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1.
2.
Jianing Sun 《Journal of Fourier Analysis and Applications》2009,15(5):739-752
In this paper, we construct a new family of Hermite-type interpolating scaling vectors with compact support, of which the
Hermite interpolation property generalizes the existing results of interpolating scaling vectors and Hermite interpolants.
In terms of the Hermite interpolatory mask, we characterize the Hermite interpolation property, approximation property and
symmetry property in detail. To illustrate these results, several examples with compact support and high smoothness are exhibited
at the end of this paper. 相似文献
3.
对具有任意伸缩矩阵A的插值加细函数,给出对应于L^2(R^s)中的小波包的一个构造方法.采样空间被直接分解来取代对加细函数的符号分解.按照这个方法构造的插值小波包能对基插值空间提供较为精细的分解,因而对自适应的插值给出较好的局部化. 相似文献
4.
Compactly supported (bi)orthogonal wavelets generated by interpolatory refinable functions 总被引:7,自引:0,他引:7
This paper provides several constructions of compactly supported wavelets generated by interpolatory refinable functions.
It was shown in [7] that there is no real compactly supported orthonormal symmetric dyadic refinable function, except the
trivial case; and also shown in [10,18] that there is no compactly supported interpolatory orthonormal dyadic refinable function.
Hence, for the dyadic dilation case, compactly supported wavelets generated by interpolatory refinable functions have to be
biorthogonal wavelets. The key step to construct the biorthogonal wavelets is to construct a compactly supported dual function
for a given interpolatory refinable function. We provide two explicit iterative constructions of such dual functions with
desired regularity. When the dilation factors are larger than 3, we provide several examples of compactly supported interpolatory
orthonormal symmetric refinable functions from a general method. This leads to several examples of orthogonal symmetric (anti‐symmetric)
wavelets generated by interpolatory refinable functions.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
5.
Martin Ehler 《Journal of Fourier Analysis and Applications》2007,13(5):511-532
In this article, we construct compactly supported multivariate pairs of dual wavelet frames, shortly called bi-frames, for
an arbitrary dilation matrix. Our construction is based on the mixed oblique extension principle, and it provides bi-frames
with few wavelets. In the examples, we obtain optimal bi-frames, i.e., primal and dual wavelets are constructed from a single
fundamental refinable function, whose mask size is minimal w.r.t. sum rule order and smoothness. Moreover, the wavelets reach
the maximal approximation orderw.r.t. the underlying refinable function. For special dilation matrices, we derive very simple
but optimal arbitrarily smooth bi-frames in arbitrary dimensions with only two primal and dual wavelets. 相似文献
6.
We investigate full rank interpolatory vector subdivision schemes whose masks are positive definite on the unit circle except
the point z=1. Such masks are known to give rise to convergent schemes with a cardinal limit function in the scalar case.
In the full rank vector case, we show that there also exists a cardinal refinable function based on this mask, however, with
respect to a different notion of refinability which nevertheless also leads to an iterative scheme for the computation of
vector fields. Moreover, we show the existence of orthogonal scaling functions for multichannel wavelets and give a constructive
method to obtain these scaling functions.
AMS subject classification (2000) 42C40, 65T60, 65D05 相似文献
7.
Costanza Conti Mariantonia Cotronei Tomas Sauer 《Journal of Approximation Theory》2010,162(3):559-575
We extend our previous work on interpolatory vector subdivision schemes to the multivariate case. As in the univariate case we show that the diagonal and off-diagonal elements of such a scheme have a significantly different structure and that under certain circumstances symmetry of the mask can increase the polynomial reproduction power of the subdivision scheme. Moreover, we briefly point out how tensor product constructions for vector subdivision schemes can be obtained. 相似文献
8.
This short note presents four examples of compactly supported symmetric refinable componentwise polynomial functions: (i) a componentwise constant interpolatory continuous refinable function and its derived symmetric tight wavelet frame; (ii) a componentwise constant continuous orthonormal and interpolatory refinable function and its associated symmetric orthonormal wavelet basis; (iii) a differentiable symmetric componentwise linear polynomial orthonormal refinable function; (iv) a symmetric refinable componentwise linear polynomial which is interpolatory and differentiable. 相似文献
9.
Karsten Koch 《Journal of Fourier Analysis and Applications》2009,15(1):1-30
In this paper we introduce an algorithm for the construction of compactly supported interpolating scaling vectors on ℝ
d
with certain symmetry properties. In addition, we give an explicit construction method for corresponding symmetric dual scaling
vectors and multiwavelets. As the main ingredients of our recipe we derive some implementable conditions for accuracy, symmetry,
and biorthogonality of a scaling vector in terms of its mask. Our method is substantiated by several bivariate examples for
quincunx and box-spline dilation matrices.
相似文献
10.
Approximation properties of multivariate wavelets 总被引:12,自引:0,他引:12
Rong-Qing Jia. 《Mathematics of Computation》1998,67(222):647-665
Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask. We connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators. Finally, we demonstrate that a refinable function in provides approximation order .
11.
Regularity of Multivariate Refinable Functions 总被引:1,自引:0,他引:1
The regularity of a univariate compactly supported refinable function is known to be related to the spectral properties of
an associated transfer operator. In the case of multivariate refinable functions with a general dilation matrix A , although factorization techniques, which are typically used in the univariate setting, are no longer applicable, we derive
similar results that also depend on the spectral properties of A .
September 30, 1996. Dates revised: December 1, 1996; February 14, 1997; August 1, 1997; November 11, 1997. Date accepted:
November 14, 1997. 相似文献
12.
Thomas A. Hogan 《Journal of Fourier Analysis and Applications》1997,3(6):757-774
Typical constructions of wavelets depend on the stability of the shifts of an underlying refinable function. Unfortunately,
several desirable properties are not available with compactly supported orthogonal wavelets, e.g., symmetry and piecewise
polynomial structure. Presently, multiwavelets seem to offer a satisfactory alternative. The study of multiwavelets involves
the consideration of the properties of several (simultaneously) refinable functions. In Section 2 of this article, we characterize
stability and linear independence of the shifts of a finite refinable function set in terms of the refinement mask. Several
illustrative examples are provided.
The characterizations given in Section 2 actually require that the refinable functions be minimal in some sense. This notion
of minimality is made clear in Section 3, where we provide sufficient conditions on the mask to ensure minimality. The conditions
are shown to be necessary also under further assumptions on the refinement mask. An example is provided illustrating how the
software package MAPLE can be used to investigate at least the case of two simultaneously refinable functions. 相似文献
13.
A construction of interpolating wavelets on invariant sets 总被引:8,自引:0,他引:8
We introduce the concept of a refinable set relative to a family of contractive mappings on a metric space, and demonstrate how such sets are useful to recursively construct interpolants which have a multiscale structure. The notion of a refinable set parallels that of a refinable function, which is the basis of wavelet construction. The interpolation points we recursively generate from a refinable set by a set-theoretic multiresolution are analogous to multiresolution for functions used in wavelet construction. We then use this recursive structure for the points to construct multiscale interpolants. Several concrete examples of refinable sets which can be used for generating interpolatory wavelets are included.
14.
V. Paulauskas 《Lithuanian Mathematical Journal》2009,49(4):426-445
In the paper, we approximate the distribution function of a sum of independent nonidentically distributed bivariate random
vectors by the distribution function of a stable vector and estimate the accuracy of such an approximation. The obtained general
result is only a little bit worse when compared with known estimates for the case of multivariate independent and identically
distributed random vectors or univariate nonidentically distributed summands. We also apply the result obtained to a specific
scheme arising when considering the so-called Increment-Ratio Statistics. 相似文献
15.
Arne Barinka Titus Barsch Stephan Dahlke Mario Mommer Michael Konik 《Journal of Computational Analysis and Applications》2002,4(4):339-361
This paper is concerned with the construction and the analysis of Gauss quadrature formulas for computing integrals of (smooth) functions against refinable functions and wavelets. The main goal of this paper is to develop rigorous error estimates for these formulas. For the univariate setting, we derive asymptotic error bounds for a huge class of weight functions including spline functions. We also discuss multivariate quadrature rules and present error estimates for specific nonseparable refinable functions, i.e., for some special box splines. 相似文献
16.
In this paper, we shall investigate the symmetry property of a multivariate orthogonal M-refinable function with a general dilation matrix M. For an orthogonal M-refinable function such that is symmetric about a point (centro-symmetric) and provides the approximation order k, we show that must be an orthogonal M-refinable function that generates a generalized coiflet of order k. Next, we show that there does not exist a real-valued compactly supported orthogonal 2Is-refinable function in any dimension such that is symmetric about a point and generates a classical coiflet. Finally, we prove that if a real-valued compactly supported orthogonal dyadic refinable function L2(Rs) has the axis symmetry, then cannot be a continuous function and can provide the approximation order at most one. The results in this paper may provide a better picture about symmetric multivariate orthogonal refinable functions. In particular, one of the results in this paper settles a conjecture in [D. Stanhill, Y.Y. Zeevi, IEEE Trans. Signal Process. 46 (1998), 183–190] about symmetric orthogonal dyadic refinable functions. 相似文献
17.
The objective of this paper is to introduce a general procedure for deriving interpolatory surface subdivision schemes with “symmetric subdivision templates” (SSTs) for regular vertices. While the precise definition of “symmetry” will be clarified in the paper, the property of SSTs is instrumental to facilitate application of the standard procedure for finding symmetric weights for taking weighted averages to accommodate extraordinary (or irregular) vertices in surface subdivisions, a topic to be studied in a continuation paper. By allowing the use of matrices as weights, the SSTs introduced in this paper may be constructed to overcome the size barrier limited to scalar-valued interpolatory subdivision templates, and thus avoiding the unnecessary surface oscillation artifacts. On the other hand, while the old vertices in a (scalar) interpolatory subdivision scheme do not require a subdivision template, we will see that this is not the case for the matrix-valued setting. Here, we employ the same definition of interpolation subdivisions as in the usual scalar consideration, simply by requiring the old vertices to be stationary in the definition of matrix-valued interpolatory subdivisions. Hence, there would be another complication when the templates are extended to accommodate extraordinary vertices if the template sizes are not small. In this paper, we show that even for C2 interpolatory subdivisions, only one “ring” is sufficient in general, for both old and new vertices. For example, for 1-to-4 split C2 interpolatory surface subdivisions, we obtain matrix-valued symmetric interpolatory subdivision templates (SISTs) for both triangular and quadrilateral meshes with sizes that agree with those of the Loop and Catmull–Clark schemes, respectively. Matrix-valued SISTs of similar sizes are also constructed for C2 interpolatory and subdivision schemes in this paper. In addition to small template sizes, an obvious feature of matrix-valued weights is the flexibility for introducing shape-control parameters. Another significance is that, in contrast to the usual scalar setting, matrix-valued SISTs can be formulated in terms of the coefficient sequence of some vector refinement equation of interpolating bivariate C2 splines with small support. For example, by modifying the spline function vectors introduced in our previous work [C.K. Chui, Q.T. Jiang, Surface subdivision schemes generated by refinable bivariate spline function vectors, Appl. Comput. Harmon. Anal. 15 (2003) 147–162; C.K. Chui, Q.T. Jiang, Refinable bivariate quartic and quintic C2-splines for quadrilateral subdivisions, Preprint, 2004], C2 symmetric interpolatory subdivision schemes associated with refinement equations of C2 cubic and quartic splines on the 6-directional and 4-directional meshes, respectively, are also constructed in this paper. 相似文献
18.
Di-Rong Chen 《Journal of Approximation Theory》1999,97(2):685
Subdivision operators play an important role in wavelet analysis. This paper studies the algebraic properties of subdivision operators with matrix mask, especially their action on polynomial sequences and on some of their invariant subspaces. As an application, we characterize, under a mild condition, the approximation order provided by refinable vectors in terms of the eigenvalues and eigenvectors of polynomial sequences of the associated subdivision operator. Moreover, some necessary conditions, in terms of nondegeneracy and simplicity of eigenvalues of a matrix related to the subdivision operator for the refinable vector to be smooth are given. The main results are new even in the scalar case 相似文献
19.
Multivariate Compactly Supported Fundamental Refinable Functions, Duals, and Biorthogonal Wavelets 总被引:8,自引:0,他引:8
In areas of geometric modeling and wavelets, one often needs to construct a compactly supported refinable function φ which has sufficient regularity and which is fundamental for interpolation [that means, φ(0)=1 and φ(α)=0 for all α∈ Z s ∖{0}].
Low regularity examples of such functions have been obtained numerically by several authors, and a more general numerical scheme was given in [1]. This article presents several schemes to construct compactly supported fundamental refinable functions, which have higher regularity, directly from a given, continuous, compactly supported, refinable fundamental function φ. Asymptotic regularity analyses of the functions generated by the constructions are given.The constructions provide the basis for multivariate interpolatory subdivision algorithms that generate highly smooth surfaces.
A very important consequence of the constructions is a natural formation of pairs of dual refinable functions, a necessary element in constructing biorthogonal wavelets. Combined with the biorthogonal wavelet construction algorithm for a pair of dual refinable functions given in [2], we are able to obtain symmetrical compactly supported multivariate biorthogonal wavelets which have arbitrarily high regularity. Several examples are computed. 相似文献
Low regularity examples of such functions have been obtained numerically by several authors, and a more general numerical scheme was given in [1]. This article presents several schemes to construct compactly supported fundamental refinable functions, which have higher regularity, directly from a given, continuous, compactly supported, refinable fundamental function φ. Asymptotic regularity analyses of the functions generated by the constructions are given.The constructions provide the basis for multivariate interpolatory subdivision algorithms that generate highly smooth surfaces.
A very important consequence of the constructions is a natural formation of pairs of dual refinable functions, a necessary element in constructing biorthogonal wavelets. Combined with the biorthogonal wavelet construction algorithm for a pair of dual refinable functions given in [2], we are able to obtain symmetrical compactly supported multivariate biorthogonal wavelets which have arbitrarily high regularity. Several examples are computed. 相似文献
20.
Costanza Conti Luca Gemignani Lucia Romani 《Advances in Computational Mathematics》2011,35(2-4):217-241
In this paper we describe a general, computationally feasible strategy to deduce a family of interpolatory non-stationary subdivision schemes from a symmetric non-stationary, non-interpolatory one satisfying quite mild assumptions. To achieve this result we extend our previous work (Conti et al., Linear Algebra Appl 431(10):1971?C1987, 2009) to full generality by removing additional assumptions on the input symbols. For the so obtained interpolatory schemes we prove that they are capable of reproducing the same space of exponential polynomials as the one generated by the original approximating scheme. Moreover, we specialize the computational methods for the case of symbols obtained by shifted non-stationary affine combinations of exponential B-splines, that are at the basis of most non-stationary subdivision schemes. In this case we find that the associated family of interpolatory symbols can be determined to satisfy a suitable set of generalized interpolating conditions at the set of the zeros (with reversed signs) of the input symbol. Finally, we discuss some computational examples by showing that the proposed approach can yield novel smooth non-stationary interpolatory subdivision schemes possessing very interesting reproduction properties. 相似文献