Dyadic unit cells

The set of dyadic unit cells is a partition of p-dimensional space and is defined as follows: With

{\ displaystyle {\ mathcal {W}} _ {n, \, (k_ {1}, \ ldots, k_ {p})}: = \ left \ {(x_ {1}, \ ldots, x_ {p} ) \ in \ mathbb {R} ^ {p}: {\ frac {k_ {i}} {2 ^ {n}}} \ leq x_ {i} <{\ frac {k_ {i} +1} {2 ^ {n}}} \ right \}, \; k_ {i} \ in \ mathbb {Z}, n \ in \ mathbb {N}}

you define a half-open cube in which has the length of the edge . ${\ displaystyle \ mathbb {R} ^ {p}}$ ${\ displaystyle 2 ^ {- n}}$

${\ displaystyle {\ mathcal {D}} _ {n}}$ denotes the set of dyadic unit cells of the order : ${\ displaystyle n}$

{\ displaystyle {\ mathcal {D}} _ {n}: = \ left \ {{\ mathcal {W}} _ {n, \, (k_ {1}, \ ldots, k_ {p})}: k_ {i} \ in \ mathbb {Z} \ right \}, \; n \ in \ mathbb {N} \ ;.}

Unit cells of the same order are disjoint and separated from each other by a grid.

The set of all dyadic unit cells im is then denoted by: ${\ displaystyle \ mathbb {R} ^ {p}}$ ${\ displaystyle {\ mathcal {D}}}$

{\ displaystyle {\ mathcal {D}}: = \ left \ {{\ mathcal {W}} _ {n, \, (k_ {1}, \ ldots, k_ {p})}: k_ {i} \ in \ mathbb {Z}, n \ in \ mathbb {N} \ right \} \ ;.}

The set of corner points of the dyadic unit cells is called the dyadic grid . ${\ displaystyle \ left \ {\ left ({\ frac {k_ {1}} {2 ^ {n}}}, \ ldots, {\ frac {k_ {p}} {2 ^ {n}}} \ right ): k_ {i} \ in \ mathbb {Z}, n \ in \ mathbb {N} \ right \}}$

meaning

The amount of the dyadic unit cells is a half-ring , and generates the Borel σ algebra of . Since is countable , is a separable σ-algebra . ${\ displaystyle {\ mathcal {D}}}$ ${\ displaystyle {\ mathcal {B}}}$ ${\ displaystyle \ mathbb {R} ^ {p}}$ ${\ displaystyle {\ mathcal {D}}}$ ${\ displaystyle {\ mathcal {B}}}$

Examples

${\ displaystyle p = 1}$ : Unit cells are half-open intervals .
${\ displaystyle p = 2}$ : Unit cells are squares .
${\ displaystyle p = 3}$ : Unit cells are cubes .

Dyadic unit cells

meaning

Examples

See also