Say I have a vector field with non-zero curl, therefore the potential function depends on the path I choose to integrate.In this paper the authors proposed to project the vector field into a gradient field of Fourier basis functions in order to make the potential function independent of the path of integration.I understand the idea but cannot understand the math of what that projection means. What I think the authors are doing is to approximate the given data using Fourier basis functions (that's why I was asking about Fourier series in another thread) in order to have a smooth function that best fits the given points and complies with the zero-curl of a gradient field. What I don't see is how many basis functions they use in order to approximate the data points, it could be $y_i = C_0 + C_1e^{jwx_i}$ or $y_i = C_0 + C_1e^{jwx_i} + C_2e^{2jwx_i}$ or any other. Moreover, it seems to me that they compute the FFT of the vector field and use all the coefficients of it, so, the function will fit all the points but still will be non-integrable.I'm sure I am missing something and mixing maybe some concepts, can someone help me unwrapping it all to understand what the idea of this projection is?
Projection of vector field on to a gradient field
fourier analysis;projection