Constrained mock-Chebyshev least squares approximation for Hermite interpolation

TL;DR

The paper tackles the problem of approximating a function on $[-1,1]$ from equispaced nodes where Runge phenomenon undermines polynomial interpolation. It extends the constrained mock-Chebyshev least squares framework to Hermite data by incorporating derivative information up to order $k$, resulting in data size $\tilde{n}=(k+1)(n+1)$ and target degree $\tilde{r}=(k+1)r=(k+1)(m+p+1)$ with $m=\left\lfloor \pi \sqrt{n/2} \right\rfloor$. The method constructs a KKT system involving matrices $\Lambda$ and $\Xi$ to compute coefficients in a basis $\mathcal{B}_{\tilde{r}}$ such that $\hat{H}_{\tilde{r},n,k}^{(\ell)}(f,x_i)=f^{(\ell)}(x_i)$ for $i\le m$ and $\ell\le k$, thereby generalizing CMCLS when $k=0$. Numerical experiments on smooth and oscillatory test functions demonstrate that the Hermite CMCLS operator and its derivative outperform standard CMCLS in accuracy as $n$ grows, albeit with worsening conditioning as $k$ increases. The results extend mock-Chebyshev strategies to Hermite data, offering a robust, derivative-informed tool for high-accuracy polynomial approximation using equispaced nodes.

Abstract

This paper addresses the challenge of function approximation using Hermite interpolation on equally spaced nodes. In this setting, standard polynomial interpolation suffers from the Runge phenomenon. To mitigate this issue, we propose an extension of the constrained mock-Chebyshev least squares approximation technique to Hermite interpolation. This approach leverages both function and derivative evaluations, resulting in more accurate approximations. Numerical experiments are implemented in order to illustrate the effectiveness of the proposed method.

Figures (7)

• Figure 1: Trend of the condition number of the KKT matrix relative to the Hermite problem with $k=1$.
• Figure 2: Mean approximation error produced by approximating the function $f_1$ (left) and $f_1^{\prime}$ (right) using the operator $\hat{H}_{\widetilde{r},n,1}$ and its first derivative $\hat{H}_{\widetilde{r},n,1}^{\prime}$, respectively. The parameter $n$ varies from $n=100$ to $n=1000$ with stepsize $50$. This is compared against the standard constrained mock-Chebyshev least squares approximation using the same number of data points.
• Figure 3: Mean approximation error produced by approximating the function $f_2$ (left) and $f_2^{\prime}$ (right) using the operator $\hat{H}_{\widetilde{r},n,1}$ and its first derivative $\hat{H}_{\widetilde{r},n,1}^{\prime}$, respectively. The parameter $n$ varies from $n=100$ to $n=1000$ in steps of $50$. This is compared against the standard constrained mock-Chebyshev least squares approximation using the same number of data points.
• Figure 4: Mean approximation error produced by approximating the function $f_3$ (left) and $f_3^{\prime}$ (right) using the operator $\hat{H}_{\widetilde{r},n,1}$ and its first derivative $\hat{H}_{\widetilde{r},n,1}^{\prime}$, respectively. The parameter $n$ varies from $n=100$ to $n=1000$ in steps of $50$. This is compared against the standard constrained mock-Chebyshev least squares approximation using the same number of data points.
• Figure 5: Mean approximation error produced by approximating the function $f_4$ (left) and $f_4^{\prime}$ (right) using the operator $\hat{H}_{\widetilde{r},n,1}$ and its first derivative $\hat{H}_{\widetilde{r},n,1}^{\prime}$, respectively. The parameter $n$ varies from $n=100$ to $n=1000$ in steps of $50$. This is compared against the standard constrained mock-Chebyshev least squares approximation using the same number of data points.

Theorems & Definitions (11)

• theorem 1
• theorem 2
• remark thmcounterremark
• remark thmcounterremark
• theorem 3
• proof
• remark thmcounterremark
• remark thmcounterremark
• remark thmcounterremark
• theorem 4