Compactly Supported Tight Frames Associated with Refinable Functions

Compactly Supported Tight Frames Associated with Refinable Functions

It is well known that in applied and computational mathematics, cardinal B-splines play an important role in geometric modeling (in computeraided geometric design), statistical data representation (or modeling), solution of differential equations (in numerical analysis), and so forth. More recently, in the development of wavelet analysis, cardinal B-splines also serve as a canonical example of ...

Compactly Supported Refinable Functions with Infinite Masks

A compactly supported scaling function can come from a refinement equation with infinitely many nonzero coefficients (an infinite mask). In this case we prove that the symbol of the mask must have the special rational form ã(Z) = b̃(Z)c̃(Z)/b̃(Z). Any finite combination of the shifts of a refinable function will have such a mask, and will be refinable. We also study compactly supported solutions o...

Construction of Compactly Supported Tight Wavelets Frames

Compactly supported tight and sibling frames with maximum vanishing moments

The notion of vanishing-moment recovery (VMR) functions is introduced in this paper for the construction of compactly supported tight frames with two generators having the maximum order of vanishing moments as determined by the given refinable function, such as the mth order cardinal B-spline Nm. Tight frames are also extended to “sibling frames” to allow additional properties, such as symmetry...

Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

Let A ∈ Rd×d, d ≥ 1 be a dilation matrix with integer entries and | detA| = 2. We construct several families of compactly supported Parseval framelets associated to A having any desired number of vanishing moments. The first family has a single generator and its construction is based on refinable functions associated to Daubechies low pass filters and a theorem of Bownik. For the construction o...