One of the important capabilities for a subdivision scheme is the reproducing property of circular shapes or parts of conics that are important analytical shapes in geometrical modelling. In this regards, the first goal of this study is to provide necessary and sufficient conditions for a non-stationary subdivision to have the reproducing property of… (More)
We start with a characterization of a pair of frames to be orthogonal in a shift-invariant space and then give a simple construction of a pair of orthogonal shift-invariant frames. This is applied to obtain a construction of a pair of Ga-bor orthogonal frames as an example. This is also developed further to obtain constructions of a pair of orthogonal… (More)
We analyze the internal structure of the multiresolution analyses of L 2 (R d) defined by the unitary extension principle (UEP) of Ron and Shen. Suppose we have a wavelet tight frame defined by the UEP. Define V 0 to be the closed linear span of the shifts of the scaling function and W 0 that of the shifts of the wavelets. Finally, define V 1 to be the… (More)
We show that the 'centered' Battle-Lemarié scaling function and wavelet of order n converge in L q (2 ≤ q ≤ ∞), uniformly in particular, to the Shannon scaling function and wavelet as n tends to the infinity.
The precise Sobolev exponent s ∞ (ϕ n) of the Butterworth refinable function ϕ n associated with the Butterworth filter of order n, bn(ξ) := cos 2n (ξ/2) cos 2n (ξ/2)+sin 2n (ξ/2) , is shown to be s∞(ϕn) = n log 2 3 + log 2 (1 + 3 −n). This recovers the previously given asymptotic estimate of s ∞ (ϕ n) of Fan and Sun , and gives more accurate regularity… (More)
Consider a continuous function g ∈ L 2 (R) that is supported on [−1, 1] and generates a Gabor frame with translation parameter 1 and modulation parameter 0 < b < 2N 2N +1 for some N ∈ N. Under an extra condition on the zeroset of the window g we show that there exists a continuous dual window supported on [−N, N ]. We also show that this result is optimal:… (More)