Chebyshev Interpolation Scheme

• Chebyshev interpolation is a numerical method that approximates functions using Chebyshev polynomials evaluated at optimally placed nodes, reducing maximum error and avoiding the Runge phenomenon.
• It guarantees spectral convergence for analytic functions through nonuniform node distribution, leading to exponential error decay and near-best approximation rates.
• Efficient digital implementations via discrete cosine transforms make the method vital for high-performance computing, signal reconstruction, and solving differential equations.

The Chebyshev interpolation scheme is a numerical method for approximating functions using polynomials constructed at a set of optimally placed nodes derived from the roots or extrema of Chebyshev polynomials. These nodes minimize maximum interpolation error on the interval and underpin the spectral convergence observed for analytic target functions. Chebyshev interpolation is central to numerous computational mathematics applications, including signal reconstruction, rootfinding, numerical quadrature, multidimensional approximation, and high-performance scientific computing architectures.

1. Mathematical Foundation of Chebyshev Interpolation

The Chebyshev polynomials of the first kind, $T_n(x)$, are defined recursively by

$$T_0(x) = 1,\quad T_1(x) = x,\quad T_{n+1}(x) = 2x\, T_n(x) - T_{n-1}(x)$$

and have the explicit trigonometric form

$T_n(x) = \cos(n \arccos x),\quad x\in[-1,1].$

Chebyshev interpolation, in its standard form, approximates $f(x)$ on $[-1,1]$ by a truncated Chebyshev series:

$f(x) \approx \sum_{k=0}^n a_k T_k(x)$

where the coefficients $a_k$ are given by the (discrete) orthogonality relation

$a_k = \frac{2}{n}\sum_{j=1}^n f(x_j) T_k(x_j)$

with the nodes $x_j$ being Chebyshev nodes:

$$x_j = \cos\left(\frac{2j - 1}{2n}\pi\right),\quad j=1,2,\ldots,n.$$

The nodes are nonuniform (clustered at the endpoints), yielding an optimal distribution for polynomial interpolation that avoids the Runge phenomenon. For general intervals $[a,b]$, an affine mapping is used.

For multivariate domains, the tensor product extension is given by

$$I_{\vec N}(f)(\vec x) = \sum_{\vec j\in J} c_{\vec j}\, T_{\vec j}(\vec x), \quad T_{\vec j}(\vec x) = \prod_{i=1}^D T_{j_i}(x_i).$$

2. Key Properties: Error, Convergence, and Lebesgue Constants

The spectral convergence of Chebyshev interpolation arises from the analytic properties of the interpolated function. For $f$ analytic in a region containing $[-1,1]$, the $L_\infty$ error satisfies

$$\max_{x\in[-1,1]} |f(x) - I_n(f)(x)| \le C \rho^{-n}$$

where $\rho>1$ depends on the size of the Bernstein ellipse in which $f$ can be analytically continued (Glau et al., 2016, Gaß et al., 2015). Multivariate error bounds are sharpened by minimizing over permutations of dimensions and account for exponential error decay in each coordinate direction.

A central element in interpolation analysis is the Lebesgue constant $\Lambda_n$, which quantifies the worst-case amplification of data errors by the interpolation process. For interpolation at Chebyshev nodes, $\Lambda_n$ grows only logarithmically with $n$, in contrast to the exponential growth observed for equispaced nodes. However, for classical polynomial interpolation in the uniform norm, $\Lambda_n$ can still be significant. Filtered Chebyshev interpolation using de la Vallée Poussin means produces interpolants with uniformly bounded Lebesgue constants in weighted norms (e.g., Jacobi weights), provided necessary and sufficient inequalities on the exponents hold (Occorsio et al., 2020):

$0 \le \gamma \le 1,\quad 0 \le \delta \le 1$

for the standard Chebyshev weight.

The uniform boundedness of the Lebesgue constant ensures near-best approximation order: the error is always within a constant factor of the best polynomial approximation error of the considered degree.

3. Digital Architectures and Efficient Computation

Chebyshev interpolation is not only theoretically optimal for approximation but also amenable to efficient digital implementation. The discrete cosine transform (DCT) computes the Chebyshev coefficients from sampled data:

$\mathbf{c} = C \mathbf{f}$

where $C$ is a cosine matrix defined explicitly in terms of the Chebyshev nodes (Tulabandhula, 2010). Digital architectures exploit systolic arrays for efficiently performing matrix-vector multiplications and recursive evaluation of Chebyshev polynomials.

Pipelined, word-serial designs enable real-time processing and 100% hardware utilization, allowing Chebyshev interpolation to be integrated directly into ADC-based systems. Furthermore, nonuniform Chebyshev sampling enables a hybrid ADC architecture: slow, low-power SAR ADCs are used for large-interval samples, and fast, high-power flash ADCs only for dense clusterings near interval edges, yielding as much as a $1/3$ reduction in total power consumption compared to equispaced sampling (Tulabandhula, 2010).

4. Comparison with Equispaced and Classical Methods

The clustered nature of Chebyshev nodes provides significant benefits over polynomial interpolation at equispaced nodes. Specifically:

• Reconstruction Error: For a given number of nodes, the maximum interpolation error with Chebyshev nodes is orders of magnitude less than that with equispaced nodes, especially for oscillatory or highly regular functions. Equispaced interpolation may encounter the Runge phenomenon, whereas Chebyshev interpolation remains stable (Tulabandhula, 2010).
• Sample Efficiency: For a target approximation error, fewer Chebyshev samples are needed than equispaced samples.
• Robustness: Chebyshev nodes minimize error both in the uniform and weighted norms (after filtering), enhancing robustness even in the presence of function singularities or low regularity.
• Numerical Stability: The well-conditioned nature of the Chebyshev basis and the logarithmic (or bounded, post-filtering) growth of Lebesgue constants provide numerical stability not shared by classical (Lagrange) interpolation.

Filtered Chebyshev interpolation with de la Vallée Poussin means further improves upon standard Lagrange interpolation by bounding the Lebesgue constant and controlling the Gibbs phenomenon, yielding errors that are comparable to the best polynomial approximation rate even in uniform norms (Occorsio et al., 2020, Bonis et al., 2020).

5. Applications and Extensions

The theoretical and computational properties of Chebyshev interpolation have led to a wide range of applications:

• Signal Reconstruction: Used both in direct digital signal processing systems and for the design of energy-efficient ADC architectures (Tulabandhula, 2010, Amartey, 30 Mar 2024).
• Multivariate Approximation: Tensorized or non-tensorial Chebyshev grids enable efficient high-dimensional interpolation with error bounds optimized using properties of analyticity in polydisks or generalized Bernstein ellipses (Glau et al., 2016, Gaß et al., 2015).
• Numerical Solution of Differential Equations: Chebyshev collocation forms the basis of pseudospectral methods for ODEs and PDEs. Filtered Chebyshev interpolants yield optimally convergent schemes for integro-differential equations such as the Prandtl equation, with fast algorithms based on banded linear systems and uniformly bounded condition numbers (Bonis et al., 2020).
• Scientific Computing: In kernel methods (e.g., fast multipole methods), Chebyshev interpolation provides a kernel-independent, spectrally convergent mechanism for function and operator approximation, facilitating low-rank compression and fast evaluation via optimized BLAS-level algorithms (Messner et al., 2012).
• Adaptive and Partition of Unity Methods: When functions have nearby singularities, overlapping domain decompositions with smooth partition of unity weights allow for spectrally accurate, globally smooth interpolants by combining local Chebyshev interpolants (Aiton et al., 2017).
• Sparse Interpolation and Error Correction: In settings where the function is (or is assumed) sparse in the Chebyshev basis, efficient sparse interpolation with robust error correction is possible, reducing required evaluations and providing error resilience (Kaltofen et al., 2019, Hubert et al., 2020).

6. Advanced Filtering and the Gibbs Phenomenon

Classical Chebyshev or Lagrange interpolation at Chebyshev nodes may still display Gibbs phenomenon near discontinuities. The filtered Chebyshev interpolation scheme, employing de la Vallée Poussin filters, defines the interpolant through a weighted sum of Chebyshev (or orthogonal polynomial) basis functions:

$$V_{n}^{m}f(x) = \sum_{k=1}^{n} f(x_k) \varphi_{n, k}^{m}(x)$$

with filter coefficients $\mu_{n, j}^{m}$ that decay linearly in a high-frequency transition band. This approach retains the interpolation property (interpolatory at nodes), ensures uniform convergence under Jacobi-weighted norms with simple constraints on exponents, and provides controlled suppression of spurious oscillations. The method realizes nearly best uniform approximation error and strongly reduces Gibbs oscillations, as evidenced by numerical experiments (Occorsio et al., 2020).

The main results specify:

• Necessary and Sufficient Conditions for Uniform Boundedness: Explicit inequalities on Jacobi weight exponents, e.g., $0 \leq \gamma, \delta \leq 1$ for Chebyshev weight $w_1$.
• Optimal Convergence Rate: The VP scheme achieves the error of best degree-$n$ polynomial approximation, with a rate improving with the function's smoothness.
• Comparison with Lagrange: Lagrange interpolation at Chebyshev nodes has unbounded Lebesgue constants in the uniform norm (typically growing logarithmically with $n$), whereas the VP filter can render these constants uniformly bounded, eliminating the log-factor penalty in approximation error bounds (Bonis et al., 2020).

7. Numerical Experiments and Practical Considerations

Numerical studies corroborate the theoretical advantages of Chebyshev interpolation and its filtered variants:

• For very smooth functions, filtered (VP) and Lagrange interpolants have similar accuracy; for lower regularity, VP schemes show considerably smaller errors, especially near singularities.
• The Gibbs phenomenon is not only diminished near discontinuities but also suppressed along the entire approximation interval for VP interpolants.
• The VP filtered interpolation operator, constructed with parameters $m = \lfloor \theta n \rfloor$, provides additional tunability: higher $m$ (larger proportion of filtering) yields more localized basis functions, further reducing overshoots (Occorsio et al., 2020).

Tables and figures in the cited literature demonstrate the uniform boundedness of the Lebesgue constant under proper weighting, the near-best convergence behavior of filtered Chebyshev interpolants, and the superior localization properties responsible for reduction of artifacts such as the Gibbs effect.

Summary

Chebyshev interpolation, particularly in its advanced, filtered forms, combines the theoretical virtues of near-minimax polynomial approximation, practical algorithmic stability, and hardware efficiency. The use of Chebyshev nodes ensures exponential convergence for analytic functions and suppresses major sources of numerical instability. Filtered Chebyshev interpolants, employing de la Vallée Poussin means, preserve the interpolatory property while enforcing uniform convergence in Jacobi-weighted norms, with explicit necessary and sufficient conditions for stability and error control. These schemes are effective in both classical approximation problems and high-performance computation, providing a robust foundation for modern scientific, engineering, and applied mathematical applications (Occorsio et al., 2020, Bonis et al., 2020).