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Spaces of holomorphic functions

In complex analysis, the delta function enters via Cauchy's integral formula, which asserts that if D is a domain in the complex plane with smooth boundary, then

f (z) = 1 2 π i ∮ ∂ D f (ζ) d ζ ζ − z, z ∈ D {\displaystyle f(z)={\frac {1}{2\pi i}}\oint _{\partial D}{\frac {f(\zeta)\,d\zeta }{\zeta -z}},\quad z\in D}

for all holomorphic functions f in D that are continuous on the closure of D. As a result, the delta function δ is represented in this class of holomorphic functions by the Cauchy integral:

δ z = f (z) = 1 2 π i ∮ ∂ D f (ζ) d ζ ζ − z. {\displaystyle \delta _{z}=f(z)={\frac {1}{2\pi i}}\oint _{\partial D}{\frac {f(\zeta)\,d\zeta }{\zeta -z}}.}

Moreover, let H (∂ D) be the Hardy space consisting of the closure in L (∂ D) of all holomorphic functions in D continuous up to the boundary of D. Then functions in H (∂ D) uniquely extend to holomorphic functions in D, and the Cauchy integral formula continues to hold. In particular for z ∈ D, the delta function δ is a continuous linear functional on H (∂ D). This is a special case of the situation in several complex variables in which, for smooth domains D, the Szegő kernel plays the role of the Cauchy integral.

Another representation of the delta function in a space of holomorphic functions is on the space H (D) ∩ L 2 (D) {\displaystyle H(D)\cap L^{2}(D)} of square-integrable holomorphic functions in an open set D ⊂ C n {\displaystyle D\subset \mathbb {C} ^{n}}. This is a closed subspace of L 2 (D) {\displaystyle L^{2}(D)}, and therefore is a Hilbert space. On the other hand, the functional that evaluates a holomorphic function in H (D) ∩ L 2 (D) {\displaystyle H(D)\cap L^{2}(D)} at a point z {\displaystyle z} of D {\displaystyle D} is a continuous functional, and so by the Riesz representation theorem, is represented by integration against a kernel K z (ζ) {\displaystyle K_{z}(\zeta)}, the Bergman kernel. This kernel is the analog of the delta function in this Hilbert space. A Hilbert space having such a kernel is called a reproducing kernel Hilbert space. In the special case of the unit disc, one has δ w = f (w) = 1 π ∬ | z | < 1 f (z) d x d y (1 − z ¯ w) 2. {\displaystyle \delta _{w}=f(w)={\frac {1}{\pi }}\iint _{|z|<1}{\frac {f(z)\,dx\,dy}{(1-{\bar {z}}w)^{2}}}.}