1,071 research outputs found
Subdivisions with infinitely supported mask
AbstractIn this paper we investigate the convergence of subdivision schemes associated with masks being polynomially decay sequences. Two-scale vector refinement equations are the formφ(x)=∑α∈Za(α)φ(2x-α),x∈R,where the vector of functions φ=(φ1,…,φr)T is in (L2(R))r and a≕(a(α))α∈Z is polynomially decay sequence of r×r matrices called refinement mask. Associated with the mask a is a linear operator on (L2(R))r given byQaf(x)≔∑α∈Za(α)f(2x-α),x∈R,f=(f1,…,fr)T∈(L2(R))r.By using same methods in [B. Han, R. Q. Jia, Characterization of Riesz bases of wavelets generated from multiresolution analysis, manuscript]; [B. Han, Refinable functions and cascade algorithms in weighted spaces with infinitely supported masks, manuscript]; [R.Q. Jia, Q.T. Jiang, Z.W. Shen, Convergence of cascade algorithms associated with nonhomogeneous refinement equations, Proc. Amer. Math. Soc. 129 (2001) 415–427]; [R.Q. Jia, Convergence of vector subdivision schemes and construction of biorthogonal multiple wavelets, in: Advances in Wavelet, Hong Kong,1997, Springer, Singapore, 1998, pp. 199–227], a characterization of convergence of the sequences (Qanf)n=1,2,… in the L2-norm is given, which extends the main results in [R.Q. Jia, S.D. Riemenschneider, D.X. Zhou, Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998) 1533–1563] on convergence of the subdivision schemes associated with a finitely supported mask to the case in which mask a is polynomially decay sequence. As an application, we also obtain a characterization of smoothness of solutions of the refinement equation mentioned above for the case r=1
Numerics and Fractals
Local iterated function systems are an important generalisation of the
standard (global) iterated function systems (IFSs). For a particular class of
mappings, their fixed points are the graphs of local fractal functions and
these functions themselves are known to be the fixed points of an associated
Read-Bajactarevi\'c operator. This paper establishes existence and properties
of local fractal functions and discusses how they are computed. In particular,
it is shown that piecewise polynomials are a special case of local fractal
functions. Finally, we develop a method to compute the components of a local
IFS from data or (partial differential) equations.Comment: version 2: minor updates and section 6.1 rewritten, arXiv admin note:
substantial text overlap with arXiv:1309.0243. text overlap with
arXiv:1309.024
Non-equispaced B-spline wavelets
This paper has three main contributions. The first is the construction of
wavelet transforms from B-spline scaling functions defined on a grid of
non-equispaced knots. The new construction extends the equispaced,
biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new
construction is based on the factorisation of wavelet transforms into lifting
steps. The second and third contributions are new insights on how to use these
and other wavelets in statistical applications. The second contribution is
related to the bias of a wavelet representation. It is investigated how the
fine scaling coefficients should be derived from the observations. In the
context of equispaced data, it is common practice to simply take the
observations as fine scale coefficients. It is argued in this paper that this
is not acceptable for non-interpolating wavelets on non-equidistant data.
Finally, the third contribution is the study of the variance in a
non-orthogonal wavelet transform in a new framework, replacing the numerical
condition as a measure for non-orthogonality. By controlling the variances of
the reconstruction from the wavelet coefficients, the new framework allows us
to design wavelet transforms on irregular point sets with a focus on their use
for smoothing or other applications in statistics.Comment: 42 pages, 2 figure
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