|
In mathematics, the dyadic cubes are a collection of cubes in R''n'' of different sizes or scales such that the set of cubes of each scale partition R''n'' and each cube in one scale may be written as a union of cubes of a smaller scale. These are frequently used in mathematics (particularly harmonic analysis) as a way of discretizing objects in order to make computations or analysis easier. For example, to study an arbitrary subset of ''A'' of Euclidean space, one may instead replace it by a union of dyadic cubes of a particular size that cover the set. One can consider this set as a pixelized version of the original set, and as smaller cubes are used one gets a clearer image of the set ''A''. Most notable appearances of dyadic cubes include the Whitney extension theorem and the Calderón–Zygmund lemma. ==Dyadic cubes in Euclidean space== In Euclidean space, dyadic cubes may be constructed as follows: for each integer ''k'' let Δ''k'' be the set of cubes in R''n'' of sidelength 2−''k'' and corners in the set : and let Δ be the union of all the Δ''k''. The most important features of these cubes are the following: # For each integer ''k'', Δ''k'' partitions R''n''. # All cubes in Δ''k'' have the same sidelength, namely 2−''k''. # If the interiors of two cubes ''Q'' and ''R'' in Δ''k'' have nonempty intersection, then either ''Q'' is contained in ''R'' or ''R'' is contained in ''Q''. # Each ''Q'' in Δ''k'' may be written as a union of 2''n'' cubes in Δ''k''+1 with disjoint interiors. We use the word "partition" somewhat loosely: for although their union is all of R''n'', the cubes in Δ''k'' can overlap at their boundaries. These overlaps, however, have zero Lebesgue measure, and so in most applications this slightly weaker form of partition is no hindrance. It may also seem odd that larger ''k'' corresponds to smaller cubes. One can think of ''k'' as the degree of magnification. In practice, however, letting Δ''k'' be the set of cubes of sidelength 2''k'' or 2−''k'' is a matter of preference or convenience. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dyadic cubes」の詳細全文を読む スポンサード リンク
|