On compactly supported spline wavelets and a duality principle

Abstract

Abstract. Let • • • C K_ ] c Vq c Vx c • • • be a multiresolution analysis ofL2 generated by the mth order 5-spline Nm{x). In this paper, we exhibit a compactly supported basic wavelet i//m(x) that generates the corresponding orthogonal complementary wavelet subspaces ... , W_ \, Wo, Wx, ... . Consequently, the two finite sequences that describe the two-scale relations of Nm(x) and i//m(x) in terms of Nm(2x -j), ;6Z, yield an efficient reconstruction algorithm. To give an efficient wavelet decomposition algorithm based on these two finite sequences, we derive a duality principle, which also happens to yield the dual bases {Nm(x -j)} and {y/m(x -j)} , relative to {Nm(x -j)} and {y/m(x -j)}, respectively.

Spatiotemporal multi-resolution approximation of the Amari type neural field model

“…B-spline wavelet and scaling functions were formulated independently by Wang (1992a), Chui andWang (1992b) and Unser et al (1993). The scaling functions are m th order B-splines and the compact support wavelet functions are a linear combination of scaling functions.…”

“…The m th order cardinal B-spline function is defined by the recurrence relation (Chui and Wang, 1992b)…”

“…For B-spline scaling and wavelet functions of order m the two-scale relation pair takes the form of (Chui and Wang, 1992b)…”

An Effective Hybrid Method for Electro- Magnetic Scattering from Inhomogeneous Objects

“…Recently, wavelets [12][13][14][15] have been widely studied and applied by researchers in various engineering areas. Their applications in electromagnetics are getting increasing attention [16][17][18].…”

Trigonometric wavelets for Hermite interpolation

“…First, in §5, the inner product matrix of the scaling functions is explicitly computed as well as the entries of its inverse, which are the coefficients of the biorthogonal bases of dual functions. The usefulness of these dual functions -as described in [3] and [5] for functions on the real axis -can be seen in §6, where they are used to establish the more complicated decomposition relations. Finally, §7 provides a short numerical example illustrating practical results and offers a discussion of open questions.…”