1 convolution.pdf - CONVOLUTION AND APPROXIMATE IDENTITIES Definition 0.1 A function u Rn \u2192 R is integrable if Z Z |u(x)| dx:= lim |u(x)| dx < \u221e

# 1 convolution.pdf - CONVOLUTION AND APPROXIMATE IDENTITIES...

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CONVOLUTION AND APPROXIMATE IDENTITIES Definition 0.1. A function u : R n R is integrable if Z R n | u ( x ) | dx := lim R →∞ Z [ - R,R ] n | u ( x ) | dx < . Definition 0.2. Let u, v be integrable functions on R n . Define the convolution u * v of u and v by u * v ( x ) := Z R n u ( x - y ) v ( y ) dy Lemma 0.1. Properties of convolution. u * v = v * u . If u is differentiable, then u * v is differentiable, and ( u * v ) = ( ∂u ) * v . Proof. For each x , we use the change of variables Φ( y ) = x - y (which satisfies J Φ = - I and therefore | det J Φ | = 1) to obtain u * v ( x ) = Z R n u ( x - y ) v ( y ) = Z R n u ( y ) v ( x - y ) dy = v * u ( x ) . In particular, if v is smooth (i.e. infinitely differentiable), then so is u * v even if u * v is not. Convolution is used to approximate rough (possibly non-differentiable or discontinuous) functions by smooth functions. Definition 0.3. Let η be smooth, rapidly decreasing on R n and satisfy R R n η ( x ) dx = 1. For each 0 < h 1, define η h ( x ) := h - n η ( h - 1 x ); note that R R n η h ( x ) dx = 1 as well by a change of variable.

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