Is there any theorem about this question? I have found the book "Analysis of Boolean function" which introduces a "Fourier expansion" :functions as multilinear polynomials. I think this may help.enter image description here
2026-03-30 23:19:07.1774912747
How to transform a general polynomial function into a multilinear function?
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You understood the question posed in the image wrongly. It is ill-posed in your title.
I think that it is an application of a general result in Fourier theory: On every Hilbert space of functions, you can (1) write every function uniquely (2) as a fourier expansion. (with respect to a given orthonormal basis. Note that such a basis exists in every vector space if you assume the Axiom of Choice and apply the Gram-Schmidt process)