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A formula for Hermite multivariate interpolation problem and partial fraction decomposition

Hakop Hakopian

TL;DR

The paper develops a new formula for Hermite multivariate interpolation based on Chung-Yao hyperplane intersections, linking high-dimensional interpolation to univariate polynomial Taylor data. It defines a Taylor-based interpolation representation $p_f(x)=\sum_{k=1}^s\phi_k(x)\mathcal{T}_{f/\phi_k, x^{(k)}, m_k-1}(x)$ using $\phi=\prod L_k$ and $\phi_i$, enabling explicit Hermite interpolation on admissible hyperplane configurations. Building on this, it presents a comprehensive univariate framework for partial fraction decomposition via Lagrange-Newton and Lagrange-Taylor formulas, providing explicit coefficients for general, multiple-root, and real rational functions. The results yield concrete, implementable formulas for real partial fraction decompositions, leveraging derivatives of reduced rational functions and organized pole structures. Overall, the work bridges multivariate interpolation with classical univariate decomposition, offering direct computational tools for both Hermite interpolation in several variables and rational function decomposition.

Abstract

We present a new formula for the Hermite multivariate interpolation corresponding to the Chung-Yao interpolation. With the help of the respective univariate interpolation formula we give a direct and explicit solution to the problem of partial fraction decomposition of rational functions.

A formula for Hermite multivariate interpolation problem and partial fraction decomposition

TL;DR

The paper develops a new formula for Hermite multivariate interpolation based on Chung-Yao hyperplane intersections, linking high-dimensional interpolation to univariate polynomial Taylor data. It defines a Taylor-based interpolation representation using and , enabling explicit Hermite interpolation on admissible hyperplane configurations. Building on this, it presents a comprehensive univariate framework for partial fraction decomposition via Lagrange-Newton and Lagrange-Taylor formulas, providing explicit coefficients for general, multiple-root, and real rational functions. The results yield concrete, implementable formulas for real partial fraction decompositions, leveraging derivatives of reduced rational functions and organized pole structures. Overall, the work bridges multivariate interpolation with classical univariate decomposition, offering direct computational tools for both Hermite interpolation in several variables and rational function decomposition.

Abstract

We present a new formula for the Hermite multivariate interpolation corresponding to the Chung-Yao interpolation. With the help of the respective univariate interpolation formula we give a direct and explicit solution to the problem of partial fraction decomposition of rational functions.
Paper Structure (9 sections, 3 theorems, 76 equations)

This paper contains 9 sections, 3 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. The Chung-Yao multivariate interpolation
  3. The Hermite multivariate interpolation
  4. A new formula for Hermite multivariate interpolation problem
  5. An application to the univariate case: partial fraction decomposition
  6. Introduction
  7. The univariate Lagrange-Newton and Lagrange-Taylor formulas
  8. The decomposition of rational functions in the general case
  9. The decomposition of real rational functions

Key Result

Theorem 1.2

Assume that the set of hyperplanes $\mathcal{L}_{n+k}$ is in general position. Then for any data set $\left\{c_\alpha\ :\ \alpha\in \mathbb I_k^{n+k}\right\}$ there exists a unique polynomial $p\in\Pi_n^k$ such that

Theorems & Definitions (5)

  • Definition 1.1
  • Theorem 1.2: ChungYao
  • Theorem 1.3: Hak3
  • Proposition 1.4
  • proof