Generalized interpolating refinable function vectors

Abstract

Interpolating 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 property ϕ ℓ ( m r + k ) = δ k δ ℓ − 1 − m , ∀ k ∈ Z , m = 0 , … , r − 1 , ℓ = 1 , … , r , where 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 by f ̃ = ∑ ℓ = 1 r ∑ k ∈ Z f ( ℓ − 1 r + k ) ϕ ℓ ( ⋅ − k ) = ∑ k ∈ Z [ f ( k ) , f ( 1 r + k ) , … , f ( r − 1 r + k ) ] ϕ ( ⋅ − k ) . Since 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 ℓ ( ⋅ − k ) , ϕ ℓ ′ ( ⋅ − k ′ ) 〉 = r − 1 δ ℓ − ℓ ′ δ k − k ′ for all k , k ′ ∈ Z and 1 ⩽ ℓ , ℓ ′ ⩽ 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.