Multivariate symmetric refinable functions and function vectors

For any symmetry group [Formula: see text], any appropriate matrix dilation (compatible with [Formula: see text]) and any appropriate symmetry center [Formula: see text] we give an explicit method for the construction of [Formula: see text]-symmetric with respect to the center [Formula: see text] refinable masks which have sum rule of an arbitrary order [Formula: see text]. Moreover, we give a description of all these masks. For any symmetry group [Formula: see text], any appropriate matrix dilation (compatible with [Formula: see text]) and any appropriate row of symmetry centers [Formula: see text] we give two explicit methods for the construction of [Formula: see text]-symmetric with respect to the row of centers [Formula: see text] refinable matrix masks which have sum rule of an arbitrary order [Formula: see text]. A description of all such matrix masks is also presented.

Multivariate sampling-type approximation

Approximation properties of the expansions [Formula: see text], where [Formula: see text] is a linear differential operator and [Formula: see text] is a matrix dilation, are studied. The sampling expansions are a special case of such differential expansions. Error estimations in [Formula: see text]-norm, [Formula: see text], are given in terms of the Fourier transform of [Formula: see text]. The approximation order depends on the smoothness of [Formula: see text], the order of [Formula: see text], the order of Strang–Fix condition for [Formula: see text] and [Formula: see text]. A wide class of [Formula: see text] including both band-limited and compactly supported functions is considered, but a special condition of compatibility [Formula: see text] with [Formula: see text] is required. Such differential expansions may be useful for engineers.

The Traits of Orthogonal Quarternary Small Function Wraps according to Quantity Matrix Dilation

The advantages of wavelet packets and their promising featu-res in various application have attracted a lot of interest and effort in recent years. In this paper, we propose the notion of quarternary small function wraps according to a quantity matrix dilation, which are gener-alization of univariate wavelet wraps. A nice procedure for designing the vector-valued quarternary small-wave wraps is provided. Their cha-racteristics are researched by virtue of time-frequency analysis method, wavelet transform and curvelet coefficients. The orthogonality formulas concerning these small-wave wraps are established. Furthermore, it is shown how to draw new orthogonal bases of space from the small-wave wraps. A method for designing a class of affine quarternary dual frames in four-dimensional space is presented. The results we obtain gains much improvement