Formula for Hermite multivariate interpolation and partial fraction decomposition

Technical Summary: Formula for Hermite Multivariate Interpolation and Partial Fraction Decomposition

Problem Statement
The paper addresses two interconnected problems in approximation theory and algebra:

1. Multivariate Hermite Interpolation: Finding an explicit formula for the unique polynomial $p \in \Pi_k^n$ (polynomials of total degree at most $n$ in $k$ variables) that satisfies Hermite interpolation conditions (values and partial derivatives) at a set of points defined by the intersections of a collection of hyperplanes.
2. Partial Fraction Decomposition: Providing a direct, explicit solution for decomposing rational functions $p(x)/q(x)$ into partial fractions, specifically handling cases with multiple roots and real coefficients (including complex conjugate pairs), without resorting to solving large linear systems.

Methodology
The author, Hakop Hakopian, utilizes the framework of Chung–Yao interpolation, which relies on the geometric arrangement of hyperplanes.

• Geometric Setup: Let $L_{n+k} = \{L_1, \dots, L_{n+k}\}$ be a collection of hyperplanes in $\mathbb{R}^k$. The set is assumed to be "admissible," meaning the intersection of any $k$ distinct hyperplanes is a single point, and the intersection of any $k+1$ is empty.

• Multiplicity Definition: For each distinct intersection point $x^{(i)}$, a multiplicity $m_i$ is defined based on the number of hyperplanes passing through it. The interpolation conditions require matching the function and its derivatives up to order $m_i - 1$ at each $x^{(i)}$.

• The Lagrange–Taylor Formula: The core methodological contribution is the derivation of a "Lagrange–Taylor" formula. The author defines a global vanishing polynomial $\phi(x) = \prod L_j(x)$ and local vanishing polynomials $\phi_i(x)$ that vanish at all intersection points except $x^{(i)}$.
  The interpolant $p_f$ is constructed as a sum of terms:
  $$p_f(x) = \sum_{i=1}^s \phi_i(x) \cdot T_{f/\phi_i, x^{(i)}, m_i-1}(x)$$
  where $T_{g, c, m}$ denotes the Taylor polynomial of degree $m$ for a function $g$ at point $c$. The proof relies on the multivariate Leibniz rule to show that this construction satisfies both the vanishing conditions at other nodes and the derivative reproduction conditions at the target node.

• Application to Univariate Decomposition: The paper applies this multivariate framework to the univariate case ($k=1$). By treating the roots of the denominator $q(x)$ as the intersection points of linear factors, the author derives explicit formulas for the coefficients of the partial fraction decomposition.

  • Complex Roots: The formula utilizes divided differences and Taylor expansions to handle multiple roots.
  • Real Decomposition: For real polynomials where non-real roots appear in conjugate pairs, the author groups these roots. The resulting formula expresses the decomposition using real quadratic factors $(x^2 + u_\nu x + v_\nu)$ and real coefficients derived from divided differences of the rational function components.

Key Contributions and Results

1. Explicit Hermite Formula: The paper presents a new, explicit formula for multivariate Hermite interpolation within the Chung–Yao setting. Unlike traditional approaches that might require solving linear systems, this formula is direct and relies on Taylor expansions of the quotient $f/\phi_i$.
2. Direct Partial Fraction Decomposition: The author derives explicit formulas for the coefficients in the partial fraction decomposition of rational functions with multiple roots.
   • For distinct roots, the formula recovers the standard result involving $q'(x_i)$.
   • For multiple roots, the coefficients are given explicitly via derivatives (for real roots) or divided differences (for complex conjugate pairs), avoiding the need to solve a linear system of equations.
3. Real Case Handling: A specific contribution is the explicit treatment of real rational functions with complex conjugate roots. The paper provides formulas for the coefficients of the terms involving irreducible quadratic factors, ensuring the decomposition remains within the field of real numbers.

Significance and Claims
The paper claims to provide a direct and explicit solution to the classical partial fraction decomposition problem, including the real case. By leveraging the univariate analogue of the newly derived multivariate Lagrange–Taylor formula, the author demonstrates that the coefficients of the decomposition can be computed directly from the function and its derivatives (or divided differences) at the roots, rather than through indirect algebraic manipulation or linear algebra.

The work unifies the geometric approach of Chung–Yao interpolation with the algebraic problem of rational function decomposition, offering a theoretical framework that yields computable, closed-form expressions for interpolation data and decomposition coefficients. The author notes that this approach simplifies the procedure for determining constants in partial fraction expansions, particularly in cases involving multiple roots.