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Derivative moments of a compact radial kernel
Statement
For W from Scaled symmetric compactly supported kernel family and coordinate indices j,k,l,
Also m₃=∫|z|³|∇W(z)|dz is finite.
Facts & Assumptions
Given: A radial C² compactly supported kernel W with integral one.
Scaled symmetric compactly supported kernel family: W and its first derivatives are continuous and compactly supported, and ∫W=1.
Symmetry cancels the first kernel moment: ∫z_jW(z)dz=0 in every coordinate.
Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections: for a continuous function on a product rectangle, the compact multiple Riemann integral equals either iterated integral.
Integration by parts for continuous factors with Riemann-integrable extensions of their interior derivatives: if F,G are continuous on a compact interval and their interior derivatives have integrable extensions, then ∫Fg=[FG]−∫F'G.
A continuous real function on a compact Jordan measurable set is Riemann integrable over that set: a continuous function on a compact Jordan set is Riemann integrable.
Proof
Choose a box with the support of W in its interior. Every integrand below is continuous on this box by [F1] and integrable by [F5]. Fix all coordinates except z_k. Applying [F4] with F=W and G=1 gives ∫∂kW dz_k=0 because W vanishes near the two endpoints. Applying it with G=z_j gives ∫z_j∂kW dz_k=−δ{jk}∫W dz_k. Integrating the section identities over the remaining coordinates by [F3] gives the first identity and ∫z_j∂kW dz=−δ{jk}∫W dz=−δ{jk}.
With G=z_jz_l, [F4] gives ∫z_jz_l∂kW dz=−δ{jk}∫z_lW dz−δ_{lk}∫z_jW dz after [F3] integrates the other coordinates. Both moments vanish by [F2]. Finally |z|³|∇W| is continuous and compactly supported, so [F5] makes its integral finite; this is m₃.
Depends on
- Scaled symmetric compactly supported kernel family
- Symmetry cancels the first kernel moment
- Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections
- Integration by parts for continuous factors with Riemann-integrable extensions of their interior derivatives
- A continuous real function on a compact Jordan measurable set is Riemann integrable over that set
Used by
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Sources
- Volker Springel, Smoothed Particle Hydrodynamics in Astrophysics, Annual Review of Astronomy and Astrophysics 48 (2010), author full text (standard reference, not scraped)
- Benjamin Seibold, MIT OpenCourseWare 18.336, Numerical Methods for Partial Differential Equations, Lecture 25: Particle Methods (Spring 2009) (standard reference, not scraped)