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 $$H_2^m(\mathbb {R}^d)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , and this paper gives the correct rate $$m-d/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>-</mml:mo> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> that arises as convergence like $$h^{m-d/2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>h</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>-</mml:mo> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> for interpolation at meshwidth $$h\rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> or a blow-up like $$r^{-(m-d/2)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>-</mml:mo> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> for norms of compactly supported functions with support radius $$r\rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . 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.