Hardy space
Let "e""n"(θ) = "e""in"θ be the standard orthonormal basis of L2(T). Let H2(T) be Hardy space, the closed subspace spanned by the "e""n" with "n" ≥ 0. Let "P" be the orthogonal projection onto Hardy space and set "T" = 2"P" - "I". The operator "H" = "iT" is the Hilbert transform on the circle and can be written as a singular integral operator.