a b s t r a c tInterpolating scalar refinable functions with compact support are of interest in several applications such as sampling theory, signal processing, computer graphics, and numerical algorithms. In this paper, we shall generalize the notion of interpolating scalar refinable functions to compactly supported interpolating d-refinable function vectors with any multiplicity r and dilation factor d. More precisely, we are interested in a d-refinable function vector φ = [φ 1 , . . . , φ r ] T such that φ is an r × 1 column vector of compactly supported continuous functions with the following interpolation propertywhere δ 0 = 1 and δ k = 0 for k = 0. Now for any function f : R → C, the function f can be interpolated and approximated bỹSince φ is interpolating,f (k/r) = f (k/r) for all k ∈ Z, that is,f agrees with f on r −1 Z.Moreover, for r 2 or d > 2, such interpolating refinable function vectors can have the additional orthogonality property: φ (· − k), φ (· − k ) = r −1 δ − δ k−k for all k, k ∈ Z and 1 , r, while it is well-known that there does not exist a compactly supported scalar 2-refinable function with both the interpolation and orthogonality properties simultaneously. In this paper, we shall characterize both interpolating d-refinable function vectors and orthogonal interpolating d-refinable function vectors in terms of their masks. We shall study their approximation properties and present a family of interpolatory masks, for compactly supported interpolating d-refinable function vectors, of type (d, r) with increasing orders of sum rules. To illustrate the results in this paper, we also present several examples of compactly supported (orthogonal) interpolating refinable function vectors and biorthogonal multiwavelets derived from such interpolating refinable function vectors.

Compactly supported Riesz wavelets are of interest in several applications such as image processing, computer graphics and numerical algorithms. In this paper, we shall investigate compactly supported MRA Riesz multiwavelet bases in L 2 (R). An algorithm is presented to derive Riesz multiwavelet bases from refinable function vectors. To illustrate our algorithm and results in this paper, we present several examples of Riesz multiwavelet bases with short support in L 2 (R).

Two-direction multiscaling functions φ and two-direction multiwavelets ψ associated with φ are a more general and more flexible setting than one-direction multiscaling functions and multiwavelets. In this paper, we derive two methods for computing continuous moments of orthogonal two-direction multiscaling functions φ and orthogonal two-direction multiwavelets ψ associated with φ. The first method is by doubling and the second method is by separation. Two examples for both methods are given.2010 Mathematics Subject Classification. 42C15.

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The accuracy of the wavelet approximation at resolution h Å 2 0n to a smooth function f is limited by O(h N), where N is the number of vanishing moments of the mother wavelet c. For any positive integer p, we derive a new approximation formula which allows us to recover a smooth f from its wavelet coefficients with accuracy O(h p). Related formulas for recovering derivatives of f are also given.

Abstract. A two-direction multiscaling function φ satisfies a recursion relation that uses scaled and translated versions of both itself and its reverse. This offers a more general and flexible setting than standard one-direction wavelet theory. In this paper, we investigate how to find and normalize point values and derivative values of two-direction multiscaling and multiwavelet functions. Determination of point values is based on the eigenvalue approach. Normalization is based on normalizing conditions for the continuous moments of φ. Examples for illustrating the general theory are given.

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