Proof of Double Fourier Transform and Classical Fourier Transform

We show that the classical complex Fourier series is a special case of a Fourier series defined over the double complex algebra. Using idempotent decomposition, any double complex function can be decomposed into two orthogonal complex components, and the convergence of the double complex Fourier series becomes equivalent to the simultaneous convergence of its two complex projections. This places classical Fourier analysis within a more fundamental algebraic framework, revealing that the Fourier transform is not primitive, but a projection of a complete double complex structure