Multivariate Hermite Interpolation

• Multivariate Hermite interpolation is a technique that constructs polynomial interpolants matching prescribed function values and partial derivatives at multidimensional nodes.
• The approach leverages algebraic tools such as Gröbner bases and residue theory to derive closed-form solutions and analyze singularity and uniqueness conditions.
• Advanced methods extend the framework to noncommutative polynomial rings and Riemannian manifolds, enabling efficient, high-dimensional, and structure-preserving computations.

Multivariate Hermite interpolation concerns the construction and analysis of interpolants that match prescribed function values and partial derivatives of various orders at given sets of nodes in multidimensional space. The field encompasses a wide variety of analytic, algebraic, and geometric frameworks—including coordinate-rectilinear, total-degree, algebraic-geometric, noncommutative, and manifold-valued generalizations—and is central to approximation theory, computational algebra, and geometric modeling.

1. General Definitions and Algebraic Foundations

Let $$A = \{a^{(1)}, \dots, a^{(s)}\} \subset \mathbb{K}^n$$ be a finite set of nodes, and let $$\nu(a^{(j)}) = (\nu_1(a_1^{(j)}), \dots, \nu_n(a_n^{(j)})) \in \mathbb{N}^n$$ be a multiplicity vector prescribing, for each node, the maximal order of partial derivatives in each coordinate to be interpolated. The multivariate Hermite interpolation problem is: for prescribed data $t_{a^{(j)}}^{(k)} \in \mathbb{K}$, $0 \leq k_i < \nu_i(a_i^{(j)})$, find a polynomial $p(x) \in \mathbb{K}[x_1, \dots, x_n]$ such that

$$\partial^k p(a^{(j)}) = t_{a^{(j)}}^{(k)} \quad \text{for all } 1 \leq j \leq s,\ k \in [0, \nu(a^{(j)}) - 1].$$

Classical existence and uniqueness depend on the dimension of the candidate space of polynomials relative to the total number of Hermite conditions imposed. In coordinatewise Hermite problems on $n$-dimensional rectilinear grids, this leads to an interpolation space of coordinatewise degree bounds.

Algebraic structure is provided by ideals generated by products of $(x_i - a_i^{(j)})^{\nu_i(a_i^{(j)})}$ and algebraic methods such as Gröbner bases and the theory of multivariate residues are used to develop closed-form interpolants and remainder expressions (Kechriniotis et al., 2023, Durakov et al., 2022).

2. Explicit Multivariate Hermite Interpolation on Grids

The canonical coordinate-grid setting with grid nodes $A = A_1 \times \cdots \times A_n$ and multiplicity functions $\nu_i: A_i \to \mathbb{N}$ admits a compact, constructive solution. For each node $$\nu(a^{(j)}) = (\nu_1(a_1^{(j)}), \dots, \nu_n(a_n^{(j)})) \in \mathbb{N}^n$$0 and multi-index $$\nu(a^{(j)}) = (\nu_1(a_1^{(j)}), \dots, \nu_n(a_n^{(j)})) \in \mathbb{N}^n$$1, define univariate “node-polynomials” $$\nu(a^{(j)}) = (\nu_1(a_1^{(j)}), \dots, \nu_n(a_n^{(j)})) \in \mathbb{N}^n$$2 and local Hermite basis elements

$$\nu(a^{(j)}) = (\nu_1(a_1^{(j)}), \dots, \nu_n(a_n^{(j)})) \in \mathbb{N}^n$$3

Collect the $$\nu(a^{(j)}) = (\nu_1(a_1^{(j)}), \dots, \nu_n(a_n^{(j)})) \in \mathbb{N}^n$$4 basis and data into $$\nu(a^{(j)}) = (\nu_1(a_1^{(j)}), \dots, \nu_n(a_n^{(j)})) \in \mathbb{N}^n$$5 and $$\nu(a^{(j)}) = (\nu_1(a_1^{(j)}), \dots, \nu_n(a_n^{(j)})) \in \mathbb{N}^n$$6. Construct a local triangular “moment matrix” $$\nu(a^{(j)}) = (\nu_1(a_1^{(j)}), \dots, \nu_n(a_n^{(j)})) \in \mathbb{N}^n$$7 with entries $$\nu(a^{(j)}) = (\nu_1(a_1^{(j)}), \dots, \nu_n(a_n^{(j)})) \in \mathbb{N}^n$$8, $$\nu(a^{(j)}) = (\nu_1(a_1^{(j)}), \dots, \nu_n(a_n^{(j)})) \in \mathbb{N}^n$$9. The unique interpolant is given by the single-sum formula (Kechriniotis et al., 2023): $t_{a^{(j)}}^{(k)} \in \mathbb{K}$0 This representation generalizes the classical 1D Hermite–Taylor expansion and readily extends to non-uniform and anisotropic grids. The polynomial can be applied locally to form piecewise “spline-like” interpolants possessing strong $t_{a^{(j)}}^{(k)} \in \mathbb{K}$1 continuity properties across cell interfaces, and the interpolation operation is an algebraic projector onto the nullspace of the interpolation ideal $t_{a^{(j)}}^{(k)} \in \mathbb{K}$2.

In computational applications (e.g., image zooming (Delibasis et al., 2024)), the separable structure admits efficient convolutional kernel representations, enabling highly parallel implementations with superior accuracy over standard splines, especially when high-order derivative information is available.

3. Total-Degree, Singularity, and Classification Results

An alternative paradigm is Hermite interpolation by polynomials of total degree $t_{a^{(j)}}^{(k)} \in \mathbb{K}$3 at nodes $t_{a^{(j)}}^{(k)} \in \mathbb{K}$4 with prescribed multi-index orders $t_{a^{(j)}}^{(k)} \in \mathbb{K}$5. The system is called regular if the interpolation matrix defined by $t_{a^{(j)}}^{(k)} \in \mathbb{K}$6 nodes and corresponding Hermite conditions is invertible. The singularity depends intricately on $t_{a^{(j)}}^{(k)} \in \mathbb{K}$7.

Key results include (Meng et al., 2013):

• For $t_{a^{(j)}}^{(k)} \in \mathbb{K}$8 nodes in $t_{a^{(j)}}^{(k)} \in \mathbb{K}$9, all total-degree Hermite schemes are singular if $0 \leq k_i < \nu_i(a_i^{(j)})$0.
• For $0 \leq k_i < \nu_i(a_i^{(j)})$1, all nontrivial Hermite schemes are singular for $0 \leq k_i < \nu_i(a_i^{(j)})$2.
• For $0 \leq k_i < \nu_i(a_i^{(j)})$3 ($0 \leq k_i < \nu_i(a_i^{(j)})$4), only a few exceptional orderings yield regular schemes, classified explicitly in terms of $0 \leq k_i < \nu_i(a_i^{(j)})$5.
• Regularity is governed via the vanishing ideal intersection and the computation of the quotient space $0 \leq k_i < \nu_i(a_i^{(j)})$6 with $0 \leq k_i < \nu_i(a_i^{(j)})$7.

These singularity results reflect the geometric and combinatorial obstructions imposed by the dimension-to-node count and form the basis for geometric classification of solvable Hermite schemes.

4. Algebraic-Geometric, Symmetric, and Residue-Based Approaches

Multivariate Hermite interpolation admits several strictly algebraic and symmetric formulations:

• Chung–Yao Framework: Given $0 \leq k_i < \nu_i(a_i^{(j)})$8 affine hyperplanes in $0 \leq k_i < \nu_i(a_i^{(j)})$9, either in general or admissible position, Hermite interpolation matches data at all intersection points (possibly of higher multiplicity). The closed form is (Hakopian, 2 Feb 2026):

$p(x) \in \mathbb{K}[x_1, \dots, x_n]$0

where $p(x) \in \mathbb{K}[x_1, \dots, x_n]$1 vanishes to order $p(x) \in \mathbb{K}[x_1, \dots, x_n]$2 at $p(x) \in \mathbb{K}[x_1, \dots, x_n]$3 and $p(x) \in \mathbb{K}[x_1, \dots, x_n]$4 denotes multivariate Taylor expansion.

• Residue Theory: Using systems $p(x) \in \mathbb{K}[x_1, \dots, x_n]$5 with a finite set of isolated common zeros $p(x) \in \mathbb{K}[x_1, \dots, x_n]$6, and local multiplicity vectors $p(x) \in \mathbb{K}[x_1, \dots, x_n]$7, the unique solution interpolant is constructed by a multidimensional residue formula (Durakov et al., 2022):

$p(x) \in \mathbb{K}[x_1, \dots, x_n]$8

where $p(x) \in \mathbb{K}[x_1, \dots, x_n]$9 is a residue kernel built from the derivatives and the local factorization of $$\partial^k p(a^{(j)}) = t_{a^{(j)}}^{(k)} \quad \text{for all } 1 \leq j \leq s,\ k \in [0, \nu(a^{(j)}) - 1].$$0 at $$\partial^k p(a^{(j)}) = t_{a^{(j)}}^{(k)} \quad \text{for all } 1 \leq j \leq s,\ k \in [0, \nu(a^{(j)}) - 1].$$1.

• Symmetric Hermite Interpolation: For symmetric functions, the Krick–Szántó and Roy–Szpirglas machinery provides Hermite interpolation in the basis of multivariate generalized Vandermonde determinants. The symmetric Hermite interpolant is explicit as a sum over signed minors:

$$\partial^k p(a^{(j)}) = t_{a^{(j)}}^{(k)} \quad \text{for all } 1 \leq j \leq s,\ k \in [0, \nu(a^{(j)}) - 1].$$2

respecting coalescence of nodes and linking directly to resultants, Schur polynomials, and subresultant theory (Krick et al., 30 Jan 2025, Roy et al., 2018).

5. Hermite Interpolation in Noncommutative and Structured Polynomial Rings

Multivariate Hermite interpolation extends to skew-polynomial rings $$\partial^k p(a^{(j)}) = t_{a^{(j)}}^{(k)} \quad \text{for all } 1 \leq j \leq s,\ k \in [0, \nu(a^{(j)}) - 1].$$3 over division rings, relevant for applications in coding theory and noncommutative algebra. In this context (Donoso et al., 2022):

• Right and left $$\partial^k p(a^{(j)}) = t_{a^{(j)}}^{(k)} \quad \text{for all } 1 \leq j \leq s,\ k \in [0, \nu(a^{(j)}) - 1].$$4-partial derivatives generalize commutative differences.
• Existence-uniqueness holds under a “derivative-polynomial independence” (DP-independence) condition, characterized constructively via a division algorithm and via the invertibility of confluent skew-Vandermonde matrices.
• The unique interpolant is built as a linear combination of specially constructed skew-polynomial basis elements $$\partial^k p(a^{(j)}) = t_{a^{(j)}}^{(k)} \quad \text{for all } 1 \leq j \leq s,\ k \in [0, \nu(a^{(j)}) - 1].$$5 matching the prescribed mixed derivatives at each node.

This structure allows algorithmic construction and explicit dimension control in noncommutative settings.

6. Multivariate Hermite Interpolation on Manifolds

When interpolating data valued in Riemannian manifolds, two paradigms have been introduced (Zimmermann et al., 2022):

1. Barycentric Hermite Interpolation (BHI):
   • The interpolant at $$\partial^k p(a^{(j)}) = t_{a^{(j)}}^{(k)} \quad \text{for all } 1 \leq j \leq s,\ k \in [0, \nu(a^{(j)}) - 1].$$6 is defined as the minimum of the weighted sum of squared Riemannian distances:

   $$\partial^k p(a^{(j)}) = t_{a^{(j)}}^{(k)} \quad \text{for all } 1 \leq j \leq s,\ k \in [0, \nu(a^{(j)}) - 1].$$7

   with $$\partial^k p(a^{(j)}) = t_{a^{(j)}}^{(k)} \quad \text{for all } 1 \leq j \leq s,\ k \in [0, \nu(a^{(j)}) - 1].$$8 the barycentric weight functions satisfying Hermite conditions via constraints on their partial derivatives. All Hermite derivative information is incorporated intrinsically at each sample point, requiring only the solution of small linear systems for the partials of $$\partial^k p(a^{(j)}) = t_{a^{(j)}}^{(k)} \quad \text{for all } 1 \leq j \leq s,\ k \in [0, \nu(a^{(j)}) - 1].$$9.

2. Tangent Space Hermite Interpolation (THI):

   • All data is mapped to a single tangent space $n$0 via the Riemannian logarithm and vector transport, and classical Hermite interpolation is performed in Euclidean space, with the result mapped back to the manifold by the exponential map. The base point $n$1 can be arbitrary (e.g., the Riemannian barycenter of the samples).

Both approaches permit interpolation of manifold-valued functions with derivative matching, and numerical comparisons confirm that THI achieves higher accuracy at higher computational cost, while BHI is intrinsic and base-point independent.

7. Compressive and High-Dimensional Hermite Interpolation

For high-dimensional domains and sparse approximation, gradient-augmented (Hermite) compressive interpolation achieves optimal or near-optimal sample complexity by combining function and gradient samples via weighted $n$2 minimization in an orthonormal polynomial basis (Adcock et al., 2017). The advantages include:

• Error bounds in a stronger Sobolev norm ($n$3 rather than $n$4), due to the incorporation of derivative information.
• Sample complexity that scales algebraically with the sparsity $n$5 and at most logarithmically in the ambient dimension $n$6.
• Significant empirical gains in accuracy and sample efficiency, particularly for partial or mixed sampling strategies.

The approach is consequential for uncertainty quantification, high-dimensional approximation, and compressed sensing in computational mathematics.

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In summary, multivariate Hermite interpolation underpins a spectrum of theories—from explicit coordinate-wise closed formulas and spline constructions to symmetric, algebraic-geometric, and manifold-valued generalizations. The diversity of settings—structured grids, arbitrary nodes, algebraic varieties, noncommutative rings, Riemannian manifolds, and high-dimensional domains—has fostered a multiplicity of analytic, algorithmic, and algebraic tools, with both exact and approximate construction methods and a rich landscape of existence, uniqueness, and singularity theorems (Kechriniotis et al., 2023, Zimmermann et al., 2022, Meng et al., 2013, Krick et al., 30 Jan 2025, Durakov et al., 2022, Adcock et al., 2017).