In ℝᵈ any boundary point, the near region can be put in very simple canonical form by choosing appropriate coordinates, so that the region is represented through functions and mapped onto the half-space with a vanishing hyperplane for a boundary. To transfer this intuition to complex holomorphic functions, the new coordinates need to be expressed through holomorphic functions. These are referred to as "holomorphic coordinates". Near points on the boundary, holomorphic coordinates {z} can be introduced, so that they are centered at that point, and the area is
This is facilitated directly by a Hermitian matrix which, when diagonalized becomes the sum in the above formula. When in the summed form, it's referred to as the "Levi Form" of Ω at the boundary point. The transformed unit vectors are tangent to the area boundary, so with ρ(z) = φ(z', xₙ) - yₙ, it can be written in a quadratic form.
A biholonomic map ω near the origin creates a holomorphic coordinate system, so the differential maps tangent vectors to tangent vectors of one lesser order. A boundary point of Ω is "pseudo-convex" if the Levi-form is non-negative, and "strongly pseudo-convex" if it's strictly positive. If this is true for the entire boundary, then Ω inherits the property. For n > 1, the Levi form can experience a "local" maximum principle in ℂⁿ. For an open ball around Ω with a boundary in ℂ² and an open ball B centered at a boundary point z⁰. All points in the intersection between ball and boundary have at least one strictly positive eigenvalue of the Levi-form. A smaller ball B' exists in B, centered in z⁰, so that holomorphic functions F on Ω ∩ B, continuous on its completion have
This is true in the more general case for a local hypersurface instead of the boundary.
The classical Weierstrass approximation theorem may be restated so that given a continuous function on a compact segment on the real axis in complex space, f can be uniformly approximated by polynomials in coordinates of z = x + iy. A complex n-hypersurface M with n = 2 or larger, will haven open ball B', B centered along M with the closure of B' in B, so that if a continuous function F in the intersection of M and B satisfies the tangential Cauchy-Riemann equations in the weak sense, then F can be uniformly approximated on the intersection of M and the closure of B' through polynomials in {z}. This is true for all strictly positive integers n. If n = 1, there are no tangential Cauchy-Riemann equations, so it's trivially valid. For n > 1, there are no requirements for the Levi-form of M. If the Levi-form has at least one strictly positive eigenvalue for all points on M, then for all centers of balls B' in M with continuous functions F satisfying the tangential Cauchy-Riemann equations in the weak sense, then there are holomorphic functions G in the lower half-sphere of B' continuous on the closure of the lower half-sphere of B', so that G and F coincide on the intersection of M and B'.
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