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Über dieses Buch

These proceedings were prepared in connection with the international conference Approximation Theory XIII, which was held March 7–10, 2010 in San Antonio, Texas. The conference was the thirteenth in a series of meetings in Approximation Theory held at various locations in the United States, and was attended by 144 participants. Previous conferences in the series were held in Austin, Texas (1973, 1976, 1980, 1992), College Station, Texas (1983, 1986, 1989, 1995), Nashville, Tennessee (1998), St. Louis, Missouri (2001), Gatlinburg, Tennessee (2004), and San Antonio, Texas (2007).

Along with the many plenary speakers, the contributors to this proceedings provided inspiring talks and set a high standard of exposition in their descriptions of new directions for research. Many relevant topics in approximation theory are included in this book, such as abstract approximation, approximation with constraints, interpolation and smoothing, wavelets and frames, shearlets, orthogonal polynomials, univariate and multivariate splines, and complex approximation.

Inhaltsverzeichnis

Frontmatter

An Asymptotic Equivalence Between Two Frame Perturbation Theorems

Abstract
In this paper, two stability results regarding exponential frames are compared. The theorems, (one proven herein, and the other in Sun and Zhou (J. Math. Anal. Appl. 235:159–167, 1999)), each give a constant such that if \({\sup }_{n\in {\mathbb{Z}}^{}}\|{\epsilon {}_{n}\|}_{\infty } < C\), and (ei⟨ ⋅,t n ) n d is a frame for L 2[−π,π] d , then (ei⟨ ⋅,t n n ) n d is a frame for L 2[−π,π] d . These two constants are shown to be asymptotically equivalent for large values of d.
B. A. Bailey

Growth Behavior and Zero Distribution of Maximally Convergent Rational Approximants

Abstract
Given a compact set E in \(\mathbb{C}\) and a function f holomorphic on E, we investigate the distribution of zeros of rational uniform approximants \(\{{r}_{n,{m}_{n}}\}\) with numerator degree≤n and denominator degree≤m n , where \({m}_{n} = o(n/\log n)\) as n. We obtain a Jentzsch–Szegő type result, i.e., the zero distribution converges weakly to the equilibrium distribution of the maximal Green domain E ρ(f) of meromorphy of f if f has a singularity of multivalued character on the boundary ∂E ρ(f). Further, we show that any singular point of f on the boundary ∂E ρ(f), that is not a pole, is a limit point of zeros of the sequence \(\{{r}_{n,{m}_{n}}\}\).
Hans-Peter Blatt, René Grothmann, Ralitza K. Kovacheva

Generalization of Polynomial Interpolation at Chebyshev Nodes

Abstract
Previously, we generalized the Lagrange polynomial interpolation at Chebyshev nodes and studied the Lagrange polynomial interpolation at a special class of sets of nodes. This special class includes some well-known sets of nodes, such as zeros of the Chebyshev polynomials of first and second kinds, Chebyshev extrema, and equidistant nodes. In this paper, we view our previous work from a different perspective and further generalize and study the Lagrange polynomial interpolation at a larger class of sets of nodes. In particular, the set of optimal nodes is included in this extended class.
Debao Chen

Green’s Functions: Taking Another Look at Kernel Approximation, Radial Basis Functions, and Splines

Abstract
The theories for radial basis functions (RBFs) as well as piecewise polynomial splines have reached a stage of relative maturity as is demonstrated by the recent publication of a number of monographs in either field. However, there remain a number of issues that deserve to be investigated further. For instance, it is well known that both splines and radial basis functions yield “optimal” interpolants, which in the case of radial basis functions are discussed within the so-called native space setting. It is also known that the theory of reproducing kernels provides a common framework for the interpretation of both RBFs and splines. However, the associated reproducing kernel Hilbert spaces (or native spaces) are often not that well understood — especially in the case of radial basis functions. By linking (conditionally) positive definite kernels to Green’s functions of differential operators we obtain new insights that enable us to better understand the nature of the native space as a generalized Sobolev space. An additional feature that appears when viewing things from this perspective is the notion of scale built into the definition of these function spaces. Furthermore, the eigenfunction expansion of a positive definite kernel via Mercer’s theorem provides a tool for making progress on such important questions as stable computation with flat radial basis functions and dimension independent error bounds.
Gregory E. Fasshauer

Sparse Recovery Algorithms: Sufficient Conditions in Terms of Restricted Isometry Constants

Abstract
We review three recovery algorithms used in Compressive Sensing for the reconstruction s-sparse vectors x N from the mere knowledge of linear measurements y=A x m , m<N. For each of the algorithms, we derive improved conditions on the restricted isometry constants of the measurement matrix A that guarantee the success of the reconstruction. These conditions are δ2s <0.4652 for basis pursuit, δ3s <0.5 and δ2s <0.25 for iterative hard thresholding, and δ4s <0.3843 for compressive sampling matching pursuit. The arguments also applies to almost sparse vectors and corrupted measurements. The analysis of iterative hard thresholding is surprisingly simple. The analysis of basis pursuit features a new inequality that encompasses several inequalities encountered in Compressive Sensing.
Simon Foucart

Lagrange Interpolation and New Asymptotic Formulae for the Riemann Zeta Function

Abstract
An asymptotic representation for the Riemann zeta function ζ(s) in terms of the Lagrange interpolation error of some function f s,2N at the Chebyshev nodes is found. The representation is based on new error formulae for the Lagrange polynomial interpolation to a function of the form \(f(y) ={ \int \nolimits \nolimits }_{\mathbb{R}} \frac{\varphi (t)} {t-iy}\mathrm{d}t.\) As the major application of this result, new criteria for ζ(s)=0 and ζ(s)≠0 in the critical strip 0<Re s<1 are given.
Michael I. Ganzburg

Active Geometric Wavelets

Abstract
We present an algorithm for highly geometric sparse representation. The algorithm combines the adaptive Geometric Wavelets method with the Active Contour segmentation to overcome limitations of both algorithms. It generalizes the Geometric Wavelets by allowing to adaptively construct wavelets supported on curved domains. It also improves upon the Active Contour method that can only be used to segment a limited number of objects. We show applications of this new method in medical image segmentation.
Itai Gershtansky, Shai Dekel

Interpolating Composite Systems

Abstract
Composite systems are multiscale decompositions similar to wavelet MRAs but with several different dilation operations for each scale. A notable example of such a system is given by the shearlet transform. In this paper we construct interpolating wavelet-type decompositions for such systems and study their approximation properties. As a main application we give an example of an interpolating shearlet transform.
Philipp Grohs

Wavelets and Framelets Within the Framework of Nonhomogeneous Wavelet Systems

Abstract
In this paper, we shall discuss recent developments in the basic theory of wavelets and framelets within the framework of nonhomogeneous wavelet systems in a natural and simple way. We shall see that nonhomogeneous wavelet systems naturally link many aspects of wavelet analysis together. There are two fundamental issues of the basic theory of wavelets and framelets: frequency-based nonhomogeneous dual framelets in the distribution space and stability of nonhomogeneous wavelet systems in a general function space. For example, without any a priori condition, we show that every dual framelet filter bank derived via the oblique extension principle (OEP) always has an underlying frequency-based nonhomogeneous dual framelet in the distribution space. We show that directional representations to capture edge singularities in high dimensions can be easily achieved by constructing nonstationary nonhomogeneous tight framelets in \({L}_{2}(\mathbb{R})\) with the dilation matrix \(2{I}_{mathbbD}\). Moreover, such directional tight framelets are derived from tight framelet filter banks derived via OEP. We also address the algorithmic aspects of wavelets and framelets such as discrete wavelet/framelet transform and its basic properties in the discrete sequence setting. We provide the reader in this paper a more or less complete picture so far on wavelets and framelets with the framework of nonhomogeneous wavelet systems.
Bin Han

Compactly Supported Shearlets

Abstract
Shearlet theory has become a central tool in analyzing and representing 2D data with anisotropic features. Shearlet systems are systems of functions generated by one single generator with parabolic scaling, shearing, and translation operators applied to it, in much the same way wavelet systems are dyadic scalings and translations of a single function, but including a precise control of directionality. Of the many directional representation systems proposed in the last decade, shearlets are among the most versatile and successful systems. The reason for this being an extensive list of desirable properties: shearlet systems can be generated by one function, they provide precise resolution of wavefront sets, they allow compactly supported analyzing elements, they are associated with fast decomposition algorithms, and they provide a unified treatment of the continuum and the digital realm. The aim of this paper is to introduce some key concepts in directional representation systems and to shed some light on the success of shearlet systems as directional representation systems. In particular, we will give an overview of the different paths taken in shearlet theory with focus on separable and compactly supported shearlets in 2D and 3D. We will present constructions of compactly supported shearlet frames in those dimensions as well as discuss recent results on the ability of compactly supported shearlet frames satisfying weak decay, smoothness, and directional moment conditions to provide optimally sparse approximations of cartoon-like images in 2D as well as in 3D. Finally, we will show that these compactly supported shearlet systems provide optimally sparse approximations of an even generalized model of cartoon-like images comprising of C 2 functions that are smooth apart from piecewise C 2 discontinuity edges.
Gitta Kutyniok, Jakob Lemvig, Wang-Q Lim

Shearlets on Bounded Domains

Abstract
Shearlet systems have so far been only considered as a means to analyze L 2-functions defined on \({\mathbb{R}}^{2}\), which exhibit curvilinear singularities. However, in applications such as image processing or numerical solvers of partial differential equations the function to be analyzed or efficiently encoded is typically defined on a non-rectangular shaped bounded domain. Motivated by these applications, in this paper, we first introduce a novel model for cartoon-like images defined on a bounded domain. We then prove that compactly supported shearlet frames satisfying some weak decay and smoothness conditions, when orthogonally projected onto the bounded domain, do provide (almost) optimally sparse approximations of elements belonging to this model class.
Gitta Kutyniok, Wang-Q Lim

On Christoffel Functions and Related Quantities for Compactly Supported Measures

Abstract
Let μ be a compactly supported positive measure on the real line, with associated orthogonal polynomials p n . Without any global restrictions such as regularity, we discuss convergence in measure for
We also establish convergence a.e.for sufficiently sparse subsequences of Christoffel function ratios.
D. S. Lubinsky

Exact Solutions of Some Extremal Problems of Approximation Theory

Abstract
F. Peherstorfer and R. Steinbauer introduced the complex T-polynomials. Recently, their rational analogues appear as Chebyshev – Markov rational functions on arcs with zeros on these arcs. Explicit representation and detailed proof for the particular case of one arc is given here. Author’s reminiscences about Franz Peherstorfer are included.
A. L. Lukashov

A Lagrange Interpolation Method by Trivariate Cubic C 1 Splines of Low Locality

Abstract
We develop a local Lagrange interpolation method for trivariate cubic C 1 splines. The splines are constructed on a uniform partition consisting of octahedra (with one additional edge) and tetrahedra. The method is 2-local and stable and therefore yields optimal approximation order. The numerical results and visualizations confirm the efficiency of the method.
G. Nürnberger, G. Schneider

Approximation of Besov Vectors by Paley–Wiener Vectors in Hilbert Spaces

Abstract
We develop an approximation theory in Hilbert spaces that generalizes the classical theory of approximation by entire functions of exponential type. The results advance harmonic analysis on manifolds and graphs, thus facilitating data representation, compression, denoising and visualization. These tasks are of great importance to machine learning, complex data analysis and computer vision.
Isaac Z. Pesenson, Meyer Z. Pesenson

A Subclass of the Length 12 Parameterized Wavelets

Abstract
In this paper, a subclass of the length 12 parameterized wavelets is given. This subclass is a parameterization of the coefficients of a subset of the trigonometric polynomials, m(ω), that satisfy the necessary conditions for orthogonality, that is m(0)=1 and \(\vert m(\omega ){\vert }^{2} + \vert m(\omega + \pi ){\vert }^{2} = 1\), but is not sufficient to represent all possible trigonometric polynomials satisfying these constraints. This parameterization has three free parameters whereas the general parameterization would have five free parameters. Finally, we graph some example scaling functions from the parameterization and conclude with a numerical experiment.
David W. Roach

Geometric Properties of Inverse Polynomial Images

Abstract
Given a polynomial \({\mathcal{T}}_{n}\) of degree n, consider the inverse image of \(\mathbb{R}\) and [−1,1], denoted by \({\mathcal{T}}_{n}^{-1}(\mathbb{R})\) and \({\mathcal{T}}_{n}^{-1}([-1,1])\), respectively. It is well known that \({\mathcal{T}}_{n}^{-1}(\mathbb{R})\) consists of n analytic Jordan arcs moving from to . In this paper, we give a necessary and sufficient condition such that (1)\({\mathcal{T}}_{n}^{-1}([-1,1])\) consists of ν analytic Jordan arcs and (2)\({\mathcal{T}}_{n}^{-1}([-1,1])\) is connected, respectively.
Klaus Schiefermayr

On Symbolic Computation of Ideal Projectors and Inverse Systems

Abstract
A zero-dimensional ideal J in the ring \(\mathbb{k}[\mathbf{x}]\) of polynomials in d variables is often given in terms of its “border basis”; that is a particular finite set of polynomials that generate the ideal. We produce a convenient formula for symbolic computation of the space of functionals on \(\mathbb{k}[\mathbf{x}]\) that annihilate J. The formula is particularly useful for computing an explicit form of an ideal projector from its values on a certain finite set of polynomials.
Boris Shekhtman

The Dimension of the Space of Smooth Splines of Degree 8 on Tetrahedral Partitions

Abstract
Let \(\Omega \subset {\mathbb{R}}^{3}\) be a connected polyhedral domain that is allowed to contain polyhedral holes and Δ be a tetrahedral partition of Ω. Given 0≤rd, we define
$${S}_{d}^{r}(\Delta ) =\{ s \in {C}^{r}(\Omega );\ s{\vert }_{ \sigma } \in {P}_{d}\mbox{ for any tetrahedron}\sigma \in \Delta )\},$$
the spline space of degree d and smoothness r, where P d is the trivariate polynomial space of total degree not exceeding d. In this paper, we obtained the following result. Theorem
$$\begin{array}{rll} \mbox{ dim}{S}_{8}^{1}(\Delta )& =&{\sum }_{\mathbf{v}\in V }\mbox{ dim}{S}_{3}^{1}(\mbox{ Star}(\mathbf{v})) + 5\vert E\vert + 9\vert F\vert + \vert T\vert + 3\vert {E}_{b}\vert + 3\vert {E}_{\delta }\vert, \end{array}$$
where V,E,F,T,E b , and E δ are the sets of vertices, edges, triangles, tetrahedra, boundary edges, and singular edges of Δ, respectively.
Xiquan Shi, Ben Kamau, Fengshan Liu, Baocai Yin

On Simultaneous Approximation in Function Spaces

Abstract
The problem of simultaneous approximation in function spaces has attracted many researchers recently. Major results on the space of vector-valued continuous functions started to appear early nineties. In 2002, results on simultaneous approximation in p-Bochner integrable function spaces were published. The objective of this paper is to give a characterization for some subspaces of Bochner integrable functions space to be simultaneously proximinal.
Eyad Abu-Sirhan

Chalmers–Metcalf Operator and Uniqueness of Minimal Projections in ℓ n ∞ and ℓ n 1 Spaces

Abstract
We construct the Chalmers–Metcalf operator for minimal projections onto hyperplanes in n and 1 n and prove it is uniquely determined. We show how we can use Chalmers–Metcalf operator to obtain uniqueness of minimal projections. The main advantage of our approach is that it is purely algebraical and does not require consideration of the min–max problems.
Lesław Skrzypek

The Polynomial Inverse Image Method

Abstract
In this survey, we discuss how to transfer results from an interval or the unit circle to more general sets. At the basis of the method is taking polynomial inverse images.
Vilmos Totik

On Approximation of Periodic Analytic Functions by Linear Combinations of Complex Exponents

Abstract
For a 2π-periodic function f, analytic on I = [0, 2π], we solve the minimization problem
$${E}_{n}(f) {=\min }_{{c}_{j}\in R}\|f + {c}_{1}f + \cdots + {c}_{n}{f{}^{(n)}\|}_{{ L}_{2}(I)}^{2},$$
and establish the convergence lim n E n (f) = 0. In case of even n i.e. 2n all of the zeros, ł j , j = 1, , 2n, of the corresponding characteristic polynomial \(1 + {c}_{1}\lambda + \cdots + {c}_{2n}{\lambda }^{2n}\) are purely imaginary, \({l}_{-j} = -{l}_{j}\), and we prove the estimate
$${ \max }_{t\in I}\vert f(t) -{\sum \nolimits }_{j=-n}^{n}{b}_{ j}{\mathrm{e}}^{\mathrm{i}{l}_{j}t}\vert \leq {\left ( \frac{2\pi } {{l}_{1}^{2}}{E}_{n}(f)\right )}^{1/2},$$
where l 1 has the smallest absolute value among all of the l’s.
Vesselin Vatchev

Matrix Extension with Symmetry and Its Applications

Abstract
In this paper, we are interested in the problems of matrix extension with symmetry, more precisely, the extensions of submatrices of Laurent polynomials satisfying some conditions to square matrices of Laurent polynomials with certain symmetry patterns, which are closely related to the construction of (bi)orthogonal multiwavelets in wavelet analysis and filter banks with the perfect reconstruction property in electronic engineering. We satisfactorily solve the matrix extension problems with respect to both orthogonal and biorthogonal settings. Our results show that the extension matrices do possess certain symmetry patterns and their coefficient supports can be controlled by the given submatrices in certain sense. Moreover, we provide step-by-step algorithms to derive the desired extension matrices. We show that our extension algorithms can be applied not only to the construction of (bi)orthogonal multiwavelets with symmetry, but also to the construction of tight framelets with symmetry and with high order of vanishing moments. Several examples are presented to illustrate the results in this paper.
Xiaosheng Zhuang
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