285 research outputs found
Smoothness of multivariate refinable functions with infinitely supported masks
AbstractIn this paper, we investigate the smoothness of multivariate refinable functions with infinitely supported masks and an isotropic dilation matrix. Using some methods as in [R.Q. Jia, Characterization of smoothness of multivariate refinable functions in Sobolev spaces, Trans. Amer. Math. Soc. 351 (1999) 4089–4112], we characterize the optimal smoothness of multivariate refinable functions with polynomially decaying masks and an isotropic dilation matrix. Our characterizations extend some of the main results of the above mentioned paper with finitely supported masks to the case in which masks are infinitely supported
Vector Subdivision Schemes for Arbitrary Matrix Masks
Employing a matrix mask, a vector subdivision scheme is a fast iterative
averaging algorithm to compute refinable vector functions for wavelet methods
in numerical PDEs and to produce smooth curves in CAGD. In sharp contrast to
the well-studied scalar subdivision schemes, vector subdivision schemes are
much less well understood, e.g., Lagrange and (generalized) Hermite subdivision
schemes are the only studied vector subdivision schemes in the literature.
Because many wavelets used in numerical PDEs are derived from refinable vector
functions whose matrix masks are not from Hermite subdivision schemes, it is
necessary to introduce and study vector subdivision schemes for any general
matrix masks in order to compute wavelets and refinable vector functions
efficiently. For a general matrix mask, we show that there is only one
meaningful way of defining a vector subdivision scheme. Motivated by vector
cascade algorithms and recent study on Hermite subdivision schemes, we shall
define a vector subdivision scheme for any arbitrary matrix mask and then we
prove that the convergence of the newly defined vector subdivision scheme is
equivalent to the convergence of its associated vector cascade algorithm. We
also study convergence rates of vector subdivision schemes. The results of this
paper not only bridge the gaps and establish intrinsic links between vector
subdivision schemes and vector cascade algorithms but also strengthen and
generalize current known results on Lagrange and (generalized) Hermite
subdivision schemes. Several examples are provided to illustrate the results in
this paper on various types of vector subdivision schemes with convergence
rates
Stationary multivariate subdivision: Joint spectral radius and asymptotic similarity
In this paper we study scalar multivariate non-stationary subdivision schemes with
a general integer dilation matrix. We present a new numerically efficient method for
checking convergence and H ̈older regularity of such schemes. This method relies on the
concepts of approximate sum rules, asymptotic similarity and the so-called joint spectral
radius of a finite set of square matrices. The combination of these concepts allows us to
employ recent advances in linear algebra for exact computation of the joint spectral radius
that have had already a great impact on studies of stationary subdivision schemes. We
also expose the limitations of non-stationary schemes in their capability to reproduce and
generate certain function spaces. We illustrate our results with several examples
Wavelets and their use
This review paper is intended to give a useful guide for those who want to
apply discrete wavelets in their practice. The notion of wavelets and their use
in practical computing and various applications are briefly described, but
rigorous proofs of mathematical statements are omitted, and the reader is just
referred to corresponding literature. The multiresolution analysis and fast
wavelet transform became a standard procedure for dealing with discrete
wavelets. The proper choice of a wavelet and use of nonstandard matrix
multiplication are often crucial for achievement of a goal. Analysis of various
functions with the help of wavelets allows to reveal fractal structures,
singularities etc. Wavelet transform of operator expressions helps solve some
equations. In practical applications one deals often with the discretized
functions, and the problem of stability of wavelet transform and corresponding
numerical algorithms becomes important. After discussing all these topics we
turn to practical applications of the wavelet machinery. They are so numerous
that we have to limit ourselves by some examples only. The authors would be
grateful for any comments which improve this review paper and move us closer to
the goal proclaimed in the first phrase of the abstract.Comment: 63 pages with 22 ps-figures, to be published in Physics-Uspekh
- …