Inward-chord control for the scalar companion #
Full Plemelj continuity at an arbitrary smooth frontier is delicate. Along an inward chord of a convex domain, however, an interior ball gives a uniform geometric denominator estimate. If
z_r = (1-r) xi + r c,
where c is interior and xi is on the frontier, then every frontier point
sigma satisfies r * delta ≤ ‖sigma - z_r‖ for one fixed delta > 0.
The factor r exactly cancels the size of z_r - xi; this is the domination
needed to pass the cancelled scalar Cauchy contour to its radial boundary
value.
Main declarations #
continuous_eval_divByMonic_X_sub_C-- joint continuity of the polynomial divided difference in its base and evaluation points;exists_uniform_norm_eval_divByMonic_X_sub_C_frontier_le-- one bound for all pairs of frontier points;ae_boundaryParam_ne_of_mem_frontier-- a frontier point is attained only on a null set of integration parameters;smoothJordanInwardPoint-- the inward real convex combination;smoothJordanInwardPoint_mem_carrier-- it lies in the carrier for0 < r ≤ 1;exists_inward_coordinates-- every point of a bounded carrier lies on such a chord from any fixed interior center;exists_uniform_inward_boundary_denominator_bound-- one interior-ball separation constant for every pair of frontier points;exists_inward_boundary_denominator_bound-- the uniform frontier separation supplied by an interior ball;exists_uniform_inward_boundary_cross_bound-- the cancellation-ready estimate with constants uniform in the chord endpoint;exists_inward_boundary_cross_bound-- the cancellation-ready multiplied form of that estimate.exists_uniform_norm_inward_regularized_integrand_sub_le_deriv-- one continuous majorant for all endpoints of chords from a fixed center.exists_norm_inward_regularized_integrand_sub_le-- an integrable-form domination bound for the difference from the regularized self-kernel.tendsto_crouzeixPolynomialScalarCompanionRegularized_inward-- dominated convergence of the regularized companion along every inward chord.tendsto_crouzeixPolynomialScalarCompanion_inward-- the corresponding full Plemelj limit when the scalar winding kernel is normalized.tendsto_crouzeixPolynomialScalarCompanionRegularized_inward_sub_self_joint-- the Plemelj error tends uniformly locally to zero as its endpoint moves.continuousOn_crouzeixPolynomialScalarCompanionRegularized_self-- the moving regularized boundary value is continuous on the frontier.tendsto_crouzeixPolynomialScalarCompanion_inward_joint-- joint radial convergence of the full companion with moving endpoint.tendsto_crouzeixPolynomialScalarCompanionRegularized_of_inward_coordinates-- an inward-coordinate chart yields unrestricted regularized convergence.exists_inward_coordinate_functions-- global radial coordinates converge to(0, xi)near every frontier point.SmoothJordanDomain.isBounded_carrier_of_cauchyKernel_eq_one-- winding normalization and exterior decay force the carrier to be bounded.tendsto_crouzeixPolynomialScalarCompanion_of_cauchyKernel_eq_one-- unrestricted full Plemelj convergence under winding normalization alone.continuousOn_crouzeixPolynomialScalarCompanionClosedExtension_of_cauchyKernel_eq_one-- the resulting canonical closed extension is continuous.norm_companionClosedExtension_le_of_boundaryPhaseTransform_radial-- the sharp phase inequality controls that extension on the closure.
If the scalar winding kernel is normalized to one throughout the carrier, then exterior Cauchy-transform decay forces that carrier to be bounded.
The closure of a winding-normalized smooth Jordan carrier is compact.
The value of the polynomial divided difference
p /ₘ (X - C xi) at sigma depends jointly continuously on xi and
sigma.
On the compact frontier, the values of all frontier-based divided differences of one polynomial admit a single nonnegative bound.
On one fundamental integration interval, a Jordan boundary parametrization takes any prescribed frontier value only on a null set.
In a bounded smooth Jordan carrier, every interior point is an inward chord point from any fixed interior center to some frontier endpoint.
Under winding normalization, every carrier point has inward coordinates without a separate boundedness hypothesis.
One ball about an interior center gives the same linear-in-r
separation estimate simultaneously for every chord endpoint and every test
point on the frontier.
An interior ball gives a linear-in-r lower bound on the distance from
the inward chord point to every frontier point.
For a fixed interior center, compactness of the frontier also bounds the chord lengths. Consequently the multiplied cancellation estimate has constants independent of both frontier points.
The preceding denominator bound in the multiplied form used after the
resolvent identity. The chord displacement contributes exactly the same
factor r as the boundary separation, leaving a uniform domination
constant.
For a fixed interior center and polynomial, the integrand domination constant can be chosen uniformly over every frontier chord endpoint.
Joint continuity of the divided difference and the uniform chord geometry give a single integrable majorant, depending only on the fixed center and polynomial, for all frontier endpoints and inward radii.
Along inward chords, the parameterized regularized kernel minus its
self-kernel is uniformly dominated by a constant times the continuous
integrable function ‖boundaryParam'‖ * ‖q.eval boundaryParam‖, where
q = p /ₘ (X - C xi). This is the exact domination input for a radial
Plemelj limit.
The radial Plemelj error tends to zero jointly when the inward radius tends to zero and the chord endpoint varies along the frontier. The target self-value moves with the endpoint; continuity of that self-value is a separate remaining step.
The regularized self-value varies continuously along the frontier. The moving removable singularity is handled by dominated convergence, using joint continuity and the compact-uniform bound for divided differences.
The explicit Plemelj boundary value is continuous on the frontier.
Combining joint decay of the radial error with continuity of the moving self-value gives the full joint radial Plemelj limit.
With normalized winding kernel, the full companion has the corresponding joint radial limit while its frontier endpoint moves.
The joint radial limit specializes to each fixed frontier endpoint.
The full companion inherits the fixed-endpoint radial limit.
Any inward-coordinate chart whose radius and endpoint tend jointly to
(0, xi) transfers the joint radial theorem to the unrestricted interior
filter. This isolates the remaining geometry needed for a full Plemelj
theorem from the completed analytic argument.
The same inward-coordinate hypothesis supplies exactly the regularized frontier convergence consumed by the canonical Plemelj extension API.
Inward coordinates may be chosen simultaneously for all carrier points. For every frontier point, the chosen radius and endpoint tend to zero and that frontier point respectively.
Winding normalization alone supplies a global inward-coordinate choice with the correct limiting parameters at every frontier point.
On every bounded smooth Jordan carrier, the regularized scalar companion has the full unrestricted interior Plemelj limit at each frontier point.
Under winding normalization, the actual scalar companion converges from the whole carrier to its explicit Plemelj boundary value.
Winding normalization alone forces boundedness and hence supplies the unrestricted regularized Plemelj limit.
Winding normalization alone gives the unrestricted full Plemelj limit from the carrier.
Winding normalization alone makes the canonical scalar-companion extension continuous on the closure.
Under winding normalization alone, the canonical closed extension takes the explicit Plemelj value on the frontier.
On a bounded smooth Jordan carrier, winding normalization makes the canonical scalar-companion extension continuous on the closure.
On a bounded smooth Jordan carrier, the canonical closed extension takes the explicit Plemelj boundary value at every frontier point.
On a bounded normalized smooth Jordan carrier, the canonical closed scalar companion preserves polynomial addition throughout the closure.
On a bounded normalized smooth Jordan carrier, the canonical closed scalar companion is conjugate-homogeneous throughout the closure.
A uniform bound for the explicit Plemelj boundary values controls the scalar companion throughout every bounded normalized carrier.
Winding normalization alone makes the canonical closed companion additive on the closure.
Winding normalization alone makes the canonical closed companion conjugate-homogeneous on the closure.
Under winding normalization alone, a uniform Plemelj boundary bound controls the scalar companion throughout the carrier.
On a bounded normalized carrier, the sharp boundary-phase inequality immediately controls the scalar companion in the interior.
The same sharp boundary-phase inequality controls the canonical closed extension on the entire closure of a bounded normalized carrier.
Constant polynomials give the conjugate constant throughout the canonical closed extension of a bounded normalized carrier.
Adding a constant polynomial shifts the canonical closed companion by the conjugate constant throughout a bounded normalized closure.
The canonical closed companion obeys the full conjugate-affine law under winding normalization.
Shifting an approximating polynomial by the conjugate constant preserves the approximation error for the correspondingly shifted closed companion.
Any uniform polynomial approximation of a canonical companion transfers, with exactly the same error, after adding a constant to the source polynomial.
Conjugate-scaling an approximating polynomial scales the closed-companion approximation error by exactly the norm of the source scalar.
Consequently, a uniform approximation transfers under polynomial scaling with the expected multiplicative error.
Uniform approximation is therefore stable under the full conjugate-affine action on canonical companions.
The preceding affine transfer has an exact pointwise error identity.
For a constant polynomial, the canonical closed companion has exactly the conjugate-polynomial auxiliary contour; no extra Plemelj identity is needed.
Winding normalization alone makes the canonical closed companion of a constant polynomial equal to the conjugate constant.
Consequently, the auxiliary contour of a constant canonical companion is automatic under winding normalization alone.