Sorry, but the real and imaginary designation is imaginary. The Fourier transform is just a functional decomposition using the sine and cosine as basis function

Sorry, but the real and imaginary designation is imaginary. The Fourier transform is just a functional decomposition using the sine and cosine as basis functions (the Fourier basis.) They have some special and useful properties such as being the eigenfunctions of a linear system which make them the preferred basis for most work. Many other functions could be used in those tables which have the properties of orthogonality and completeness and the decomposition would still be one to one and self invertible. The wavelet transform, for example, which doesn't even pretend to be complex. It was the understanding of wavelet theory that made me realize that sin and cosine decomposition (the Fourier basis) is just one of many.