Multiresolution analysis

Definition

• Self-similarity in time demands that each subspace V_k is invariant under shifts by integer multiples of 2^k. That is, for each the function g defined as also contained in.
• Self-similarity in scale demands that all subspaces are time-scaled versions of each other, with scaling respectively dilation factor 2^k-l. I.e., for each there is a with.
• In the sequence of subspaces, for k>l the space resolution 2^l of the l-th subspace is higher than the resolution 2^k of the k-th subspace.
• Regularity demands that the model subspace V₀ be generated as the linear hull of the integer shifts of one or a finite number of generating functions or. Those integer shifts should at least form a frame for the subspace, which imposes certain conditions on the decay at infinity. The generating functions are also known as scaling functions or father wavelets. In most cases one demands of those functions to be piecewise continuous with compact support.
• Completeness demands that those nested subspaces fill the whole space, i.e., their union should be dense in, and that they are not too redundant, i.e., their intersection should only contain the zero element.

  Important conclusions