Manuscript XVI-K@1 · XVI-K.tex · lemma · aa7553bb

Lemma: Homogeneity from the Fourier families

Homogeneity, inventory, and the finite graded span · lines 969–1003 · proof included or linked

XVI-K@1#lem:homog

For positive dilations,

W_{0}^{[a]} = a^{- 2} W_{0}, (E_{0}^{A})^{[a]} = a^{- 2} E_{0}^{A},

and therefore

S^{[a]} = a^{- 2} S, (\partial^{p} S)^{[a]} = a^{- 2 - ∣ p ∣} \partial^{p} S .

Proof. For the scaled test,

ψ_{a} (t, p) = a^{- 1} ψ (t / a, a p) .

In (eq:W-fam), substitute $t = a u$ , $p = k / a$ . The time and momentum measures, the factor $1/∣ p ∣$ , and the test contribute $a$ , $a^{- 3}$ , $a$ , and $a^{- 1}$ , respectively. Their product is $a^{- 2}$ . The same substitution in (eq:E-fam) preserves $t < 0$ and turns its sine quotient into $a sin (- u ∣ k ∣) /∣ k ∣$ . It gives the same power. The derivative assertion follows from

(\partial^{p} t)^{[a]} = a^{- ∣ p ∣} \partial^{p} (t^{[a]}) .

Displayed formulas in this passage

unlabeled · line 972
$W_{0}^{[a]} = a^{- 2} W_{0}, (E_{0}^{A})^{[a]} = a^{- 2} E_{0}^{A},$
unlabeled · line 977
$S^{[a]} = a^{- 2} S, (\partial^{p} S)^{[a]} = a^{- 2 - ∣ p ∣} \partial^{p} S .$
unlabeled · line 986
$ψ_{a} (t, p) = a^{- 1} ψ (t / a, a p) .$
unlabeled · line 1000
$(\partial^{p} t)^{[a]} = a^{- ∣ p ∣} \partial^{p} (t^{[a]}) .$

Lemma: Homogeneity from the Fourier families

Displayed formulas in this passage

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