Strang Fix Research Articles

Nonseparable, Compactly Supported Interpolating Refinable Functions with Arbitrary Smoothness

In this paper we construct families of compactly supported nonseparable interpolating refinable functions with arbitrary smoothness (or regularity). The symbols for the newly constructed scaling functions are given by a simple formula related to the Bernstein polynomials. The emphasis of the paper is to show that under an easy-to-verify geometric condition these families satisfy Cohenrs condition, and they have arbitrarily high regularity. Furthermore, the constructed scaling functions satisfy, under the same geometrical condition, the Strang–Fix conditions of arbitrarily high order, which implies that corresponding interpolating schemes have arbitrarily high accuracy.

Quasi-Orthogonality and Quasi-Projections

Our main concern in this paper is the design of simplified filtering procedures for the quasi-optimal approximation of functions in subspaces of L2generated from the translates of a function ϕ(x). Examples of signal representations that fall into this framework are Schoenberg's polynomial splinesof degree n, and the various multiresolution spaces associated with the wavelet transform. After a brief review of the relation between the order of approximation of the representation and the concept of quasi-interpolation (Strang–Fix conditions), we investigate the implication of these conditions on the various basis functions and their duals (vanishing moment and quasi-interpolation properties). We then introduce the notion of quasi-duality and show how to construct quasi-orthogonal and quasi-dual basis functions that are much shorter than their exact counterparts. We also consider the corresponding quasi-orthogonal projection operator at sampling step h and derive asymptotic error formulas and bounds that are essentially the same as those associated with the exact least-squares solution. Finally, we use the idea of a perfect reproduction of polynomials of degree n to construct short kernel quasi-deconvolution filters that provide a well-behaved approximation of an oblique projection operator.

Approximation by translates of refinable functions

The functions\(f_1(x), \dots, f_r(x)\) 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\(1, x, \dots, x^{p-1}\) 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,\(r \times r\) 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 transforms\(\hat f_i(\omega)\) , but for refinable functions they can be explicitly verified from the \(c_k\).