Approximation by translates of refinable functions

Abstract

The functions f 1 (x ), . . . , f r (x) are refinable if they are combinations of the rescaled and translated functions f i (2x − k ). This is very common in scientific computing on a regular mesh. The space V 0 of approximating functions with meshwidth h = 1 is a subspace of V 1 with meshwidth h = 1/2. These refinable spaces have refinable basis functions. The accuracy of the computations depends on p, the order of approximation, which is determined by the degree of polynomials 1, x , . . . , x p−1 that lie in V 0 .Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions f i (x ) are known only through the coefficients c k in the refinement equation-scalars in the traditional case, r ×r matrices for multiwavelets. The scalar "sum rules" that determine p are well known. We find the conditions on the matrices c k that yield approximation of order p from V 0 . These are equivalent to the Strang-Fix conditions on the Fourier transformŝ f i (ω), but for refinable functions they can be explicitly verified from the c k .

On interpolatory divergence-free wavelets

“…, (see also [11,21]) and the application of Λ − 1 to ξ − i ( · 2 ) ∈ S 3,2 yields that ξ − is refinable with mask matrices ,…”

The limits of refinable functions

“…For the general theory and more examples of multiple refinable functions, we refer the reader to [5], [6], [10], [11], [12], [15], [18].…”

“…The explicit formula for the solution Φ(x) in the special case s = 3/2, t = −1/8, λ = −1/8, and µ = 1/2 was given by Heil, Strang, and Strela [6]. In this case, Φ(x) is supported on [0, 2]: 2 (x − 1) for 1 < x ≤ 2.…”

Multivariate matrix refinable functions with arbitrary matrix dilation

“…The characterization of the accuracy order of Φ in terms of the eigenvalues and eigenvector structures of the infinite matrix L were studied in [11], [25] and [17] for the case d = 1. In [1], a similar characterization of the accuracy order of Φ was obtained based on the ergodic theorem for the multivariate case with arbitrary matrix dilations M (no restriction on the diagonalization on M ), and the coefficients y β,i (κ) for the polynomial reproducing…”

“…The accuracy order of the (M, P) refinable vector Φ = t (φ 1 , · · · , φ r ) was considered in [11], [25] and [17] for the case d = 1 and M = (2), in [7] for M = 2I r and in [1] for the multivariate case with arbitrary dilation matrix. In Section 3, we will show that, under mild conditions, Φ provides approximation of order k, k ∈ Z + \{0}, if and only if the matrix refinement mask P satisfies the vanishing moment conditions of order k. We will also determine explicitly the coefficients for the polynomial reproducing under the assumption that the integer shifts of Φ (φ l (· − κ), κ ∈ Z d , l = 1, · · · , r) are linearly independent.…”