-point resolution on $[0,1]$ is just not high enough to catch those frequences without an error.


Also note that you want the frequences not on "on the interval $[0,N-1]$" as you wrote, but rather on the interval $[-M,M]$ where $M\ll N$. For decent signals, the frequences with indices close to $N$ in the discrete $FT$ actually catch the negative low frequences in the continuous FT, not the high ones. 


