Multicentric Piecewise Holomorphic Functions

• The paper introduces a method that encodes scalar piecewise holomorphic functions as vector-valued functions using polynomial or rational multicentric expansions.
• It employs Cauchy integral techniques and Hermite interpolation to develop localized power series with explicit error bounds and geometric convergence.
• The approach underpins stable spectral projection and holomorphic functional calculus applications, enabling efficient matrix function evaluations in complex domains.

The multicentric representation of piecewise holomorphic functions is a methodology that utilizes polynomial or rational variables to encode scalar holomorphic (and piecewise constant) functions as vector-valued functions, enabling convergent power series expansions that are localized to geometric regions determined by level sets of the variable. This framework supports precise decomposition, robust numerical evaluation, and direct application to spectral projection and operator theory. The approach generalizes classical holomorphic functional calculus and is deeply connected to Padé approximation theory, Hermite interpolation, and spectral projection for operators in Banach and Hilbert spaces.

1. Fundamental Construction: Polynomial and Rational Multicentric Expansions

The classical construction begins with a monic polynomial $p(z)$ of degree $d$ with simple roots $\Lambda=\{\lambda_1,\ldots,\lambda_d\}$. The scalar function $\varphi(z)$ holomorphic in a neighborhood of the lemniscate $V(\rho)=\{|p(z)|\leq\rho\}$ is encoded as

$$\varphi(z) = \sum_{j=1}^d \delta_j(z) f_j(w), \quad w=p(z)\,,$$

where basis functions (“Lagrange factors”)

$$\delta_j(z) = \frac{p(z)}{p'(\lambda_j)(z-\lambda_j)}$$

form a partition of unity: $\sum_j \delta_j(z) \equiv 1$.

Each $f_j$ is analytic in $|w|<\rho$, and admits a power series $f_j(w) = \sum_{k=0}^\infty \alpha_{j,k} w^k$, coefficients computed via contour (Cauchy) integrals or Taylor recursion. The framework extends naturally to rational variables: for $r(z)=p(z)/q(z)$ (with $q$ coprime to $p$, $q(\lambda_j)\neq0$), domains of convergence become $|r(z)|<\rho$, with modified basis functions

$$\delta_j(z) = \frac{r(z)}{r'(\lambda_j)(z-\lambda_j)}\,,$$

which inherit the partition property away from poles of $q$.

2. Cauchy Integral Decomposition and Local Power Series

For $\varphi$ holomorphic in $|p(z)|<\rho$, the Cauchy formula yields

$$\varphi(z) = \frac{1}{2\pi i} \int_{\gamma_\rho} \frac{\varphi(\lambda)}{\lambda-z} d\lambda\,,$$

where $\gamma_\rho$ is the union of oriented level curves $|p(\lambda)|=\rho$. By Lagrange interpolation,

$$\frac{1}{\lambda-z} = \sum_{j=1}^d \delta_j(z)\,K_j(\lambda,w)\,,$$

with kernels $$K_j(\lambda,w) = \frac{1}{\lambda-\lambda_j}\frac{p(\lambda)}{p(\lambda)-w}$$. Substitution yields explicit formulas for the vector coefficients: $$f_j(w) = \frac{1}{2\pi i} \int_{\gamma_\rho} K_j(\lambda,w)\,\varphi(\lambda)\,d\lambda\,,$$ which are holomorphic in $|w|<\rho$. Expanding in powers of $w$ gives

$f_j(w) = \sum_{k=0}^\infty \alpha_{j,k}w^k\,,$

with bounds

$$|\alpha_{j,k}| \leq L_{\rho,j} \rho^{-k-1}\max_{\gamma_\rho}|\varphi|\,,$$

where $$L_{\rho,j} = (2\pi)^{-1}\int_{\gamma_\rho}|d\lambda|/|\lambda-\lambda_j|$$.

3. Hermite Interpolation via Truncated Multicentric Representation

Truncation of the power series after $n$ terms yields a polynomial

$$P_n(z) = \sum_{j=1}^d \delta_j(z)\,\sum_{m=0}^{n} \alpha_{j,m} [p(z)]^m\,,$$

which interpolates $\varphi(z)$ and its derivatives up to order $n$ at each $\lambda_j$, establishing Hermite interpolation of degree $d(n+1)-1$ with $d(n+1)$ interpolation conditions (Nevanlinna et al., 10 Nov 2025). The multicentric truncated form remains numerically stable as $n$ increases—a property that distinguishes it from traditional Hermite bases, some of which (e.g., the “special” basis or parallel-friendly forms) display exponential sensitivity to $n$ or off-root evaluation.

4. Extensions to Rational Variables and Piecewise Domains

Employing $r(z)=p(z)/q(z)$ extends the representation to arbitrary compact sets not accessible via polynomial lemniscates. Provided $\varphi$ is holomorphic except for possible isolated poles and critical values (which are avoided by choosing $\gamma_\rho$ appropriately), the representation

$\varphi(z) = \sum_{j=1}^d \delta_j(z) f_j(r(z))$

remains valid throughout all components of $\{|r(z)|<\rho\}$ where $\varphi$ is holomorphic. The analytic and convergence results for power series and error bounds (radius $\rho$, geometric decay $\sim (R/\rho)^{N+1}/(\rho-R)$ for truncation) carry over identically, except the region of convergence in $z$ may now avoid critical points and poles.

5. Multicentric Padé Approximation and Capacity Convergence

In the related context of rational (Padé) approximation, multicentric schemes interpolate formal germs $f_0$ (at $z=0$) and $f_\infty$ (at $z=\infty$) simultaneously, yielding rational approximants $P_n/Q_n$ that converge in capacity outside a compact “S-property” set $F$ (Yattselev, 2021). In the two-center case, off $F$ the approximants exhibit strong asymptotics with geometric error decay dictated by model functions derived from Green’s potentials and Szegő-type constructs: $|f_\rho(z)-P_n/Q_n(z)| = (1+o(1))/w(z)\,\{\left(\frac{\phi^2(z)}{z}\right)^n(SD)^2(z)\mathbbm{1}_{D_0} - \left(\frac{z}{\phi^2(z)}\right)^n/(SD)^2(z)\mathbbm{1}_{D_\infty}\}\,,$ with $|{\phi^2}/{z}|<1$ in $D_0$, ensuring geometric convergence.

This framework generalizes to m-point Padé, with capacity and uniform convergence guaranteed under Buslaev’s S-property; error rates are precisely determined by contour geometry and potential theory.

6. Computational Considerations and Applications

The calculation of multicentric representations involves precomputing Lagrange polynomials at roots, evaluating integrals or derivative recurrences for coefficients, and combining terms efficiently (typically via Horner’s rule). The cost is $O(d(n+1))$ per evaluation for degree $d$ and truncation order $n$ (Nevanlinna et al., 10 Nov 2025). Since error due to truncation decays geometrically, $n$ can be increased to meet arbitrarily strict tolerances without incurring floating-point instability—a property not shared by alternative Hermite bases.

Applications include:

• Holomorphic functional calculus for bounded operators, using fast and stable spectral projection formulas, especially for disconnected spectral sets partitioned by polynomial or rational lemniscates (Apetrei et al., 2016).
• Computation of Riesz spectral projections and spectral projectors for matrices/operators, accommodating spectrum in multiply connected domains.
• Efficient evaluation of holomorphic matrix functions $f(A)$ corresponding to piecewise holomorphic $f$ localized to spectral subspaces.
• Parallelized Hermite polynomial evaluation in high-performance contexts.

7. Mathematical Properties, Error Bounds, and Generalization

Existence and uniqueness (away from critical values of $r(z)$) are guaranteed for the multicentric representation provided the system defined by the basis functions on preimages of $w$ is nonsingular (Andrei et al., 2021). Uniform convergence and error estimates are explicit: $$|a_{j,k}| \leq L_{\rho,j}\rho^{-k-1}\max_{|r|=\rho}|\varphi|,\qquad |\text{truncation error}| \leq \max_{|r|=\rho}|\varphi|\cdot L_{\rho,j}\cdot (R/\rho)^{N+1}/(\rho-R)\,.$$ In operator theory, the multicentric expansion produces norm bounds for evaluations $\varphi(A)$ via spectral mapping, and explicit operator-norm estimates can be derived by controlling the distance from the critical locus and the geometry of the lemniscate domain.

For multiple roots, expansions are available in powers of $(z-\lambda_j)$ up to the multiplicity, or via $p(z)^n$ variables, with each site producing $n$ local analytic functions. The uniform convergence and operator-norm bounds persist, provided the lemniscate domain avoids spectral singularities and critical points.

A plausible implication is that the multicentric representation provides a unifying algebraic and analytic framework for localized, stable approximation of holomorphic and piecewise holomorphic functions, with direct transferability to numerical analysis, spectral theory, and matrix function evaluation. The approach is robust with respect to truncation and geometric complexity, provided standard conditions on holomorphy and avoidance of poles/critical loci are met.