Hermite–Trigonometric Interpolation Methods

Updated 22 May 2026

Hermite–trigonometric interpolation is the construction of trigonometric polynomials that interpolate both periodic function values and their derivatives, ensuring spectral accuracy.
It leverages FFT-based and barycentric methods to solve small linear systems efficiently on uniform and nonuniform grids with an O(N log N) complexity.
This robust framework finds applications in computational analysis, geometric design, and data fitting, supported by rigorous convergence and error guarantees.

Hermite–trigonometric interpolation is the construction of trigonometric polynomials or splines that interpolate, at a prescribed set of nodes, both the values of a periodic function and finitely many of its derivatives. In the periodic (typically $2\pi$ -periodic) setting, it provides a natural analogue to classical Hermite polynomial interpolation, preserving spectral accuracy and exploiting the harmonic structure of the domain. The fundamental idea dates back to Hermite's 1885 cotangent formulæ and underpins a range of modern computational and analytic techniques. Implementations involve either direct Fourier-based solutions on uniform grids or barycentric and Lagrange-type constructions for nonuniform settings, and admit efficient algorithms and rigorous convergence guarantees.

1. Historical and Theoretical Foundations

The genesis of Hermite–trigonometric interpolation is traced to Hermite’s partial-fraction cotangent expansions for periodic interpolation, explicitly relating function values and derivatives at distinct points using trigonometric identities. The classical cotangent identity states that for $n$ distinct points $a_j$ modulo $\pi$ , the rational function $\cos(nz)-\cos(\sum a_j)$ divided by $\prod_{j=1}^n\sin(z-a_j)$ admits a decomposition as a sum over $\cot(z-a_j)$ times rational weights. Modern developments—motivated by applications in function theory, special functions, and numerical analysis—systematize all rational trigonometric interpolants (with or without Hermite data) as combinations of finite Fourier series plus sums over simple poles at prescribed nodes, where the residues are themselves trigonometric polynomials of bounded degree. This principle underlies the explicit formulae of both W. Chu and W. Johnson, as well as their generalizations via Meijer–Nørlund–Braaksma expansions (Dyachenko et al., 2023).

2. Problem Statement and Formulation

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a $2\pi$ -periodic function with $m$ continuous and periodic derivatives. Fix $n$ 0 (odd) and uniformly spaced nodes $n$ 1, $n$ 2. The Hermite–trigonometric interpolation problem seeks a trigonometric polynomial $n$ 3, of degree $n$ 4, such that for each node $n$ 5 and each derivative order $n$ 6, $n$ 7. By the periodicity and the dimension count (matching the total number of interpolation conditions to the degrees of freedom in the trigonometric basis), there exists a unique solution for each choice of $n$ 8 and $n$ 9 (Denysiuk, 2021, Denysiuk, 2019).

3. Methods of Construction

3.1 Fourier/FFT-Based Construction on Uniform Grids

On a uniform grid, the global Hermite–trigonometric interpolant can be represented as a single trigonometric polynomial: $a_j$ 0 or equivalently in exponential form. Substituting the grid nodes and their derivatives into this basis yields, for each Fourier frequency $a_j$ 1 $a_j$ 2, an independent $a_j$ 3 linear system coupling the coefficients at $a_j$ 4 and $a_j$ 5. For $a_j$ 6 (first-derivative matching), this reduces to a $a_j$ 7 system; for $a_j$ 8, a $a_j$ 9 block; and so forth. The right-hand sides are computed by FFTs of the sampled function and its derivatives. The coefficient blocks—depending only on $\pi$ 0—can be inverted or factorized in advance. Overall, the interpolation pipeline consists of sampling $\pi$ 1 and its derivatives, FFT computation, small linear solves per $\pi$ 2, and collection of the final coefficients. The asymptotic complexity is $\pi$ 3 (Denysiuk, 2021, Denysiuk, 2019).

Algorithmic Workflow

Step	Operation	Complexity
1. Sampling	Evaluate $\pi$ 4 and derivatives at nodes	$\pi$ 5
2. FFT	Compute discrete Fourier coefficients	$\pi$ 6
3. Linear solves	Solve $\pi$ 7 blocks $\pi$ 8	$\pi$ 9
4. Synthesis	Form interpolant's coefficients	$\cos(nz)-\cos(\sum a_j)$ 0

3.2 Barycentric and Lagrange-Type Constructions

Beyond uniform grids, barycentric trigonometric Hermite interpolation employs rational basis functions $\cos(nz)-\cos(\sum a_j)$ 1 that satisfy the Lagrange property $\cos(nz)-\cos(\sum a_j)$ 2. Hermite data are incorporated by recursive lifting using vanishing factors and iterative corrections: $\cos(nz)-\cos(\sum a_j)$ 3 with the global interpolant written via barycentric sums, sharing a common denominator. Differentiation matrices accelerate the evaluation of recursive corrections for higher derivatives. This construction is applicable for arbitrary Hermite order $\cos(nz)-\cos(\sum a_j)$ 4 and generalizes to scattered node sets. It provides barycentric stability and nearly optimal Lebesgue constant behavior for analytic $\cos(nz)-\cos(\sum a_j)$ 5 (Elefante, 2022).

4. Analytic Properties: Uniqueness, Stability, and Error

Hermite–trigonometric interpolation is unisolvent under standard regularity and bandwidth conditions, provided the number of interpolation conditions matches the dimension of the trigonometric polynomial space. For analytic $\cos(nz)-\cos(\sum a_j)$ 6, the interpolant achieves spectral (exponential) convergence as $\cos(nz)-\cos(\sum a_j)$ 7 increases for $\cos(nz)-\cos(\sum a_j)$ 8; for $\cos(nz)-\cos(\sum a_j)$ 9, empirical rates slow to algebraic $\prod_{j=1}^n\sin(z-a_j)$ 0. The FFT-based methods, as well as the barycentric approach, are numerically stable provided that the kernels or differentiation matrices are well-conditioned for the regime $\prod_{j=1}^n\sin(z-a_j)$ 1 and the grid avoids clustering-associated ill-conditioning. The error decays as $\prod_{j=1}^n\sin(z-a_j)$ 2 for $\prod_{j=1}^n\sin(z-a_j)$ 3 (Denysiuk, 2021, Elefante, 2022).

5. Explicit Formulae and Generalized Interpolation Identities

All Hermite–type trigonometric interpolation identities can be cast as rational decompositions using trigonometric partial fractions. The general principle is that the ratio of a trigonometric (or exponential-Laurent) numerator to a product of sines at nodes admits a decomposition as a finite Fourier polynomial plus a sum over simple poles at those nodes, with weights/rational functions determined by residue calculation or combinatorial algebra. Examples include Hermite's cotangent expansion, Chu’s Laurent polynomial formulation, and Johnson’s two-variable polynomial ratios. These cover both classical (Lagrange-type) and Hermite (multiple-derivative) interpolation (Dyachenko et al., 2023).

Source/Author	Formulation	Setting
Hermite (1885)	Cotangent/sine expansion	General trigonometric polynomials
Meijer, Nørlund, Braaksma	Barnes, hypergeometric	Mellin–Barnes integrals, hypergeom.
Chu, Johnson	Partial fractions	Polynomial/Fourier numerators
Dyachenko–Karp (2023)	General residue–Fourier form	Arbitrary Laurent numerators and nodes

6. Applications and Extensions

Hermite–trigonometric interpolation underpins the construction of ellipse-preserving subdivision schemes and exponential Hermite splines. Explicit compactly supported basis functions in spaces of exponential polynomials $\prod_{j=1}^n\sin(z-a_j)$ 4 enable exact reproduction of sinusoids and affine invariance of interpolated curves. These bases yield nonstationary and stationary vector subdivision algorithms converging to $\prod_{j=1}^n\sin(z-a_j)$ 5 continuous, fourth-order accurate limit curves. The interpolants serve in geometric design and data fitting, especially for periodic/closed data, and can be cast equivalently in exponential Bézier or Hermite form (Conti et al., 2014).

7. Practical Considerations and Numerical Implementation

Efficient Hermite–trigonometric interpolation is dominated by FFT cost and small-dimension block solves, admitting precomputation of all kernel matrices for fixed grid sizes and Hermite order, further accelerating repeated interpolations for variable data sets. For barycentric schemes, all differentiation and correction matrices can be precomputed for fixed node configurations. Stability is maintained for moderate Hermite order $\prod_{j=1}^n\sin(z-a_j)$ 6 compared with node count $\prod_{j=1}^n\sin(z-a_j)$ 7. Grid and node choice interacts with aliasing and convergence behavior; for analytic or bandlimited $\prod_{j=1}^n\sin(z-a_j)$ 8, moderate $\prod_{j=1}^n\sin(z-a_j)$ 9 and oversampling safeguards stability and accuracy. Practical implementations are documented, for example, in Denysiuk–Hryshko and others (Denysiuk, 2021, Denysiuk, 2019, Elefante, 2022).