Examples of Refinable Componentwise Polynomials

This short note presents four examples of compactly supported symmetric refinable componentwise polynomial functions: (i) a componentwise constant interpolatory continuous refinable function and its derived symmetric tight wavelet frame; (ii) a componentwise constant continuous orthonormal and interpolatory refinable function and its associated symmetric orthonormal wavelet basis; (iii) a differentiable symmetric componentwise linear polynomial orthonormal refinable function; (iv) a symmetric refinable componentwise linear polynomial which is interpolatory and differentiable. This note presents four examples of compactly supported symmetric refinable functions with some special properties such as the componentwise polynomial property, which is defined to be Definition. We say that a function φ : R 7→ C is a componentwise polynomial if there exists an open set G such that the Lebesgue measure of R\G is zero and the restriction of φ on every connected component of G coincides with a polynomial. Of courses, on different components φ may coincide with different polynomials. It is clear that a compactly supported piecewise polynomial (i.e. the open set G has only finitely many connected components), which is called a spline, is a componentwise polynomial. Therefore, although a componentwise polynomial is generally not a spline, it is closely related to a spline and generalizes the concept of a spline. The difference between a componentwise polynomial and a spline lies in that it can have infinitely many “pieces” and the “knots” could consist of a compact set, which may have cluster points and therefore, not knots any more in the sense of the theory of splines. For example, a nontrivial compactly supported componentwise constant could be continuous, as shown in Example 1. Componentwise polynomials were first introduced in [1, 10] under the name of local polynomials. Some basic properties of componentwise polynomials can be found in [1, 10]. It was shown in [1, 10] that a compactly supported refinable componentwise polynomial has an analytic form. In particular, an iteration formula is given in [1, Lemma 2] to compute the polynomial on each component. We say that a function φ is interpolatory if φ is continuous and satisfies φ(0) = 1 and φ(k) = 0 for all k ∈ Z\{0}. We say that φ is orthonormal if {φ(· − k) : k ∈ Z} is an orthonormal system (sequence) in L2(R). It is proven in [7] that a compactly supported refinable spline whose shifts form a Riesz system must be a B-spline function, up to an integer shift. So, the only refinable orthonormal spline is χ[0,1], the discontinuous characteristic function of [0, 1]. The only spline interpolatory refinable function φ is the hat function, which is not differentiable. Extending the concept of piecewise polynomials (that is, splines) to componentwise polynomials, we are able to construct four interesting examples: the first one is a compactly supported refinable componentwise constant which is symmetric and continuous. This immediately leads to an example of shortly supported symmetric tight wavelet frame such that each framelet is continuous. The second one is a compactly supported refinable componentwise constant which is symmetric, continuous, interpolatory and orthonormal, plus whose mask has rational coefficients. This immediately leads to a componentwise constant symmetric orthonormal wavelet basis which is 2000 Mathematics Subject Classification. 42C40.