Section 7.1 and Lemma 7.1.3 #
The definition σ(u, x)
given in Section 7.1.
We may assume u ∈ t
whenever proving things about this definition.
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The definition of `𝓙₀(𝔖), defined above Lemma 7.1.2
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The definition of `𝓙(𝔖), defined above Lemma 7.1.2
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The definition of `𝓛₀(𝔖), defined above Lemma 7.1.2
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The definition of `𝓛(𝔖), defined above Lemma 7.1.2
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The projection operator P_𝓒 f(x)
, given above Lemma 7.1.3.
In lemmas the c
will be pairwise disjoint on C
.
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The definition I_i(x)
, given above Lemma 7.1.3.
The cube of scale s
that contains x
. There is at most 1 such cube, if it exists.
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- TileStructure.Forest.cubeOf i x = Classical.epsilon fun (I : Grid X) => x ∈ I ∧ s I = i
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The definition T_𝓝^θ f(x)
, given in (7.1.3).
For convenience, the suprema are written a bit differently than in the blueprint
(avoiding cubeOf
), but this should be equivalent.
This is 0
if x
doesn't lie in a cube.
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- One or more equations did not get rendered due to their size.
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Scaled integral appearing in the definition of boundaryOperator
.
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The operator S_{1,𝔲} f(x)
, given in (7.1.4).
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- t.boundaryOperator u f x = ∑ I : Grid X, (↑I).indicator (fun (x : X) => ∑ J ∈ t.𝓙' u (c I) (s I), TileStructure.Forest.ijIntegral f I J) x
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The indexing set for the collection of balls 𝓑, defined above Lemma 7.1.3.
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The center function for the collection of balls 𝓑.
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- TileStructure.Forest.c𝓑 z = c z.2.2
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The radius function for the collection of balls 𝓑.
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Lemma 7.1.1, freely translated.
Part of Lemma 7.1.2
The constant used in first_tree_pointwise
.
Has value 10 * 2 ^ (104 * a ^ 3)
in the blueprint.
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Lemma 7.1.4
Lemma 7.1.5
The constant used in third_tree_pointwise
.
Has value 2 ^ (151 * a ^ 3)
in the blueprint.
Equations
- TileStructure.Forest.C7_1_6 a = 2 ^ (151 * ↑a ^ 3)
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Lemma 7.1.6
The constant used in pointwise_tree_estimate
.
Has value 2 ^ (151 * a ^ 3)
in the blueprint.
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Lemma 7.1.3.