[Paper Review] Convergence properties of spline-like cardinal interpolation operators acting on $l^p$ data

Jeff Ledford|arXiv (Cornell University)|Dec 14, 2013

Advanced Numerical Analysis Techniques4 references4 citations

TL;DR

This paper establishes the convergence of spline-like cardinal interpolation operators to reconstruct functions in $L^p(\mathbb{R})$ from their integer samples, under a general class of interpolants satisfying specific Fourier-domain conditions. It proves that for functions in a Paley-Wiener-type space with bounded variation measures, the $L^p$ norm of the interpolation error tends to zero as the parameter $\alpha \to \infty$, extending classical sampling theorems to a broad class of kernels including odd-degree splines, Gaussians, Poisson kernels, and multiquadrics.

ABSTRACT

If $f\in \{f\in L^p(\mathbb{R}): f(x)=\int_{-π}^πe^{ixξ}dβ(ξ), β\in B.V.([-π,π]) \}$, then $f$ is determined by its samples on the integers by taking an appropriate limit. Specifically, $\| f - L_{ϕ_α}f \|_{L^p(\mathbb{R})} o 0$ as $α o\infty$ provided that $\{ϕ_α: α\in A\}$ is what we call a spline-like family of cardinal interpolators.

Motivation & Objective

• To extend the classical sampling theorem to a general class of cardinal interpolation operators beyond splines and Gaussians.
• To establish $L^p$ convergence of interpolation operators acting on $\ell^p$ data for functions in a Paley-Wiener-type space with bounded variation spectral measures.
• To define and characterize a new class of interpolants—'spline-like cardinal interpolators'—that satisfy uniform boundedness and decay conditions on their Fourier transforms.
• To verify that key families such as odd-degree splines, Gaussians, Poisson kernels, and multiquadrics belong to this class and satisfy the convergence conditions.
• To provide a unified framework for convergence of interpolation operators in $L^p$ based on spectral and functional analytic properties of the kernel's Fourier transform.

Proposed method

• Define a family of interpolators $\phi_\alpha$ satisfying four conditions: slow growth, positive and bounded Fourier transform on $[-\pi,\pi]$, $C^1$ regularity away from zero, and polynomial decay at infinity.
• Introduce the fundamental function $L_{\phi_\alpha}$ via its Fourier transform: $\hat{L}_{\phi_\alpha}(\xi) = (2\pi)^{-1/2} \frac{\hat{\phi}_\alpha(\xi)}{\sum_{j\in\mathbb{Z}} \hat{\phi}_\alpha(\xi + 2\pi j)}$, ensuring it interpolates the Kronecker delta at integers.
• Construct the interpolation operator $\mathscr{I}_{\phi_\alpha}[f](x) = \sum_{j\in\mathbb{Z}} f(j) L_{\phi_\alpha}(x - j)$ for $f \in \ell^p$.
• Define auxiliary functions $\mathscr{M}[\hat{\phi}_\alpha]_j(\xi) = \frac{\hat{\phi}_\alpha(\xi + 2\pi j)}{\hat{\phi}_\alpha(\xi)}$ for $|\xi| \leq \pi$, used to analyze spectral decay and uniform bounds.
• Establish the 'spline-like family' condition via four axioms: uniform $L^1$ bounds on mixed Hilbert-type sums (B2), pointwise a.e. decay (B3), uniform majorization (B4), and individual kernel conditions (B1).
• Use estimates on modified Bessel functions and exponential decay to verify the conditions for specific families like multiquadrics $\{(x^2 + c^2)^k\}$ and Poisson kernels.

Experimental results

Research questions

• RQ1 Does the interpolation error $\|f - L_{\phi_\alpha}f\|_{L^p(\mathbb{R})}$ converge to zero as $\alpha \to \infty$ for functions in the Paley-Wiener space with bounded variation spectral measures?
• RQ2 Can the classical sampling theorem be generalized to a broad class of non-spline, non-Gaussian kernels such as multiquadrics and Poisson kernels?
• RQ3 What spectral and functional conditions on the kernel $\phi_\alpha$ ensure uniform $L^p$ convergence of the interpolation operator across $\ell^p$ data?
• RQ4 Are the Poisson kernel and the multiquadric family $\{(x^2 + c^2)^{k-1/2}\}$, $k \in \mathbb{N}$, valid members of the spline-like interpolator class with convergence guarantees?
• RQ5 How do the auxiliary functions $\mathscr{M}[\hat{\phi}_\alpha]_j(\xi)$ control the convergence behavior through uniform bounds and decay?

Key findings

• The $L^p$ norm of the interpolation error $\|f - \mathscr{I}_{\phi_\alpha}[f]\|_{L^p(\mathbb{R})}$ converges to zero as $\alpha \to \infty$ for all $f$ in the space $\mathcal{A} = \{f \in L^p(\mathbb{R}) : f(x) = (2\pi)^{-1/2} \int_{-\pi}^{\pi} e^{ix\xi} d\beta(\xi), \beta \in BV([-π,\pi])\}$.
• The class of spline-like cardinal interpolators includes odd-degree cardinal splines, Gaussians, Poisson kernels, and multiquadrics $\{(x^2 + c^2)^{k-1/2}\}$, $k \in \mathbb{N}$, all of which satisfy the convergence conditions.
• For the multiquadric family $\{(x^2 + c^2)^{k-1/2}\}$, the convergence result is new and established via uniform bounds on $\mathscr{M}[\hat{\phi}_\alpha]_j(\xi)$ and decay estimates using modified Bessel functions.
• The Poisson kernel is shown to satisfy the spline-like family conditions, and its $L^p$ convergence is a new result in this context.
• The uniform $L^1$ bound in condition (B2) is verified for multiquadrics and Poisson kernels using exponential decay and polynomial growth estimates on $\mathscr{M}[\hat{\phi}_\alpha]_j(\xi)$.
• The decay condition (B3) holds a.e. for all families considered, as $\mathscr{M}[\hat{\phi}_\alpha]_j(\xi) \to 0$ as $\alpha \to \infty$ for $j \neq 0$, ensuring asymptotic orthogonality of spectral components.