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Construction of vector-valued multivariate wavelet frame packets. (English) Zbl 1253.42039

Summary: In this paper, the splitting trick coined by D.-R. Chen [SIAM J. Math. Anal. 31, No. 4, 726–739 (2000; Zbl 0966.42024)] is used to construct vector-valued multivariate wavelet frame packets with an arbitrary dilation matrix A. It is shown that, as long as finitely many splitting steps are applied, the resulting sequence of functions is a frame of \(L^2{(\mathbb{R}^d)}^r\). If the matrix \(Q(\xi)\) associated with the splitting is unitary, then the splitting can be applied infinitely many times to prove the existence of frame with the frame bounds as shown in Theorem 3.3.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15 General harmonic expansions, frames
65T60 Numerical methods for wavelets

Citations:

Zbl 0966.42024