Master Dissertation

Proposition 1.12. We state some often occurring affine transformations.
In the following u ∈ D ′ (R n ) and φ ∈ C(R n ).
1. Reflection: 〈u(−x), φ(x)〉 = 〈u(x), φ(−x)〉.
2. Scaling: 〈u(tx), φ(x)〉 = t −n 〈u(x), φ(x/t)〉 = t −n 〈u(x), λtφ(x)〉, letting
λtφ(x) := φ(x/t).
Proof. 1. and 2. follow directly from (1.2) by choosing A = −I and A = tI
respectively.
Note that λt : S(R n ) → S(R n ) is continuous. This follows since
λtφα,β = sup |x α D β (λt(φ(x)))| = sup |x α t −|β| (D β φ)(x/t)|
Further we define
= t |α|−|β| sup |(x/t) α (D β φ)(x/t)|
= t |α|−|β| φα,β.
Definition 1.13. Let u ∈ D ′ (R n ), h ∈ R n and φ ∈ C(R n ). We define the
translation τh of u by
〈u(x − h), φ(x)〉 = 〈τhu, φ〉 = 〈u, τ−hφ〉 = 〈u(x), φ(x + h)〉.
Clearly this defines a distribution.
Recall that a function f is said to be homogeneous of order λ ∈ C if
f(tx) = t λ f(x) for all t > 0 and x ∈ R n . We need to define the similar
property for distributions.
Definition 1.14. A distribution u is said to be homogeneous of degree
λ ∈ C if
〈u(tx), φ(x)〉 = 〈t λ u(x), φ(x)〉,
for all t > 0 and φ ∈ C ∞ 0 .
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