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by denial
1074 days ago
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Singular integrals are a lovely topic, especially from a Fourier analytic perspective. Taking the Fourier transform, F(H(f))(x) = -i * sgn(x) * F(f)(x), which implies H^2 = - I. H has the Fourier multiplier -i * sgn(x). The Riesz transforms R_j are a higher dimensional generalization of the Hilbert transform with Fourier multipliers -i * x_j/|x_j|, which leads to the nice property sum R_j^2 = -I. |
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