In mathematics, singular integral operators on closed curves arise in problems in analysis, in particular complex analysis and harmonic analysis. The two main singular integral operators, the Hilbert transform and the Cauchy transform, can be defined for any smooth Jordan curve in the complex plane and are related by a simple algebraic formula. In the special case of Fourier series for the unit circle, the operators become the classical Cauchy transform, the orthogonal projection onto Hardy space, and the Hilbert transform a real orthogonal linear complex structure. In general the Cauchy transform is a non-self-adjoint idempotent and the Hilbert transform a non-orthogonal complex structure. The range of the Cauchy transform is the Hardy space of the bounded region enclosed by the Jordan curve. The theory for the original curve can be deduced from that of the unit circle, where, because of rotational symmetry, both operators are classical singular integral operators of convolution type. The Hilbert transform satisfies the jump relations of Plemelj and Sokhotski, which express the original function as the difference between the boundary values of holomorphic functions on the region and its complement. Singular integral operators have been studied on various classes of functions, including Hölder spaces, Lp spaces and Sobolev spaces. In the case of L2 spaces—the case treated in detail below—other operators associated with the closed curve, such as the Szegő projection onto Hardy space and the Neumann–Poincaré operator, can be expressed in terms of the Cauchy transform and its adjoint.
Operators on the unit circle
If f is in L2(T), then it has a Fourier series expansion
f ( θ ) = ∑ n ∈ Z a n e i n θ . {\displaystyle \displaystyle {f(\theta )=\sum _{n\in {\mathbf {Z} }}a_{n}e^{in\theta }.}}
Hardy space H2(T) consists of the functions for which the negative coefficients vanish, an = 0 for n < 0. These are precisely the square-integrable functions that arise as boundary values of holomorphic functions in the unit disk |z| < 1. Indeed, f is the boundary value of the function
F ( z ) = ∑ n ≥ 0 a n z n , {\displaystyle \displaystyle {F(z)=\sum _{n\geq 0}a_{n}z^{n},}}
in the sense that the functions
f r ( θ ) = F ( r e i θ ) , {\displaystyle \displaystyle {f_{r}(\theta )=F(re^{i\theta })},}
defined by the restriction of F to the concentric circles |z| = r, satisfy
‖ f r − f ‖ 2 → 0 {\displaystyle \displaystyle {\|f_{r}-f\|_{2}\rightarrow 0}} as r → 1 {\displaystyle \displaystyle {r\rightarrow 1}} . The orthogonal projection P of L2(T) onto H2(T) is called the Szegő projection. It is a bounded operator on L2(T) with operator norm 1. By Cauchy's theorem
F ( z ) = 1 2 π i ∫ | ζ | = 1 f ( ζ ) ζ − z d ζ = 1 2 π ∫ − π π f ( θ ) 1 − e − i θ z d θ . {\displaystyle \displaystyle {F(z)={1 \over 2\pi i}\int _{|\zeta |=1}{f(\zeta ) \over \zeta -z}\,d\zeta ={1 \over 2\pi }\int _{-\pi }^{\pi }{f(\theta ) \over 1-e^{-i\theta }z}\,d\theta .}}
Thus
… excerpt ends here. Continue reading the full article.
