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• W1565463375 abstract "Abstract The notion of orthonormal wavelets is extended to that of tight wavelet frames to allow more flexibility for wavelet construction and redundancy for certain applications. For cardinal splines, the flexibility indeed permits the existence and construction of compactly supported wavelets. However, while the “matrix extension” approach for the construction of orthonormal wavelets is a natural route for constructing such tight wavelet frames of cardinal splines of order greater than 1, by using two or more Laurent polynomials to extend a square matrix of dimension two to a rectangular matrix, it happens that at least one of these Laurent polynomials could not be divisible by (1 − z)2. In other words, the spline-wavelet with this particular Laurent polynomial as its two-scale symbol has only one vanishing moment. To increase the number of vanishing moments, the notion of vanishing moment recovery (VMR) functions was introduced. A significant portion of this article is devoted to this relatively recent development. Of course, dilation by 2 can be extended to arbitrary integer dilations while preserving the VMR functionality. This is another topic of discussion in this survey paper. This extension, as well as extension to vector-valued (i.e. multi-) wavelets, can be considered as special cases of a more general consideration of tight spline-wavelet frames with arbitrary nested knot sequences that allow multiple (i.e. stacked) knots. In particular, when m knots are stacked at x = α for mth order spline functions, we have splines and spline-wavelet tight frames on a half interval [a, ∞), and if, in addition, another m knots are stacked at x = b > a, the theory applies to a bounded interval [a, b]. This study can be considered as the spline approach to construction of non-stationary wavelets on bounded intervals. It is a summary of our recent joint work with W. He. Most of the results presented in this paper are valid for bi-frames, and particularly, sibling frames (i.e. bi-frames associated with the same refinable function), when tightness is sacrificed to achieve certain additional desirable properties. We will only consider the properties of symmetry, shift-invariance, and inter-orthogonality." @default.
• W1565463375 created "2016-06-24" @default.
• W1565463375 creator A5033634172 @default.
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• W1565463375 date "2003-01-01" @default.
• W1565463375 modified "2023-09-25" @default.
• W1565463375 title "Recent Development of Spline Wavelet Frames with Compact Support" @default.
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• W1565463375 doi "https://doi.org/10.1016/s1570-579x(03)80035-6" @default.
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