Local polynomial property and linear independence of refinable distributions

Abstract

In this paper, local polynomial property, global linear independence, and local linear dependence of the convolution of a B-spline and a refinable distribution supported on a Cantor-like set are studied.

Total positivity and refinable functions with general dilation

“…Thus, the symmetric orthogonal wavelets (SOW) and cardinal orthogonal wavelets (COW) for M 3, which are not possible for M = 2, are constructed (see [7] for SOW with M = 3, [14] for SOW with M = 4, [2] for SOW with M 3, and [3] for COW with M 3). Also there are examples of refinable functions whose integer translates are globally but not locally linearly independent ( [8,10] for M = 3 and [9] for M 3), a property which is again not possible for M = 2. (We shall give an example of a refinable ripplet with this property in Section 3.)…”

Summation and intersection of refinable shift invariant spaces

“…Furthermore, linear independence is also a necessary condition for orthogonality, or bi-orthogonality, of refinable functions in the construction of various wavelets. There is a long list of publications on the linear independence of distributions and refinable distributions (see, for instance, [9,10,14,18,22,26]). …”

On Refinable Sets

“…There are in fact T that are not tiles, let alone self-affine tiles, such that χ T are refinable. A simple example is T = [0, 2] ∪ [3,5] ∪ [6,8], for which f (x) = χ T (x) satisfies f (x) = f (2x) + f (2x − 2) − f (2x − 3) − f (2x − 5) + f (2x − 6) + f (2x − 8) a.e.…”