Chebyshev Interpolation

• Chebyshev interpolation is a polynomial approximation method that uses non-uniform Chebyshev nodes to suppress the Runge phenomenon and achieve near-minimax uniform accuracy.
• It employs Chebyshev polynomials with FFT-accelerated algorithms and cosine expansions to deliver exponential convergence and efficient error control across multiple dimensions.
• The method's numerical stability, adaptability through filtering techniques, and applications in spectral methods, uncertainty quantification, and computational finance make it a robust tool in modern numerical analysis.

Chebyshev interpolation is a polynomial interpolation scheme based on evaluation at specially chosen, non-uniform Chebyshev nodes, resulting in near-minimax uniform approximation and stable numerical properties. The method utilizes the structure of Chebyshev polynomials and their extremal distribution of nodes to suppress the Runge phenomenon and achieve exponential convergence for analytic functions. Chebyshev interpolation generalizes efficiently to higher dimensions and serves as the foundation for numerous modern algorithms in spectral methods, uncertainty quantification, numerical analysis, and computational finance. This article details the mathematical structure, algorithmic implementation, theoretical guarantees, practical extensions, and modern research related to Chebyshev interpolation.

1. Mathematical Foundations and Node Selection

Chebyshev interpolation exploits the properties of Chebyshev polynomials of the first kind, defined for $x \in [-1, 1]$ as $T_n(x) = \cos(n\,\arccos x)$. These polynomials have explicit extremal and optimality structures on $[-1,1]$:

• The zeros of $T_{n+1}$ give the Chebyshev–Gauss nodes, while the extrema provide the Chebyshev–Gauss–Lobatto nodes.
• For degree $N$, Chebyshev–Gauss nodes are $x_k = \cos\left(\frac{2k+1}{2N+2}\pi\right)$, $k=0,1,\dots,N$ (Diehl et al., 2023).
• Chebyshev nodes cluster quadratically near the endpoints, a property central for controlling large oscillations in high-degree interpolation (Runge phenomenon) (Amartey, 2024).

The optimality of Chebyshev or scaled Chebyshev grids follows from classical minimax theory; among all node distributions interpolating in $C_M^{s+1}[-1,1]$, scaled Chebyshev nodes minimize the max-norm of the error factor $\max_{x \in[-1,1]}|\prod_i(x-c_i)|$ (Hoang, 2013).

2. Polynomial Construction and Algorithms

Given function samples $f_k$ at Chebyshev nodes $T_n(x) = \cos(n\,\arccos x)$0, the interpolating polynomial $T_n(x) = \cos(n\,\arccos x)$1 can be expressed in several mathematically equivalent but numerically distinct forms:

• Lagrange form: $T_n(x) = \cos(n\,\arccos x)$2, where $T_n(x) = \cos(n\,\arccos x)$3 is the fundamental Lagrange basis (Gaß et al., 2015).
• Chebyshev (cosine) expansion: $T_n(x) = \cos(n\,\arccos x)$4, with coefficients computed via a discrete cosine transform (DCT) (Amartey, 2024).
• Barycentric form: The numerically preferred form,

$T_n(x) = \cos(n\,\arccos x)$5

with barycentric weights $T_n(x) = \cos(n\,\arccos x)$6 for Gauss nodes (Diehl et al., 2023, Laris et al., 2018). Evaluation is $T_n(x) = \cos(n\,\arccos x)$7, insensitive to rounding error, and vectorizable in modern implementations.

Key features:

• At node $T_n(x) = \cos(n\,\arccos x)$8, $T_n(x) = \cos(n\,\arccos x)$9 (Lagrange property).
• If $[-1,1]$0 is very close to a node, direct evaluation with $[-1,1]$1 avoids cancellation.

FFT-accelerated algorithms and DCTs enable $[-1,1]$2 coefficient computation for large-scale problems (Tulabandhula, 2010, Amartey, 2024).

3. Error Analysis and Convergence

The error of Chebyshev interpolation admits both classical and modern bounds:

• Smoothness-based (Sobolev) bound: If $[-1,1]$3, $[-1,1]$4 (Diehl et al., 2023).
• Analyticity-based (Bernstein ellipse) bound: For $[-1,1]$5 analytic in the open Bernstein ellipse of parameter $[-1,1]$6,

$[-1,1]$7

with $[-1,1]$8, yielding geometric (spectral) convergence (Laris et al., 2018, Glau et al., 2016, Amartey, 2024, Gaß et al., 2015).

The pointwise interpolation error is near-minimax: for the Chebyshev grid, the interpolant $[-1,1]$9 is within a fixed constant of the best uniform approximation polynomial of degree $T_{n+1}$0 (Hoang, 2013).

Lebesgue constants, which govern the maximal error amplification from nodal noise, grow logarithmically for Chebyshev grids: $T_{n+1}$1, as opposed to exponential growth for equispaced nodes (Amartey, 2024, Liu et al., 22 Nov 2025).

4. Multidimensional Tensorization and Generalizations

Chebyshev interpolation extends naturally to multivariate settings via tensor-product constructions:

• For $T_{n+1}$2 dimensions, nodes are defined as $T_{n+1}$3, $T_{n+1}$4 for each dimension (Diehl et al., 2023).
• The full tensor grid comprises $T_{n+1}$5 nodes.
• The interpolant is expressed as a sum over tensor-product basis functions: $T_{n+1}$6 (Gaß et al., 2015, Glau et al., 2016).

Error bounds for the multivariate case generalize the 1D result. If $T_{n+1}$7 is analytic in a poly-ellipse with radii $T_{n+1}$8, then

$T_{n+1}$9

and sharper mixed-derivative bounds (see (Glau et al., 2016) for advanced technical refinements).

High-dimensional tensor grids face the curse of dimensionality; modern approaches use adaptive, sparse, or low-rank tensor decompositions (e.g., functional Tucker expansion, as in (Dolgov et al., 2020)), and alternative node sets such as Lissajous–Chebyshev or Padua points to balance complexity and accuracy (Dencker et al., 2015, Hubert et al., 2020).

5. Stability, Filtering, and Gibbs Phenomenon

Chebyshev interpolation at classical nodes avoids the Runge phenomenon and is stable in both the uniform and weighted norms. However, classical Lagrange interpolation at Chebyshev nodes still incurs Lebesgue constants $N$0 and suffers from Gibbs oscillations near jump discontinuities.

Filtered polynomial interpolation, notably through de la Vallée–Poussin (VP) operators, combines local Lagrange interpolation with spectral filtering:

• The VP interpolant $N$1 at Chebyshev nodes applies filter coefficients $N$2 to damp high-degree modes while preserving interpolation at the nodes (Occorsio et al., 2020, Occorsio et al., 2021).
• VP interpolation uniformly bounds the Lebesgue constant—no logarithmic growth—provided Jacobi weight exponents satisfy explicit inequalities (Occorsio et al., 2020).
• Filtered Chebyshev interpolants exhibit strong suppression of the Gibbs phenomenon, preserving spectral convergence in smooth pieces and rapidly damping overshoots at discontinuities.

In the context of weighted spaces (Jacobi weights), necessary and sufficient conditions characterize when the VP operator’s Lebesgue constant is uniformly bounded, ensuring near-best approximation order, i.e., $N$3, where $N$4 is the best approximation error by degree-$N$5 polynomials (Occorsio et al., 2020).

6. Applications and Adaptive Strategies

Chebyshev interpolation underpins a diverse array of applications:

• Numerical PDEs and spectral methods: Used for constructing differentiation matrices, pseudospectral solvers, and projection operators in both regular and singularly perturbed settings (Aiton et al., 2017).
• Sparse and compressed reconstruction: Sparse Chebyshev interpolation recovers functions with few nonzero Chebyshev coefficients with sample complexity $N$6 in the multivariate setting, dramatically reducing the number of required samples relative to tensor grids (Hubert et al., 2020, Kaltofen et al., 2019).
• Signal processing: Chebyshev interpolation's minimax and cosine-series structure enables stable signal reconstruction from nonequispaced samples, outperforming equispaced and even some Fourier methods in presence of endpoint singularities or Gibbs artifacts (Amartey, 2024, Tulabandhula, 2010).
• Derivative computation (“Greeks”) in finance: Once built, the derivative of a Chebyshev interpolant is available in closed form, allowing ultra-efficient and stable sensitivity calculations for option pricing and risk metrics (Laris et al., 2018, Glau et al., 2017).
• Algorithm architectures: Digital and hardware-efficient architectures, such as systolic and word-serial DCT engines, implement Chebyshev interpolation efficiently for real-time systems, often with major reductions in sampling rate and power consumption compared to equispaced approaches (Tulabandhula, 2010).

Adaptive partition and overlapping partition of unity methods resolve non-analyticities and local singularities by locally splitting the interval and gluing together local Chebyshev interpolants with smooth weights, preserving spectral convergence globally and reducing node counts by orders of magnitude compared to a global interpolant (Aiton et al., 2017).

7. Modern Developments, Generalizations, and Stability

Recent research addresses both theoretical and practical aspects:

• Stable polynomial interpolation on nonequispaced nodes: Grünwald-type operators and symmetric-wave interpolation establish strong convergence and stability even for certain perturbed or equidistant node sets, provided filtering or node warping is applied (C, 21 Nov 2025, Liu et al., 22 Nov 2025).
• Generalized Chebyshev bases: Extensions to generalized Chebyshev polynomials associated with root systems and Weyl groups facilitate sparse interpolation for high-dimensional function spaces with algebraic or symmetry constraints (Hubert et al., 2020, Dencker et al., 2015).
• Error bounds in high dimensions: Recent sharp bounds (Glau & Mahlstedt (Glau et al., 2016)) provide tight, dimension-adapted error estimates for tensor-product Chebyshev interpolation.
• Practical guidelines: For high-accuracy or high-dimensional tasks, careful selection of node distribution (e.g., scaled Chebyshev, adaptive partitioning) and the use of filtering (VP or similar) optimizes both approximation order and computational stability (Hoang, 2013, Occorsio et al., 2020, Occorsio et al., 2021).

Filtered and adaptively constructed Chebyshev interpolants, as well as low-rank tensorized surrogates, remain the methods of choice whenever uniform accuracy, high stability, and computational efficiency are critical.

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References: All results and constructions detailed above are grounded in the cited arXiv works (Diehl et al., 2023, Hoang, 2013, Occorsio et al., 2020, Occorsio et al., 2021, Glau et al., 2016, Hubert et al., 2020, Liu et al., 22 Nov 2025, C, 21 Nov 2025, Dencker et al., 2015, Laris et al., 2018, Gaß et al., 2015, Glau et al., 2017, Tulabandhula, 2010, Amartey, 2024, Aiton et al., 2017, Dolgov et al., 2020, Sun, 2018, Kaltofen et al., 2019, 1605.02776).