Optimal Compactly Supported Functions in Sobolev Spaces

Abstract

This paper constructs unique compactly supported functions in Sobolev spaces that have minimal norm, maximal support, and maximal central value, under certain renormalizations. They may serve as optimized basis functions in interpolation or approximation, or as shape functions in meshless methods for PDE solving. Their norm is useful for proving upper bounds for convergence rates of interpolation in Sobolev spaces $H_2^m(\R^d)$, and this paper gives the correct rate $m-d/2$ that arises as convergence like $h^{m-d/2}$ for interpolation at meshwidth $h\to 0$ or a blow-up like $r^{-(m-d/2)}$ for norms of compactly supported functions with support radius $r\to 0$. In Hilbert spaces with infinitely smooth reproducing kernels, like Gaussians or inverse multiquadrics, there are no compactly supported functions at all, but in spaces with limited smoothness, compactly supported functions exist and can be optimized in the above way. The construction is described in Hilbert space via projections, and analytically via trace operators. Numerical examples are provided.

Figures (4)

• Figure 1: Kernel translates $g_{1,x}$ for the 1D exponential kernel for varying $x$.
• Figure 2: Bump functions for the 1D exponential kernel for varying $r$. Left: $r=1,\,1.5,\,2,\,2.5,\,3$, right: $r=1,\ldots,50$. The red line is the kernel $K(0,y)=\exp(-|y|)$ occurring numerically in the limit.
• Figure 3: Bump functions (solid) for the exponential kernel, $r=1$ (left) and $r=10$ (right). The red curves are for $W_2^{3/2}(\mathbb{R}^2)$, the blue curves for $W_2^1(\mathbb{R}^1)$. The corresponding Wendland functions are dashed.
• Figure 4: Left: optimal bump functions for the 2D exponential kernel for $W_2^{5/2}(\mathbb{R}^2)$ with varying $r$. The red line is the normalized kernel occurring in the limit. Right: the case $r=1$ with the Wendland function $\phi_{3,1}$ in cyan and dashed.

Theorems & Definitions (29)

• Definition 1
• Definition 2
• Theorem 1
• proof
• Lemma 1
• proof
• Theorem 2
• Corollary 1
• Theorem 3
• proof