Quincunx fundamental refinable functions and quincunx biorthogonal wavelets

We analyze the approximation and smoothness properties of quincunx fundamental refinable functions. In particular, we provide a general way for the construction of quincunx interpolatory refinement masks associated with the quincunx lattice in . Their corresponding quincunx fundamental refinable functions attain the optimal approximation order and smoothness order. In addition, these examples are minimally supported with symmetry. For two special families of such quincunx interpolatory masks, we prove that their symbols are nonnegative. Finally, a general way of constructing quincunx biorthogonal wavelets is presented. Several examples of quincunx interpolatory masks and quincunx biorthogonal wavelets are explicitly computed.

     

Interpolation Wavelets in Boundary Value Problems

We propose and validate a simple numerical method that finds an approximate solution with any given accuracy to the Dirichlet boundary value problem in a disk for a homogeneous equation with the Laplace operator. There are many known numerical methods that solve this problem, starting with the approximate calculation of the Poisson integral, which gives an exact representation of the solution inside the disk in terms of the given boundary values of the required functions. We employ the idea of approximating a given 2π-periodic boundary function by trigonometric polynomials, since it is easy to extend them to harmonic polynomials inside the disk so that the deviation from the required harmonic function does not exceed the error of approximation of the boundary function. The approximating trigonometric polynomials are constructed by means of an interpolation projection to subspaces of a multiresolution analysis (approximation) with basis 2π-periodic scaling functions (more exactly, their binary rational compressions and shifts). Such functions were constructed by the authors earlier on the basis of Meyer-type wavelets; they are either orthogonal and at the same time interpolating on uniform grids of the corresponding scale or only interpolating. The bounds on the rate of approximation of the solution to the boundary value problem are based on the property ofMeyer wavelets to preserve trigonometric polynomials of certain (large) orders; this property was used for other purposes in the first two papers listed in the references. Since a numerical bound of the approximation error is very important for the practical application of the method, a considerable portion of the paper is devoted to this issue, more exactly, to the explicit calculation of the constants in the order bounds of the error known earlier.     

Approximation properties of multivariate wavelets

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 .

     

Trigonometric interpolation and wavelet decompositions

The aim of this paper is to describe explicit decomposition and reconstruction algorithms for nested spaces of trigonometric polynomials. The scaling functions of these spaces are defined as fundamental polynomials of Lagrange interpolation. The interpolatory conditions and the construction of dual functions are crucial for the approach presented in this paper.This paper was completed while the first author visited the Center for Approximation Theory, Texas A&M University. Research partially supported by Deutsche Forschungsgemeinschaft.     

Bi-frames with 4-fold axial symmetry for quadrilateral surface multiresolution processing

When bivariate filter banks and wavelets are used for surface multiresolution processing, it is required that the decomposition and reconstruction algorithms for regular vertices derived from them have high symmetry. This symmetry requirement makes it possible to design the corresponding multiresolution algorithms for extraordinary vertices. Recently lifting-scheme based biorthogonal bivariate wavelets with high symmetry have been constructed for surface multiresolution processing. If biorthogonal wavelets have certain smoothness, then the analysis or synthesis scaling function or both have big supports in general. In particular, when the synthesis low-pass filter is a commonly used scheme such as Loop’s scheme or Catmull-Clark’s scheme, the corresponding analysis low-pass filter has a big support and the corresponding analysis scaling function and wavelets have poor smoothness. Big supports of scaling functions, or in other words big templates of multiresolution algorithms, are undesirable for surface processing. On the other hand, a frame provides flexibility for the construction of “basis” systems. This paper concerns the construction of wavelet (or affine) bi-frames with high symmetry.In this paper we study the construction of wavelet bi-frames with 4-fold symmetry for quadrilateral surface multiresolution processing, with both the dyadic and refinements considered. The constructed bi-frames have 4 framelets (or frame generators) for the dyadic refinement, and 2 framelets for the refinement. Namely, with either the dyadic or refinement, a frame system constructed in this paper has only one more generator than a wavelet system. The constructed bi-frames have better smoothness and smaller supports than biorthogonal wavelets. Furthermore, all the frame algorithms considered in this paper are given by templates so that one can easily implement them.     

Sobolev estimates for constructive uniform-grid FFT interpolatory approximations of spherical functions

The fast Fourier transform (FFT) based matrix-free ansatz interpolatory approximations of periodic functions are fundamental for efficient realization in several applications. In this work we design, analyze, and implement similar constructive interpolatory approximations of spherical functions, using samples of the unknown functions at the poles and at the uniform spherical-polar grid locations \(\left (\frac {j\pi }{N}, \frac {k \pi }{N}\right )\), for j=1,…,N?1, k=0,…,2N?1. The spherical matrix-free interpolation operator range space consists of a selective subspace of two dimensional trigonometric polynomials which are rich enough to contain all spherical polynomials of degree less than N. Using the \({\mathcal {O}}(N^{2})\) data, the spherical interpolatory approximation is efficiently constructed by applying the FFT techniques (in both azimuthal and latitudinal variables) with only \({\mathcal {O}}(N^{2} \log N)\) complexity. We describe the construction details using the FFT operators and provide complete convergence analysis of the interpolatory approximation in the Sobolev space framework that are well suited for quantification of various computer models. We prove that the rate of spectrally accurate convergence of the interpolatory approximations in Sobolev norms (of order zero and one) are similar (up to a log term) to that of the best approximation in the finite dimensional ansatz space. Efficient interpolatory quadratures on the sphere are important for several applications including radiation transport and wave propagation computer models. We use our matrix-free interpolatory approximations to construct robust FFT-based quadrature rules for a wide class of non-, mildly-, and strongly-oscillatory integrands on the sphere. We provide numerical experiments to demonstrate fast evaluation of the algorithm and various theoretical results presented in the article.     

Polynomial wavelets on the unit circle

In this paper, we have considered polynomial wavelets on unit circle. The scaling functions are considered to be the fundamental polynomials of the Lagrange interpolants on the equally spaced nodes different from the n roots of unity, which satisfy certain interpolatory conditions.     

Biorthogonal exponential sequences with weight function on the real line and an orthogonal sequence of trigonometric functions

Some orthogonal functions can be mapped onto other orthogonal functions by the Fourier transform. In this paper, by using the Fourier transform of Stieltjes-Wigert polynomials, we derive a sequence of exponential functions that are biorthogonal with respect to a complex weight function like on . Then we restrict these introduced biorthogonal functions to a special case to obtain a sequence of trigonometric functions orthogonal with respect to the real weight function on .

     

Tight frames of multidimensional wavelets

In this paper we deal with multidimensional wavelets arising from a multiresolution analysis with an arbitrary dilation matrix A, namely we have scaling equations $$\varphi ^s (x) = \sum\limits_{k \in \mathbb{Z}^n } {h_k^s \sqrt {|\det A|} \varphi ^1 } (Ax - k) for s = 1, \ldots ,q,$$ where ?¹ is a scaling function for this multiresolution and ?², …, ?^q (q=|det A |) are wavelets. Orthogonality conditions for ?¹, …, ?^q naturally impose constraints on the scaling coefficients $\{ h_k^s \} _{k \in \mathbb{Z}^n }^{s = 1, \ldots ,q} $ , which are then called the wavelet matrix. We show how to reconstruct functions satisfying the scaling equations above and show that ?², …, ?^q always constitute a tight frame with constant 1. Furthermore, we generalize the sufficient and necessary conditions of orthogonality given by Lawton and Cohen to the case of several dimensions and arbitrary dilation matrix A.     

Rational functions and real Schubert calculus

We single out some problems of Schubert calculus of subspaces of codimension that have the property that all their solutions are real whenever the data are real. Our arguments explore the connection between subspaces of codimension and rational functions of one variable.

     

Frames, modular functions for shift-invariant subspaces and FMRA wavelet frames

We introduce the concept of the modular function for a shift-invariant subspace that can be represented by normalized tight frame generators for the shift-invariant subspace and prove that it is independent of the selections of the frame generators for the subspace. We shall apply it to study the connections between the dimension functions of wavelet frames for any expansive integer matrix and the multiplicity functions for general multiresolution analysis (GMRA). Given a frame mutiresolution analysis (FMRA), we show that the standard construction formula for orthonormal multiresolution analysis wavelets does not yield wavelet frames unless the underlying FMRA is an MRA. A modified explicit construction formula for FMRA wavelet frames is given in terms of the frame scaling functions and the low-pass filters.

     

Orthogonal Laurent polynomials corresponding to certain strong Stieltjes distributions with applications to numerical quadratures

In this paper we shall be mainly concerned with sequences of orthogonal Laurent polynomials associated with a class of strong Stieltjes distributions introduced by A.S. Ranga. Algebraic properties of certain quadratures formulae exactly integrating Laurent polynomials along with an application to estimate weighted integrals on with nearby singularities are given. Finally, numerical examples involving interpolatory rules whose nodes are zeros of orthogonal Laurent polynomials are also presented.

     

Nonlinear multiscale decompositions: The approach of A. Harten

Datadependent interpolatory techniques can be used in the reconstruction step of a multiresolution scheme designed à la Harten. In this paper we carefully analyze the class of Essentially NonOscillatory (ENO) interpolatory techniques described in [11] and their potential to improve the compression capabilities of multiresolution schemes. When dealing with nonlinear multiresolution schemes the issue of stability also needs to be carefully considered.     

Optimal two-dimensional interpolatory ternary subdivision schemes with two-ring stencils

For any interpolatory ternary subdivision scheme with two-ring stencils for a regular triangular or quadrilateral mesh, we show that the critical Hölder smoothness exponent of its basis function cannot exceed , where the critical Hölder smoothness exponent of a function is defined to be

On the other hand, for both regular triangular and quadrilateral meshes, we present several examples of interpolatory ternary subdivision schemes with two-ring stencils such that the critical Hölder smoothness exponents of their basis functions do achieve the optimal smoothness upper bound . Consequently, we obtain optimal smoothest interpolatory ternary subdivision schemes with two-ring stencils for the regular triangular and quadrilateral meshes. Our computation and analysis of optimal multidimensional subdivision schemes are based on the projection method and the -norm joint spectral radius.

     

Biorthogonal wavelets with 4-fold axial symmetry for quadrilateral surface multiresolution processing

Surface multiresolution processing is an important subject in CAGD. It also poses many challenging problems including the design of multiresolution algorithms. Unlike images which are in general sampled on a regular square or hexagonal lattice, the meshes in surfaces processing could have an arbitrary topology, namely, they consist of not only regular vertices but also extraordinary vertices, which requires the multiresolution algorithms have high symmetry. With the idea of lifting scheme, Bertram (Computing 72(1–2):29–39, 2004) introduces a novel triangle surface multiresolution algorithm which works for both regular and extraordinary vertices. This method is also successfully used to develop multiresolution algorithms for quad surface and \(\sqrt 3\) triangle surface processing in Wang et al. (Vis Comput 22(9–11):874–884, 2006; IEEE Trans Vis Comput Graph 13(5):914–925, 2007) respectively. When considering the biorthogonality, these papers do not use the conventional \(L^2({{\rm I}\kern-.2em{\rm R}}^2)\) inner product, and they do not consider the corresponding lowpass filter, highpass filters, scaling function and wavelets. Hence, some basic properties such as smoothness and approximation power of the scaling functions and wavelets for regular vertices are unclear. On the other hand, the symmetry of subdivision masks (namely, the lowpass filters of filter banks) for surface subdivision is well studied, while the symmetry of the highpass filters for surface processing is rarely considered in the literature. In this paper we introduce the notion of 4-fold symmetry for biorthogonal filter banks. We demonstrate that 4-fold symmetric filter banks result in multiresolution algorithms with the required symmetry for quad surface processing. In addition, we provide 4-fold symmetric biorthogonal FIR filter banks and construct the associated wavelets, with both the dyadic and \(\sqrt 2\) refinements. Furthermore, we show that some filter banks constructed in this paper result in very simple multiresolution decomposition and reconstruction algorithms as those in Bertram (Computing 72(1–2):29–39, 2004) and Wang et al. (Vis Comput 22(9–11):874–884, 2006; IEEE Trans Vis Comput Graph 13(5):914–925, 2007). Our method can provide the filter banks corresponding to the multiresolution algorithms in Wang et al. (Vis Comput 22(9–11):874–884, 2006) for dyadic multiresolution quad surface processing. Therefore, the properties of the scaling functions and wavelets corresponding to those algorithms can be obtained by analyzing the corresponding filter banks.     

Propagation of normality along regular analytic Jordan arcs in analytic functions with values in a complex unital Banach algebra with continuous involution

Globevnik and Vidav have studied the propagation of normality from an open subset of a region of the complex plane for analytic functions with values in the space of bounded linear operators on a Hilbert space . We obtain a propagation of normality in the more general setting of a converging sequence located on a regular analytic Jordan arc in the complex plane for analytic functions with values in a complex unital Banach algebra with continuous involution. We show that in this more general setting, the propagation of normality does not imply functional commutativity anymore as it does in the case studied by Globevnik and Vidav. An immediate consequence of the Propagation of Normality Theorem is that the further generalization given by Wolf of Jamison's generalization of Rellich's theorem is equivalent to Jamison's result. We obtain a propagation property within Banach subspaces for analytic Banach space-valued functions. The propagation of normality differs from the propagation within Banach subspaces since the set of all normal elements does not form a Banach subspace.

     

On the existence and constructions of orthonormal wavelets on

For a multiresolution analysis of associated with the scaling matrix having determinant we prove the existence of a wavelet basis with certain desirable properties if and its real-valued counterpart if the scaling function is real-valued and . That those results cannot be extended to and respectively in general is demonstrated by Adams's theorem about vector fields on spheres. Moreover we present some new explicit constructions of wavelets, among which is a variation of Riemenschneider-Shen's method for

     

Subspaces with normalized tight frame wavelets in

In this paper we investigate the subspaces of which have normalized tight frame wavelets that are defined by set functions on some measurable subsets of called Bessel sets. We show that a subspace admitting such a normalized tight frame wavelet falls into a class of subspaces called reducing subspaces. We also consider the subspaces of that are generated by a Bessel set in a special way. We present some results concerning the relation between a Bessel set and the corresponding subspace of which either has a normalized tight frame wavelet defined by the set function on or is generated by .