Symmetric Wave Interpolation Method

• The SWI method constructs a globally smooth, polynomial-type interpolant by mapping equidistant nodes onto Chebyshev grids and then reverting to the original domain.
• It leverages discrete cosine transforms for efficient coefficient computation, resulting in faster convergence and improved endpoint stability.
• SWI is applicable in spectral methods, signal processing, and financial data fitting, offering stable exponential convergence using uniform sample sets.

The symmetric wave interpolation (SWI) method is a global interpolation technique designed to combine the practical accessibility of equidistant sampling nodes with the superior numerical stability of Chebyshev interpolation. SWI achieves stable and accurate polynomial-type approximation from uniformly spaced data, circumventing the classical Runge phenomenon, and—according to recent experimental benchmarks—often surpasses conventional Chebyshev interpolation in both error and efficiency. The method's core mechanism is a symmetric mapping that translates equidistant points onto Chebyshev grids, enabling the computation of stable Chebyshev expansions which are then "unwound" back onto the original domain, yielding globally smooth interpolants that respect the original data grid (Liu et al., 22 Nov 2025).

1. Underlying Problem and Motivation

In global interpolation, the objective is to construct a function $S_n(x)$ such that $S_n(x_i) = f(x_i)$ for a set of nodes $x_i$ in $[-1,1]$. Equidistant nodes $x_i = -1 + 2i/n$ are prevalent in applications like uniform sensor arrays and financial time-series, yet polynomial interpolation on such grids is prone to the Runge phenomenon: large oscillations and error explosions near domain endpoints as $n$ increases. Chebyshev point-based interpolants, using $z_i = \cos(πi/n)$ (first kind) or $z_i = \cos((2i+1)π/[2(n+1)])$ (second kind), mitigate these instabilities but require nonuniform node placement, which is often unavailable in practical datasets. SWI bridges this critical gap by transferring Chebyshev stability to equidistantly sampled data.

2. Symmetric Mapping: Equidistant to Chebyshev Nodes

SWI utilizes specific mappings $K_1(x)$ and $K_2(x)$ to transform equidistant nodes into Chebyshev grids:

• First-kind mapping: $K_1(x) = \cos[ π(x+1)/2 ]$ ensures $K_1(x_i) = \cos(πi/n) = z_i^{[1]}$.
• Second-kind mapping: $K_2(x) = \cos[ π(nx + n + 1) / (2(n+1)) ]$ guarantees $K_2(x_i) = \cos[(2i+1)π/(2(n+1))] = z_i^{[2]}$.

The inverses $T_1(z)$ and $T_2(z)$ are known in closed form, respectively:

• $T_1(z) = -1 + (2/π)\arccos(z)$
• $T_2(z) = [-(n+1) + (2(n+1)/π)\arccos(z)]/n$

This bidirectional correspondence permits evaluation and expansion in either coordinate set without loss of information.

3. Construction of the Symmetric Wave Interpolant

For a given smooth function $f(x)$, SWI constructs the "pulled-back" function $g^{[j]}(z) := f(T_j(z))$ on the transformed Chebyshev grid and expresses it as a degree-$n$ Chebyshev expansion:

$p_n^{[j]}(z) = \sum_{k=0}^n c_k^{[j]} T_k(z)$

where $T_k(z) = \cos(k \arccos(z))$ are Chebyshev polynomials. The interpolant on the original domain is thus

$$S_n^{[j]}(x) = p_n^{[j]}(K_j(x)) = \sum_{k=0}^n c_k^{[j]} \psi_k^{[j]}(x)$$

with $\psi_k^{[j]}(x) = \cos[k \arccos(K_j(x))]$ termed the "symmetric wave" basis (Editor's term). These basis functions inherit the oscillatory structure of Chebyshev polynomials but are defined directly on the equidistant domain.

4. Coefficient Computation and Implementation Workflow

Critical to SWI's practicality is that interpolant coefficients $c_k^{[j]}$ can be computed via discrete cosine transforms (DCTs), paralleling standard Chebyshev workflows:

• First-kind SWI:

$$c_k^{[1]} = \frac{2}{n}\sum_{i=0}^n y_i \cos\bigl(k\pi i/n\bigr) \omega_i^{[1]}$$

with $\omega_i^{[1]} = 1$ for $0 < i < n$, $1/2$ for $i=0$ or $n$.

• Second-kind SWI:

$$c_k^{[2]} = \frac{2}{n+1}\sum_{i=0}^n y_i \cos\bigl(k (2i+1)\pi/[2(n+1)]\bigr)$$

This approach is computationally efficient: SWI adds only $O(n)$ overhead to classical Chebyshev interpolation. Computing the mapping $K_j(x_i)$ is $O(n)$, the DCT is $O(n \log n)$, and evaluating $S_n^{[j]}(x)$ can be performed via Clenshaw recurrence or inverse DCT with matching complexity.

5. Theoretical Stability and Error Suppression

SWI leverages the classical error bounds of Chebyshev interpolation applied to the mapped function $g^{[j]}$. Notably,

• First-kind SWI error:

$$\|f - S_n^{[1]}\|_\infty \leq \frac{2^{-2(n+1)}}{(n+1)!} \max_{z \in [-1,1]} |(g^{[1]})^{(n+1)}(z)|$$

• Second-kind SWI error (with parameter $\epsilon > 0$):

$$\max_{x \in [-1,1]} |f(x) - S_n^{[2]}(x)| < \frac{2^{1-n}}{(n+1)!} \max_{z \in [-1,1]} |(g^{[2]})^{(n+1)}(z)| + \epsilon$$

These bounds exhibit exponential decay in $n$ and explain the suppression of Runge-type endpoint oscillations: after mapping, the problematic regions are handled within the Chebyshev grid's well-conditioned structure.

6. Benchmarks and Comparative Performance

Experimental comparisons assess SWI (both first-kind SWI₁-Equid and second-kind SWI₂-Equid) against Chebyshev interpolation (CI-Cheby) on standard test functions including the Runge function $f_1(x)=1/(1+25x^2)$, Gaussian bumps, oscillatory rationals, and asymmetric humps. Results indicate:

• SWI achieves uniformly small $\max$-norm and $L^1$ errors at elevated $n$, without endpoint oscillations.
• For target error $\mathrm{max}\ < 10^{-3}$, SWI typically requires fewer points than Chebyshev interpolation. For example:

Function	CI‐Cheby (n₁)	SWI‐Equid (n₂)
Runge f₁	20	16
Oscill. f₂	56	40

In nearly all tested cases (except for two pathological variants), SWI outperformed or matched Chebyshev in sample efficiency, with error curves consistently demonstrating faster convergence and increased endpoint stability (cf. Figure 1, (Liu et al., 22 Nov 2025)).

7. Advantages, Limitations, and Domains of Application

SWI’s key advantage is its ability to deliver Chebyshev-level stability and exponential convergence using only equidistant samples—a common data format across science and engineering. It is robust to data rounding and retains global smoothness, with no need for domain remeshing. A limitation inherited from Chebyshev theory is sensitivity to feature location: for functions with sharp variations close to $x = \pm 1$, the mapping further concentrates points near boundaries, possibly necessitating additional expansion terms for high fidelity. Adaptive mappings and hybrid, piecewise strategies are plausible remedies.

SWI is applicable wherever stable global polynomial-type approximation is required from uniform samples, with domains including spectral methods for PDEs, signal processing with uniform time steps, actuator-sensor networks, and financial data fitting.

By providing a mathematically rigorous and practically convenient tool for global function approximation on equidistant data, symmetric wave interpolation closes a persistent gap between numerical stability and point accessibility (Liu et al., 22 Nov 2025).