Manuscript XIX-E@1 · XIX-E.tex · proposition · 702f146e

Proposition: Different placement operations

Prescribed profiles, convolution, and clocks › Density multiplication and convolution do not coincide · lines 369–392 · proof included or linked

XIX-E@1#prop:placement, XIX-E@1#eq:commutator

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If $σ$ is smooth and nonconstant, the commutator $M_{σ} S_{ρ} - S_{ρ} M_{σ}$ is nonzero on $C_{c}^{\infty} (R)$ . More explicitly,

(M_{σ} S_{ρ} - S_{ρ} M_{σ}) f (t) = \int ρ (u) (σ (t) - σ (t - u)) f (t - u) d u .

Proof. The formula follows by subtraction. There is $t_{c}$ with $σ^{'} (t_{c}) \neq = 0$ . Choose a sufficiently small $u_{0} > 0$ such that $ρ (u_{0}) > 0$ and $σ (t_{c}) - σ (t_{c} - u_{0}) \neq = 0$ . The latter difference has a fixed nonzero sign on a small interval about $u_{0}$ . Choose a nonnegative nonzero $f$ supported in the corresponding interval about $t_{c} - u_{0}$ . The integral at $t_{c}$ has that strict sign.

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XIX-E@1#eq:commutator · line 374
$(M_{σ} S_{ρ} - S_{ρ} M_{σ}) f (t) = \int ρ (u) (σ (t) - σ (t - u)) f (t - u) d u .$

Proposition: Different placement operations

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