[Paper Review] A construction of linear bounded interpolatory operators on the torus

S. Chandrasekaran, H. N. Mhaskar|arXiv (Cornell University)|Nov 24, 2010

Electromagnetic Scattering and Analysis18 references3 citations

TL;DR

This paper presents a construction of linear, bounded interpolatory operators for multivariate periodic functions on the q-torus using an optimization-based approach. By solving a constrained minimization problem that controls the Sobolev norm of the interpolant, the method ensures uniform boundedness and optimal convergence rates at limit points of scattered data, overcoming issues like the Runge phenomenon in equispaced interpolation while maintaining stability and approximation accuracy.

ABSTRACT

Let $q\ge 1$ be an integer. Given $M$ samples of a smooth function of $q$ variables, $2π$--periodic in each variable, we consider the problem of constructing a $q$--variate trigonometric polynomial of spherical degree $Ø(M^{1/q})$ which interpolates the given data, remains bounded (independent of $M$) on $[-π,π]^q$, and converges to the function at an optimal rate on the set where the data becomes dense. We prove that the solution of an appropriate optimization problem leads to such an interpolant. Numerical examples are given to demonstrate that this procedure overcomes the Runge phenomenon when interpolation at equidistant nodes on $[-1,1]$ is constructed, and also provides a respectable approximation for bivariate grid data, which does not become dense on the whole domain.

Motivation & Objective

• To develop a constructive, linear method for interpolating smooth, multivariate periodic functions on the q-torus using scattered data points.
• To ensure the interpolant remains uniformly bounded (independent of the number of samples) and converges at an optimal rate at limit points of the data.
• To overcome the Runge phenomenon in equispaced univariate interpolation by extending the approach to multivariate periodic functions.
• To provide a stable, near-best approximation that satisfies interpolatory constraints without requiring data to be dense over the entire domain.
• To establish theoretical bounds on the convergence rate and operator norm using Sobolev space theory and Fourier analysis.

Proposed method

• Formulates the interpolation problem as a constrained minimization of the Sobolev norm $\|P^{(s)}\|_p$ subject to interpolation conditions at given data points.
• Uses the periodicity of the function and a change of variables to map the problem to the q-torus, enabling the use of trigonometric polynomials.
• Applies duality and Fourier coefficient comparison techniques to bound the error between the interpolant and the target function.
• Employs the Arzela–Ascoli theorem and subsequential convergence arguments to prove uniform convergence at limit points of the data.
• Establishes stability via bounds on the operator norm in Sobolev spaces, ensuring $\|\mathbb{P}_n^*\|_{W^p_s} \leq c\|f\|_{W^p_s}$.
• Uses the modulus of continuity and difference operators to derive convergence rates in Hölder-type norms based on smoothness parameters.

Experimental results

Research questions

• RQ1 Can a linear, bounded interpolatory operator be constructed for multivariate periodic functions on the torus that avoids the Runge phenomenon?
• RQ2 What is the optimal convergence rate of such an interpolant when data points accumulate at certain limit points?
• RQ3 How can the operator norm be controlled independently of the number of samples, ensuring stability?
• RQ4 Can the method provide near-best approximation in Sobolev spaces while satisfying interpolation constraints?
• RQ5 What conditions on the data distribution ensure convergence of the interpolant at limit points, even if the data are not dense globally?

Key findings

• The solution to the constrained minimization problem yields a linear, bounded interpolant with $\|\mathbb{P}_n^*\|_{W^p_s} \leq c\|f\|_{W^p_s}$, ensuring stability independent of the number of samples.
• The interpolant converges to the true function at an optimal rate $O(\delta^{s - q/p})$ at limit points of the data, where $\delta$ is the local density scale.
• The method overcomes the Runge phenomenon in univariate equispaced interpolation by ensuring boundedness and convergence in the multivariate periodic setting.
• Numerical examples confirm that the method provides stable and accurate approximations for bivariate grid data, even when data are not dense across the entire domain.
• The convergence rate is optimal in the sense that it matches the best possible approximation error in the given Sobolev space.
• The construction is valid for any scattered data set and does not require specific geometric arrangements or global density.