Real Paley-Wiener theorem for the octonion Fourier transform

For the octonion Fourier transform, we establish the real Paley-Wiener theorem. It relates the mean of derivatives of a function with the support of its octonion Fourier transform via lim(m ->infinity)parallel to partial derivative(m alpha)f parallel to(1/m)(L2(R3,O)) = (2 pi)(vertical bar alpha vertical bar)parallel to w(alpha)parallel to(L infinity(suppFO(f)),O) for any octonion-valued Sobolev function f is an element of H-infinity(R-3, O).

Automated summary

The real Paley-Wiener theorem is established for the octonion Fourier transform, relating the mean of derivatives of a function with the support of its octonion Fourier transform. This relationship holds for any octonion-valued Sobolev function in H-infinity(R-3, O).