Narrow Results

Refinement equations of the type play an exceptional role in the theory of wavelets, subdivision algorithms and computer design. It is known that the regularity of their compactly supported solutions (refinable functions) depends on the spectral properties of special N-dimensional linear operators T₀, T₁ constructed by the coefficients of the equation. In particular, the structure of kernels and of common invariant subspaces of these operators have been intensively studied in the literature. In this paper, we give a complete classification of the kernels and of all the root subspaces of T₀ and T₁, as well as of their common invariant subspaces. This result answers several open questions stated in the literature and clarifies the structure of the space spanned by the integer translates of refinable functions. This also leads to some results on the moduli of continuity of refinable functions and wavelets in various functional spaces. In particular, it is proved that the Hölder exponent of those functions is sharp whenever it is not an integer.

The notion of multiple vector-valued wavelet packets is introduced. A procedure for constructing the multiple vector-valued wavelet packets is presented. Their characteristics are investigated by means of integral transformation and operator theory, and three orthogonality formulas concerning the multiple vector-valued wavelet packets. Finally, new orthogonal bases of L²(R, C^{s × s}) are constructed from these multiple vector-valued wavelet packets.

In this paper, the notion for orthogonal vector-valued multivariate wavelet packets of space L²(Rⁿ,C^s) is introduced. A method of the construction of orthogonal vector-valued wavelet packets associated with quantity dilation matrix aI (2 ≤ a ∈ Z) is presented. The properties for vector-valued multivariate wavelet packets have been investigated. Finally, a new orthogonal basis of L²(Rⁿ,C^s) is derived from the orthogonal multivariate vector-valued wavelet packets.

In this paper, the notion of orthogonal vector-valued wavelet packets of space L²(Rⁿ,C^s) is introduced. A procedure for constructing orthogonal vector-valued wavelet packets associated with dilation matrix 2I is presented. The properties of vector-valued wavelet packets in higher dimensions have been investigated. Finally, a new orthogonal basis of L²(Rⁿ,C^s) is obtained from the orthogonal vector-valued wavelet packets in higher dimensions.

In this paper, the notion of vector-valued wavelet packets associated with arbitrary integer-valued dilation matrix A is introduced. First, the definition for vector-valued wavelet packets is given. A method for constructing the orthogonl vector-valued wavelet packets is presented. The properties for these multivariate vector-valued wavelet packets are investigated.