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Second-order convolution bias for an interior C² field
Statement
Let f∈C²(R^d) have bounded Hessian and define S_εf(x)=∫f(x−z)W_ε(z)dz. Then for every ε>0 and x∈R^d,
Facts & Assumptions
Given: The kernel family, a C² function f with bounded Hessian, and ε>0.
Scaling identities for kernel moments and derivatives and Scaled symmetric compactly supported kernel family: ∫W_ε=1, ∫zW_ε=0, and ∫|z|²W_ε=ε²m₂; the kernel definition gives W_ε≥0.
Multivariable Taylor formula with a Lagrange remainder along a line segment: for a C² function along a line segment, the degree-one Taylor formula has a second-derivative remainder at an interior point of that segment.
Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in : on a bounding rectangle, integration is linear and satisfies the absolute-value estimate and monotonicity.
Proof
For each x,z apply [F2] with increment −z. It gives f(x−z)=f(x)−∇f(x)·z+R(x,z), where |R(x,z)|≤(1/2)||D²f||∞|z|².
Integrate this identity against the nonnegative W_ε. Linearity in [F3] and [F1] make the constant term integrate to f(x) and the linear term zero. The absolute-value estimate and monotonicity in [F3], together with W_ε≥0, bound the remainder by (1/2)||D²f||∞∫|z|²W_ε=(m₂/2)ε²||D²f||∞.
Depends on
- Scaled symmetric compactly supported kernel family
- Symmetry cancels the first kernel moment
- Scaling identities for kernel moments and derivatives
- Multivariable Taylor formula with a Lagrange remainder along a line segment
- Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $\mathbb{R}^m$
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)