121 results
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2. Extensions of linear operators from hyperplanes and strong uniqueness of best approximation in L(X,W)
- Author
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Paweł Wójcik
- Subjects
Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Banach space ,010103 numerical & computational mathematics ,Codimension ,Extension (predicate logic) ,01 natural sciences ,Projection (linear algebra) ,Operator (computer programming) ,Hyperplane ,Uniqueness ,0101 mathematics ,Analysis ,Subspace topology ,Mathematics - Abstract
The aim of this paper is to present some results concerning the problem of minimal projections and extensions. Let X be a reflexive Banach space and let Y be a closed subspace of X of codimension one. Let W be a finite-dimensional Banach space. We present a new sufficient condition under which any minimal extension of an operator A ∈ L ( Y , W ) is strongly unique. In this paper we show (in some circumstances) that if 1 λ ( Y , X ) , then a minimal projection from X onto Y is a strongly unique minimal projection. Moreover, we introduce and study a new geometric property of normed spaces. In this paper we also present a result concerning the strong unicity of best approximation.
- Published
- 2019
3. Reproducing kernel orthogonal polynomials on the multinomial distribution
- Author
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Robert C. Griffiths and Persi Diaconis
- Subjects
Numerical Analysis ,Stationary distribution ,Markov chain ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Poisson kernel ,010103 numerical & computational mathematics ,Kravchuk polynomials ,01 natural sciences ,Combinatorics ,symbols.namesake ,Kernel (statistics) ,Orthogonal polynomials ,symbols ,Test statistic ,Multinomial distribution ,0101 mathematics ,Analysis ,Mathematics - Abstract
Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q n ( x , y ; N , p ) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n = 0 , 1 , … , N . The Poisson kernel ∑ n = 0 N ρ n Q n ( x , y ; N , p ) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new duplication formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The duplication formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The χ 2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.
- Published
- 2019
4. Comparison of probabilistic and deterministic point sets on the sphere
- Author
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Peter J. Grabner and T. A. Stepanyuk
- Subjects
Unit sphere ,Numerical Analysis ,Sequence ,Applied Mathematics ,General Mathematics ,Existential quantification ,010102 general mathematics ,Probabilistic logic ,Sampling (statistics) ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Point (geometry) ,0101 mathematics ,Constant (mathematics) ,Analysis ,Mathematics - Abstract
In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (especially spherical t -designs) are better or as good as probabilistic ones like the jittered sampling model. We find asymptotic equalities for the discrete Riesz s -energy of sequences of well separated t -designs on the unit sphere S d ⊂ R d + 1 , d ≥ 2 . The case d = 2 was studied in Hesse (2009) and Hesse and Leopardi (2008). In Bondarenko et al., (2015) it was established that for d ≥ 2 , there exists a constant c d , such that for every N > c d t d there exists a well-separated spherical t -design on S d with N points. This paper gives results, based on recent developments that there exists a sequence of well separated spherical t -designs such that t and N are related by N ≍ t d .
- Published
- 2019
5. On the existence of optimal meshes in every convex domain on the plane
- Author
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András Kroó
- Subjects
Numerical Analysis ,Polynomial ,Conjecture ,Degree (graph theory) ,Plane (geometry) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Polytope ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Cardinality ,Polygon mesh ,0101 mathematics ,Constant (mathematics) ,Analysis ,Mathematics - Abstract
In this paper we study the so called optimal polynomial meshes for domains in K ⊂ R d , d ≥ 2 . These meshes are discrete point sets Y n of cardinality c n d which have the property that ‖ p ‖ K ≤ A ‖ p ‖ Y n for every polynomial p of degree at most n with a constant A > 1 independent of n . It was conjectured earlier that optimal polynomial meshes exist in every convex domain. This statement was previously shown to hold for polytopes and C 2 like domains. In this paper we give a complete affirmative answer to the above conjecture when d = 2 .
- Published
- 2019
6. On the entropy numbers of the mixed smoothness function classes
- Author
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Vladimir Temlyakov
- Subjects
Numerical Analysis ,Multivariate statistics ,Nonlinear approximation ,Greedy approximation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Applied mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Analysis ,Mathematics - Abstract
Behavior of the entropy numbers of classes of multivariate functions with mixed smoothness is studied here. This problem has a long history and some fundamental problems in the area are still open. The main goal of this paper is to develop a new method of proving the upper bounds for the entropy numbers. This method is based on recent developments of nonlinear approximation, in particular, on greedy approximation. This method consists of the following two steps strategy. At the first step we obtain bounds of the best m -term approximations with respect to a dictionary. At the second step we use general inequalities relating the entropy numbers to the best m -term approximations. For the lower bounds we use the volume estimates method, which is a well known powerful method for proving the lower bounds for the entropy numbers. It was used in a number of previous papers.
- Published
- 2017
7. C0-semigroups and resolvent operators approximated by Laguerre expansions
- Author
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Pedro J. Miana and Luciano Abadias
- Subjects
Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Banach space ,Holomorphic function ,Order (ring theory) ,01 natural sciences ,Convolution ,Functional calculus ,010101 applied mathematics ,Rate of convergence ,Laguerre polynomials ,0101 mathematics ,Analysis ,Mathematics ,Resolvent - Abstract
In this paper we introduce Laguerre expansions to approximate vector-valued functions. We apply this result to approximate C 0 -semigroups and resolvent operators in abstract Banach spaces. We study certain Laguerre functions in order to estimate the rate of convergence of these expansions. Finally, we illustrate the main results of this paper with some examples: shift, convolution and holomorphic semigroups, where the rate of convergence is improved.
- Published
- 2017
8. Stability of Fredholm properties on interpolation Banach spaces
- Author
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Mieczysław Mastyło, Natan Kruglyak, and Irina Asekritova
- Subjects
Mathematics::Functional Analysis ,Numerical Analysis ,Pure mathematics ,Functor ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Banach space ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Surjective function ,Operator (computer programming) ,Interpolation space ,0101 mathematics ,Equivalence (measure theory) ,Analysis ,Interpolation ,Mathematics - Abstract
The main aim of this paper is to prove novel results on stability of the semi-Fredholm property of operators on interpolation spaces generated by interpolation functors. The methods are based on some general ideas we develop in the paper. This allows us to extend some previous work in literature to the abstract setting. We show an application to interpolation methods introduced by Cwikel–Kalton–Milman–Rochberg which includes, as special cases, the real and complex methods up to equivalence of norms and also some other well known methods of interpolation. A by-product of these results get the stability of isomorphisms on Calderon products of Banach function lattices. We also study the important characteristics in operator Banach space theory, the so-called modules of injection and surjection, and we prove interpolation estimates of these modules of operators on scales of the Calderon complex interpolation spaces.
- Published
- 2020
9. On the strong divergence of Hilbert transform approximations and a problem of Ul’yanov
- Author
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Holger Boche and Volker Pohl
- Subjects
Numerical Analysis ,Sequence ,Conjecture ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,020206 networking & telecommunications ,02 engineering and technology ,01 natural sciences ,Combinatorics ,symbols.namesake ,Uniform norm ,Subsequence ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Hilbert transform ,0101 mathematics ,Divergence (statistics) ,Finite set ,Fourier series ,Analysis ,Mathematics - Abstract
This paper studies the approximation of the Hilbert transform f ? = H f of continuous functions f with continuous conjugate f ? based on a finite number of samples. It is known that every sequence { H N f } N ? N which approximates f ? from samples of f diverges (weakly) with respect to the uniform norm. This paper conjectures that all of these approximation sequences even contain no convergent subsequence. A property which is termed strong divergence.The conjecture is supported by two results. First it is proven that the sequence of the sampled conjugate Fejer means diverges strongly. Second, it is shown that for every sample based approximation method { H N } N ? N there are functions f such that ? H N f ? ∞ exceeds any given bound for any given number of consecutive indices N .As an application, the later result is used to investigate a problem associated with a question of Ul'yanov on Fourier series which is related to the possibility to construct adaptive approximation methods to determine the Hilbert transform from sampled data. This paper shows that no such approximation method with a finite search horizon exists.
- Published
- 2016
10. On approximation properties of generalized Kantorovich-type sampling operators
- Author
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Olga Orlova and Gert Tamberg
- Subjects
Numerical Analysis ,Generalization ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Microlocal analysis ,Sampling (statistics) ,010103 numerical & computational mathematics ,Spectral theorem ,Singular integral ,Operator theory ,01 natural sciences ,Electronic mail ,Fourier integral operator ,Convolution ,Kernel (statistics) ,0101 mathematics ,Operator norm ,Analysis ,Mathematics - Abstract
In this paper, we generalize the notion of Kantorovich-type sampling operators using the Fejer-type singular integral. By means of these operators we are able to reconstruct signals (functions) which are not necessarily continuous. Moreover, our generalization allows us to take the measurement error into account. The goal of this paper is to estimate the rate of approximation by the above operators via high-order modulus of smoothness.
- Published
- 2016
11. Calculating the spectral factorization and outer functions by sampling-based approximations—Fundamental limitations
- Author
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Volker Pohl and Holger Boche
- Subjects
Numerical Analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Sampling (statistics) ,Spectral density ,010103 numerical & computational mathematics ,Function (mathematics) ,Dirichlet's energy ,Spectral theorem ,Hardy space ,Singular integral ,01 natural sciences ,symbols.namesake ,symbols ,0101 mathematics ,Closed-form expression ,Analysis ,Mathematics - Abstract
This paper considers the problem of approximating the spectral factor of continuous spectral densities with finite Dirichlet energy based on finitely many samples of these spectral densities. Although there exists a closed form expression for the spectral factor, this formula shows a very complicated behavior because of the non-linear dependency of the spectral factor from spectral density and because of a singular integral in this expression. Therefore approximation methods are usually applied to calculate the spectral factor. It is shown that there exists no sampling-based method which depends continuously on the samples and which is able to approximate the spectral factor for all densities in this set. Instead, to any sampling-based approximation method there exists a large set of spectral densities so that the approximation method does not converge to the spectral factor for every spectral density in this set as the number of available sampling points is increased. The paper will also show that the same results hold for sampling-based algorithms for the calculation of the outer function in the theory of Hardy spaces.
- Published
- 2020
12. Multivariate bounded variation functions of Jordan–Wiener type
- Author
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Alexander Brudnyi and Yu. Brudnyi
- Subjects
Pointwise ,Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Predual ,010103 numerical & computational mathematics ,Space (mathematics) ,01 natural sciences ,Linear subspace ,Separable space ,Bounded function ,Bounded variation ,Differentiable function ,0101 mathematics ,Analysis ,Mathematics - Abstract
We introduce and study spaces of multivariate functions of bounded variation generalizing the classical Jordan and Wiener spaces. Multivariate generalizations of the Jordan space were given by several prominent researchers. However, each of the proposed concepts preserves only few properties of Jordan variation which are designed to a selected application. In contrast, the multivariate generalization of the Jordan space presented in this paper preserves all known and reveals some previously unknown properties of the space. These, in turn, are special cases of the basic properties of the introduced spaces proved in the paper. Specifically, the first part of the paper describes structure properties of functions of bounded ( k , p ) -variation ( V p k functions). It includes assertions on discontinuity sets and pointwise differentiability of V p k functions and their Luzin type and C ∞ approximations. The second part presents results on Banach structure of V p k spaces, namely, atomic decomposition and constructive characterization of their predual spaces. As a result, we obtain the so-called two-stars theorems describing V p k spaces as second duals of their separable subspaces consisting of functions of “vanishing variation”.
- Published
- 2020
13. The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets
- Author
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Hein Hundal and Frank Deutsch
- Subjects
Mathematics(all) ,Alternating projections ,General Mathematics ,Convex feasibility problem ,0211 other engineering and technologies ,02 engineering and technology ,01 natural sciences ,Cyclic projections ,POCS ,Combinatorics ,symbols.namesake ,Intersection ,Angle between subspaces ,Projections onto convex sets ,0101 mathematics ,Mathematics ,Numerical Analysis ,021103 operations research ,Series (mathematics) ,Applied Mathematics ,010102 general mathematics ,Hilbert space ,Regular polygon ,Orthogonal projections ,Angle between convex sets ,Rate of convergence ,The strong conical hull intersection property (strong CHIP) ,Norm of nonlinear operators ,Iterated function ,Norm (mathematics) ,symbols ,Algorithm ,Analysis ,Regularity properties of convex sets: regular, linearly regular, boundedly regular, boundedly linearly regular, normal, weakly normal, uniformly normal - Abstract
The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the ''convex feasibility'' problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the ''angles'' between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the ''norm'' of the product of the (nonlinear) metric projections onto related convex sets. In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the ''linear regularity property'' of Bauschke and Borwein, the ''normal property'' of Jameson (as well as Bakan, Deutsch, and Li's generalization of Jameson's normal property), the ''strong conical hull intersection property'' of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.
- Published
- 2008
14. Approximation with scaled shift-invariant spaces by means of quasi-projection operators
- Author
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Rong-Qing Jia
- Subjects
Mathematics(all) ,General Mathematics ,Wavelet Analysis ,Cascade algorithms ,Moduli of smoothness ,010103 numerical & computational mathematics ,Spectral theorem ,Quasi-interpolation ,01 natural sciences ,0101 mathematics ,Mathematics ,Approximation theory ,Numerical Analysis ,Lipschitz spaces ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Operator theory ,Lipschitz continuity ,Compact operator on Hilbert space ,Approximation order ,Sobolev space ,Spline (mathematics) ,Shift-invariant spaces ,Sobolev spaces ,De Boor's algorithm ,Quasi-projection ,Analysis - Abstract
The work of de Boor and Fix on spline approximation by quasiinterpolants has had far-reaching influence in approximation theory since publication of their paper in 1973. In this paper, we further develop their idea and investigate quasi-projection operators. We give sharp estimates in terms of moduli of smoothness for approximation with scaled shift-invariant spaces by means of quasi-projection operators. In particular, we provide error analysis for approximation of quasi-projection operators with Lipschitz spaces. The study of quasi-projection operators has many applications to various areas related to approximation theory and wavelet analysis.
- Published
- 2004
- Full Text
- View/download PDF
15. Connections between the Support and Linear Independence of Refinable Distributions
- Author
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Jianzhong Wang and David K. Ruch
- Subjects
Discrete mathematics ,Mathematics(all) ,Numerical Analysis ,General Mathematics ,Multiresolution analysis ,Applied Mathematics ,010102 general mathematics ,Regular polygon ,01 natural sciences ,010101 applied mathematics ,Dilation (metric space) ,Distribution (mathematics) ,If and only if ,Linear independence ,0101 mathematics ,Independence (probability theory) ,Analysis ,Mathematics ,Integer (computer science) - Abstract
The purpose of this paper is to study the relationships between the support of a refinable distributionφand the global and local linear independence of the integer translates ofφ. It has been shown elsewhere that a compactly supported distributionφhas globally independent integer translates if and only ifφhas minimal convex support. However, such a distribution may have “holes” in its support. By insisting thatφ∈L2(R) and generates a multiresolution analysis, Lemarié and Malgouyres have ensured that no such holes can occur. In this article we generalize this result to refinable distributions. We also give a result on the local linear independence of the integer translates ofφ. We work with integer dilation factorN⩾2 throughout this paper.
- Published
- 1998
- Full Text
- View/download PDF
16. On the differential equation for the Laguerre–Sobolev polynomials
- Author
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Clemens Markett
- Subjects
Numerical Analysis ,Pure mathematics ,Differential equation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Field (mathematics) ,010103 numerical & computational mathematics ,Differential operator ,01 natural sciences ,Measure (mathematics) ,Sobolev space ,Product (mathematics) ,Orthogonal polynomials ,Laguerre polynomials ,0101 mathematics ,Analysis ,Mathematics - Abstract
The Laguerre–Sobolev polynomials form an orthogonal polynomial system with respect to a Sobolev-type inner product associated with the Laguerre measure on the positive half-axis and two point masses M , N > 0 at the origin involving functions and derivatives. These polynomials have attracted much interest over the last two decades, since they became known to satisfy, for any value of the Laguerre parameter α ∈ N 0 , a spectral differential equation of finite order 4 α + 10 . In this paper we establish a new explicit representation of the corresponding differential operator which consists of a number of elementary components depending on α , M , N . Their interaction reveals a rich structure both being useful for applications and as a model for further investigations in the field. In particular, the Laguerre–Sobolev differential operator is shown to be symmetric with respect to the Sobolev-type inner product.
- Published
- 2019
17. Central limit theorems for multivariate Bessel processes in the freezing regime II: The covariance matrices
- Author
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Michael Voit and Sergio Andraus
- Subjects
Pure mathematics ,General Mathematics ,Gaussian ,Mathematics::Classical Analysis and ODEs ,FOS: Physical sciences ,010103 numerical & computational mathematics ,01 natural sciences ,60F05, 60J60, 60B20, 70F10, 82C22, 33C67 ,symbols.namesake ,FOS: Mathematics ,Limit (mathematics) ,Representation Theory (math.RT) ,0101 mathematics ,Mathematical Physics ,Eigenvalues and eigenvectors ,Central limit theorem ,Mathematics ,Numerical Analysis ,Hermite polynomials ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Mathematical Physics (math-ph) ,Covariance ,symbols ,Laguerre polynomials ,Mathematics - Probability ,Mathematics - Representation Theory ,Analysis ,Bessel function - Abstract
Bessel processes $(X_{t,k})_{t\ge0}$ in $N$ dimensions are classified via associated root systems and multiplicity constants $k\ge0$. They describe interacting Calogero-Moser-Suther\-land particle systems with $N$ particles and are related to $\beta$-Hermite and $\beta$-Laguerre ensembles. Recently, several central limit theorems were derived for fixed $t>0$, fixed starting points, and $k\to\infty$. In this paper we extend the CLT in the A-case from start in 0 to arbitrary starting distributions by using a limit result for the corresponding Bessel functions. We also determine the eigenvalues and eigenvectors of the covariance matrices of the Gaussian limits and study applications to CLTs for the intermediate particles for $k\to\infty$ and then $N\to\infty$., Comment: 20 pages
- Published
- 2019
18. Weierstrass type approximation by weighted polynomials in Rd
- Author
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András Kroó
- Subjects
Numerical Analysis ,Polynomial ,Continuous function ,Degree (graph theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Regular polygon ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Uniform limit theorem ,Combinatorics ,Hyperplane ,0101 mathematics ,Real line ,Analysis ,Mathematics - Abstract
In this paper we consider weighted polynomial approximation on unbounded multidimensional domains in the spirit of the weighted version of the Weierstrass trigonometric theorem according to which every continuous function on the real line with equal finite limits at ± ∞ is a uniform limit on R of weighted algebraic polynomials of degree 2 n with varying weights ( 1 + t 2 ) − n . We will verify a similar statement in the multivariate setting for a general class of convex weights. We also consider the similar problem of multivariate polynomial approximation with varying weights for some typical non convex weights. In case of non convex weights of the form w α ( x ) ≔ ( 1 + | x 1 | α + . . . + | x d | α ) 1 α , 0 α 1 in order for weighted polynomial approximation to hold for a given continuous function it is necessary that the function vanishes on a certain exceptional set consisting of all coordinate hyperplanes and ∞ . Moreover, in case of rational α this condition is also sufficient for weighted polynomial approximation to hold.
- Published
- 2019
19. Fourier–Dunkl system of the second kind and Euler–Dunkl polynomials
- Author
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Antonio J. Durán, Mario Pérez, and Juan L. Varona
- Subjects
Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Generating function ,010103 numerical & computational mathematics ,Function (mathematics) ,Partial fraction decomposition ,01 natural sciences ,Exponential function ,symbols.namesake ,Fourier transform ,symbols ,Euler's formula ,0101 mathematics ,Analysis ,Quotient ,Bessel function ,Mathematics - Abstract
We prove a partial fraction decomposition of a quotient of two functions E α ( i t x ) and I α ( i t ) which are defined in terms of the Bessel functions J α and J α + 1 of the first kind. This expansion leads naturally to the introduction of an orthonormal system with respect to the measure | x | 2 α + 1 d x 2 α + 1 Γ ( α + 1 ) in [ − 1 , 1 ] , which we call the Fourier–Dunkl system of the second kind. Euler–Dunkl polynomials E n , α ( x ) of degree n are defined by considering E α ( t x ) ∕ I α ( t ) as a generating function. It is shown that the sum ∑ m = 1 ∞ 1 ∕ j m , α 2 k , where j m , α are the positive zeros of J α , is equal (up to an explicit factor) to E 2 k − 1 , α ( 1 ) . For α = 1 ∕ 2 this leads to classical results of Euler since the function E 1 ∕ 2 ( x ) is the exponential function and E n , 1 ∕ 2 ( x ) are (essentially) the Euler polynomials. In the second part of the paper a sampling theorem of Whittaker–Shannon–Kotel’nikov type is established which is strongly related to the above-mentioned partial decomposition and which holds for all functions in the Payley–Wiener space defined by the Dunkl transform in [ − 1 , 1 ] .
- Published
- 2019
20. On the strong restricted isometry property of Bernoulli random matrices
- Author
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Ran Lu
- Subjects
Numerical Analysis ,Applied Mathematics ,General Mathematics ,Gaussian ,Probability (math.PR) ,010102 general mathematics ,62G35, 42C15 ,010103 numerical & computational mathematics ,16. Peace & justice ,01 natural sciences ,Restricted isometry property ,Combinatorics ,Matrix (mathematics) ,Bernoulli's principle ,symbols.namesake ,Robustness (computer science) ,FOS: Mathematics ,symbols ,Erasure ,0101 mathematics ,Random matrix ,Random variable ,Mathematics - Probability ,Analysis ,Mathematics - Abstract
The study of the restricted isometry property (RIP) of corrupted random matrices is particularly important in the field of compressed sensing (CS) with corruptions. If a matrix still satisfies the RIP after that a certain portion of rows are erased, then we say that this matrix has the strong restricted isometry property (SRIP). In the field of compressed sensing, random matrices which satisfy certain moment conditions are of particular interest. Among these matrices, those with entries generated from i.i.d. Gaussian or symmetric Bernoulli random variables are often typically considered. Recent studies have shown that matrices with entries generated from i.i.d. Gaussian random variables satisfy the SRIP under arbitrary erasure of rows with high probability. In this paper, we study the erasure robustness property of Bernoulli random matrices. Our main result shows that with overwhelming probability, the SRIP holds for Bernoulli random matrices. Moreover, our analysis leads to a robust version of the famous Johnson–Lindenstrauss lemma for Bernoulli random matrices.
- Published
- 2019
21. Frame decomposition and radial maximal semigroup characterization of Hardy spaces associated to operators
- Author
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Lixin Yan, Liang Song, Xuan Thinh Duong, and Ji Li
- Subjects
Analytic semigroup ,Numerical Analysis ,Semigroup ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Holomorphic functional calculus ,010103 numerical & computational mathematics ,Hardy space ,01 natural sciences ,Functional calculus ,Combinatorics ,symbols.namesake ,Mathematics - Analysis of PDEs ,Bounded function ,Norm (mathematics) ,FOS: Mathematics ,symbols ,0101 mathematics ,Lp space ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
Let $L$ be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and H\"older's continuity. Also assume that $L$ has a bounded holomorphic functional calculus on $L^2(\mathbb{R}^n)$. In this paper, we construct a frame decomposition for the functions belonging to the Hardy space $H_{L}^{1}(\mathbb{R}^n)$ associated to $L$, and for functions in the Lebesgue spaces $L^p$, $1, Comment: 37 pages, to appear in Journal of Approximation Theory
- Published
- 2019
22. Local and global error estimates for Berrut’s rational interpolant at well-spaced nodes
- Author
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Jean Sidon
- Subjects
Numerical Analysis ,Class (set theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Interval (mathematics) ,01 natural sciences ,Combinatorics ,Distribution (mathematics) ,Equidistant ,0101 mathematics ,Chebyshev nodes ,Global error ,Analysis ,Mathematics - Abstract
Recently, the convergence–divergence properties of the Berrut rational interpolant (Berrut, 1988) were studied in the case of equidistant nodes (Mastroianni and Szabados, 2017). A very general class called “well-spaced nodes” was introduced in Bos et al. (2013), a class which includes essentially any distribution of nodes that satisfies a particular regularity condition, such as equidistant nodes, Chebyshev nodes as well as extended Chebyshev nodes. In this paper we show that results obtained in Mastroianni and Szabados, (2017) for the equidistant nodes hold more generally for the class of well-spaced nodes, with the restriction that the end points of the definition interval are nodes.
- Published
- 2019
23. Non-universality of the Riemann zeta function and its derivatives when σ≥1
- Author
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Takashi Nakamura and Hirofumi Nagoshi
- Subjects
Numerical Analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Universality theorem ,010103 numerical & computational mathematics ,01 natural sciences ,Riemann zeta function ,Bohr model ,Universality (dynamical systems) ,symbols.namesake ,symbols ,0101 mathematics ,Analysis ,Mathematics ,Mathematical physics - Abstract
Let ζ ( s ) be the Riemann zeta function. In 1911, Bohr showed that the set { ζ ( σ + i τ ) : σ > 1 , τ ∈ R } is dense in ℂ . By Voronin’s denseness theorems in 1972, the sets { ( ζ ( σ + i λ 1 + i τ ) , … , ζ ( σ + i λ n + i τ ) ) : σ ≥ 1 , τ ∈ R } with distinct λ 1 , … , λ n ∈ R and { ( ζ ( σ + i τ ) , ζ ′ ( σ + i τ ) , … , ζ ( n − 1 ) ( σ + i τ ) ) : σ ≥ 1 , τ ∈ R } are dense in ℂ n . By Voronin’s universality theorem, for any fixed 1 ∕ 2 σ 1 and any non-negative integer k , the set { ζ σ , τ ( k ) : τ ∈ R } is dense in C [ a , b ] , where ζ σ , τ ( k ) ( t ) ≔ ζ ( k ) ( σ + i t + i τ ) , t ∈ [ a , b ] . In the present paper, we prove that the set { ζ σ , τ ( k ) : σ ≥ 1 , τ ∈ R } ∩ C [ a , b ] is not dense in C [ a , b ] .
- Published
- 2019
24. Lattice rules with random n achieve nearly the optimal O(n−α−1∕2) error independently of the dimension
- Author
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Dirk Nuyens, Peter Kritzer, Frances Y. Kuo, and Mario Ullrich
- Subjects
Discrete mathematics ,Numerical Analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Upper and lower bounds ,Randomized algorithm ,Numerical integration ,Sobolev space ,Combinatorics ,Random variate ,Rate of convergence ,Lattice (order) ,Exponent ,0101 mathematics ,Analysis ,Mathematics - Abstract
We analyze a new random algorithm for numerical integration of d -variate functions over [ 0 , 1 ] d from a weighted Sobolev space with dominating mixed smoothness α ≥ 0 and product weights 1 ≥ γ 1 ≥ γ 2 ≥ ⋯ > 0 , where the functions are continuous and periodic when α > 1 ∕ 2 . The algorithm is based on rank-1 lattice rules with a random number of points n . For the case α > 1 ∕ 2 , we prove that the algorithm achieves almost the optimal order of convergence of O ( n − α − 1 ∕ 2 ) , where the implied constant is independent of the dimension d if the weights satisfy ∑ j = 1 ∞ γ j 1 ∕ α ∞ . The same rate of convergence holds for the more general case α > 0 by adding a random shift to the lattice rule with random n . This shows, in particular, that the exponent of strong tractability in the randomized setting equals 1 ∕ ( α + 1 ∕ 2 ) , if the weights decay fast enough. We obtain a lower bound to indicate that our results are essentially optimal. This paper is a significant advancement over previous related works with respect to the potential for implementation and the independence of error bounds on the problem dimension. Other known algorithms which achieve the optimal error bounds, such as those based on Frolov’s method, are very difficult to implement especially in high dimensions. Here we adapt a lesser-known randomization technique introduced by Bakhvalov in 1961. This algorithm is based on rank-1 lattice rules which are very easy to implement given the integer generating vectors. A simple probabilistic approach can be used to obtain suitable generating vectors.
- Published
- 2019
25. Exceptional Jacobi polynomials
- Author
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Niels Bonneux
- Subjects
Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,symbols.namesake ,Mathematics - Classical Analysis and ODEs ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,Jacobi polynomials ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this paper we present a systematic way to describe exceptional Jacobi polynomials via two partitions. We give the construction of these polynomials and restate the known aspects of these polynomials in terms of their partitions. The aim is to show that the use of partitions is an elegant way to label these polynomials. Moreover, we prove asymptotic results according to the regular and exceptional zeros of these polynomials., 40 pages, 1 figure
- Published
- 2019
26. Asymptotic expansion of orthogonal polynomials via difference equations
- Author
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Xiao-Min Huang, Li-Hua Cao, and Xiang-Sheng Wang
- Subjects
Numerical Analysis ,Recurrence relation ,Matching (graph theory) ,Differential equation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Constant of integration ,010103 numerical & computational mathematics ,01 natural sciences ,Symmetry (physics) ,Simple (abstract algebra) ,Orthogonal polynomials ,0101 mathematics ,Asymptotic expansion ,Analysis ,Mathematics - Abstract
This paper aims to develop a simple and unified technique in finding asymptotic expansion of orthogonal polynomials from their difference equations. By preserving the symmetry in the difference equation, we are able to express the higher-order terms in the asymptotic expansion as an integral whose integrand can be explicitly obtained by a recurrence relation, while the integration constant is to be determined by a matching condition that relates to the initial conditions and coefficients in the difference equation.
- Published
- 2019
27. System theory and orthogonal multi-wavelets
- Author
-
Maria Charina, Mihai Putinar, Costanza Conti, and Mariantonia Cotronei
- Subjects
Basic linear algebra ,Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Scalar (mathematics) ,Linear system ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,MathematicsofComputing_NUMERICALANALYSIS ,Univariate ,Data_CODINGANDINFORMATIONTHEORY ,010103 numerical & computational mathematics ,Unitary matrix ,01 natural sciences ,Transfer function ,Wavelet ,0101 mathematics ,Circulant matrix ,Analysis ,Mathematics - Abstract
In this paper we provide a complete and unifying characterization of compactly supported univariate scalar orthogonal wavelets and vector-valued or matrix-valued orthogonal multi-wavelets. This characterization is based on classical results from system theory and basic linear algebra. In particular, we show that the corresponding wavelet and multi-wavelet masks are identified with a transfer function F ( z ) = A + B z ( I − D z ) − 1 C , z ∈ D = { z ∈ C : | z | 1 } , of a conservative linear system. The complex matrices A , B , C , D define a block circulant unitary matrix. Our results show that there are no intrinsic differences between the elegant wavelet construction by Daubechies or any other construction of vector-valued or matrix-valued multi-wavelets. The structure of the unitary matrix defined by A , B , C , D allows us to parametrize in a systematic way all classes of possible wavelet and multi-wavelet masks together with the masks of the corresponding refinable functions.
- Published
- 2019
28. On the approximation of functions with line singularities by ridgelets
- Author
-
Axel Obermeier and Philipp Grohs
- Subjects
Numerical Analysis ,Computational complexity theory ,Advection ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Function (mathematics) ,Solver ,01 natural sciences ,Set (abstract data type) ,Line (geometry) ,Applied mathematics ,Gravitational singularity ,0101 mathematics ,Analysis ,Mathematics - Abstract
In Grohs and Obermeier (2015), the authors discussed the existence of numerically feasible solvers for advection equations that run in optimal computational complexity. In this paper, we complete the last remaining requirement to achieve this goal — by showing that ridgelets, on which the solver is based, approximate functions with line singularities (which may appear as solutions to the advection equation) with the best possible approximation rate (up to arbitrarily small δ > 0 ) if the functions decay sufficiently fast, and with a slight penalty if not. Structurally, the proof resembles Candes (2001), where a similar result was proved for a different ridgelet construction, which is however not well-suited for use in a PDE solver (and in particular, not suitable for the CDD-schemes (Cohen et al., 2001) we are interested in). Due to the differences between the two ridgelet constructions, we have to deal with quite a different set of issues, but are also able to relax the (support) conditions on the function being approximated. Finally, the proof employs a new convolution-type estimate that could be of independent interest due to its sharpness.
- Published
- 2019
29. A note on continuous sums of ridge functions
- Author
-
Vugar E. Ismailov, Aysel A. Asgarova, and Rashid A. Aliev
- Subjects
Numerical Analysis ,Multivariate statistics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Geometry ,010103 numerical & computational mathematics ,Function (mathematics) ,0101 mathematics ,Ridge (differential geometry) ,01 natural sciences ,Analysis ,Mathematics - Abstract
In this paper we prove that if a continuous multivariate function is represented by a sum of arbitrarily behaved ridge functions, then, under a suitable condition, it can be represented by a sum of continuous ridge functions.
- Published
- 2019
30. Nested sequences of rational spaces: Bernstein approximation, dimension elevation, and Pólya-type theorems on positive polynomials
- Author
-
Hermann Render, Rachid Ait-Haddou, Marie-Laurence Mazure, Cybermedia Center [Osaka] (CMC), Osaka University [Osaka], Calcul des Variations, Géométrie, Image (CVGI ), Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), and University College Dublin [Dublin] (UCD)
- Subjects
Numerical Analysis ,Pure mathematics ,Polynomial ,Sequence ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Extended Chebyshev spaces ,Context (language use) ,010103 numerical & computational mathematics ,Interval (mathematics) ,01 natural sciences ,Dimension elevation ,Approximation by rational functions ,Uniform norm ,P\'olya-type theorems ,Simple (abstract algebra) ,Bounded function ,Convergence (routing) ,Bernstein-type operators ,Positive polynomials ,[MATH]Mathematics [math] ,0101 mathematics ,Analysis ,Mathematics - Abstract
à paraître; International audience; On a given closed bounded interval, an infinite nested sequence of Extended Chebyshev spaces containing the constants automatically generates an infinite sequence of positive linear operators of Bernstein-type. Unlike the polynomial framework, this situation does not guarantee convergence of the corresponding approximation process. Obviously, convergence cannot be obtained without the density of the union of all the involved spaces in the set of continuous functions equipped with the uniform norm. The initial purpose of this work was to answer the following question: conversely, is density sufficient to guarantee convergence? Addressing this issue is all the more natural as density was indeed proved to imply convergence in the special case of nested sequences of Müntz spaces on positive intervals. In this paper we give a negative answer to the aforementioned question by considering nested sequences of rational spaces defined by infinite sequences of real poles outside the given interval. Surprisingly, in this rational context, we show that ensuring convergence is equivalent to determining all Pólya positive sequences, in the sense of all infinite sequences of positive numbers which guarantee Pólya-type results for the positivity of univariate polynomials on the non-negative axis. This interesting connection with Pólya positive sequences enables us to produce a simple necessary and sufficient condition for the poles to ensure convergence, thanks to results by Baker and Handelman on strongly positive sequences of polynomials.
- Published
- 2018
31. Elementary examples of solutions to Bochner’s problem for matrix differential operators
- Author
-
W. Riley Casper
- Subjects
Numerical Analysis ,Pure mathematics ,Sequence ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Eigenfunction ,Differential operator ,01 natural sciences ,Matrix (mathematics) ,Orthogonal matrix ,0101 mathematics ,Algebra over a field ,Analysis ,Monic polynomial ,Generating function (physics) ,Mathematics - Abstract
In this paper, we demonstrate an elementary method for constructing new solutions to Bochner’s problem for matrix differential operators from known solutions. We then describe a large family of solutions to Bochner’s problem, obtained from classical solutions, which include several examples known from the literature. By virtue of the method of construction, we show how one may explicitly identify a generating function for the associated sequence of monic w -orthogonal matrix polynomials { p ( x , n ) } , as well as the associated algebra D ( w ) of all matrix differential operators for which the { p ( x , n ) } are eigenfunctions.
- Published
- 2018
32. Approximation properties of univalent mappings on the unit ball in Cn
- Author
-
Mihai Iancu, Gabriela Kohr, Hidetaka Hamada, and Sebastian Schleißinger
- Subjects
Unit sphere ,Numerical Analysis ,Approximation property ,Applied Mathematics ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Regular polygon ,Automorphism ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Euclidean geometry ,0101 mathematics ,Representation (mathematics) ,Analysis ,Mathematics ,Loewner differential equation - Abstract
Let n ≥ 2 . In this paper, we obtain approximation properties of various families of normalized univalent mappings f on the Euclidean unit ball B n in C n by automorphisms of C n whose restrictions to B n have the same geometric property of f . First, we obtain approximation properties of spirallike, convex and g -starlike mappings f on B n by automorphisms of C n whose restrictions to B n have the same geometric property of f , respectively. Next, for a nonresonant operator A with m ( A ) > 0 , we obtain an approximation property of mappings which have A -parametric representation by automorphisms of C n whose restrictions to B n have A -parametric representation. Certain questions will be also mentioned. Finally, we obtain an approximation property by automorphisms of C n for a subset of S I n 0 ( B n ) consisting of mappings f which satisfy the condition ‖ D f ( z ) − I n ‖ 1 , z ∈ B n . Related results will be also obtained.
- Published
- 2018
33. Shape preserving properties of univariate Lototsky–Bernstein operators
- Author
-
Xiao-Wei Xu, Ron Goldman, and Xiao-Ming Zeng
- Subjects
Discrete mathematics ,Numerical Analysis ,Monomial ,Degree (graph theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Generating function ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Iterated function ,Order (group theory) ,Constant function ,0101 mathematics ,Convex function ,Analysis ,Mathematics - Abstract
The main goal of this paper is to study shape preserving properties of univariate Lototsky–Bernstein operators L n ( f ) based on Lototsky–Bernstein basis functions. The Lototsky–Bernstein basis functions b n , k ( x ) ( 0 ≤ k ≤ n ) of order n are constructed by replacing x in the i th factor of the generating function for the classical Bernstein basis functions of degree n by a continuous nondecreasing function p i ( x ) , where p i ( 0 ) = 0 and p i ( 1 ) = 1 for 1 ≤ i ≤ n . These operators L n ( f ) are positive linear operators that preserve constant functions, and a non-constant function γ n p ( x ) . If all the p i ( x ) are strictly increasing and strictly convex, then γ n p ( x ) is strictly increasing and strictly convex as well. Iterates L n M ( f ) of L n ( f ) are also considered. It is shown that L n M ( f ) converges to f ( 0 ) + ( f ( 1 ) − f ( 0 ) ) γ n p ( x ) as M → ∞ . Like classical Bernstein operators, these Lototsky–Bernstein operators enjoy many traditional shape preserving properties. For every ( 1 , γ n p ( x ) ) -convex function f ∈ C [ 0 , 1 ] , we have L n ( f ; x ) ≥ f ( x ) ; and by invoking the total positivity of the system { b n , k ( x ) } 0 ≤ k ≤ n , we show that if f is ( 1 , γ n p ( x ) ) -convex, then L n ( f ; x ) is also ( 1 , γ n p ( x ) ) -convex. Finally we show that if all the p i ( x ) are monomial functions, then for every ( 1 , γ n + 1 p ( x ) ) -convex function f , L n ( f ; x ) ≥ L n + 1 ( f ; x ) if and only if p 1 ( x ) = ⋯ = p n ( x ) = x .
- Published
- 2017
34. The optimal error bound for the method of simultaneous projections
- Author
-
Rafał Zalas and Simeon Reich
- Subjects
General Mathematics ,Context (language use) ,010103 numerical & computational mathematics ,01 natural sciences ,Projection (linear algebra) ,symbols.namesake ,Operator (computer programming) ,FOS: Mathematics ,Projection method ,Applied mathematics ,Product topology ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,Discrete mathematics ,Numerical Analysis ,Applied Mathematics ,010102 general mathematics ,Hilbert space ,Numerical Analysis (math.NA) ,41A25, 41A28, 41A44, 41A65 ,Linear subspace ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Optimization and Control (math.OC) ,symbols ,Affine transformation ,Analysis - Abstract
In this paper we find the optimal error bound (smallest possible estimate, independent of the starting point) for the linear convergence rate of the simultaneous projection method applied to closed linear subspaces in a real Hilbert space. We achieve this by computing the norm of an error operator which we also express in terms of the Friedrichs number. We compare our estimate with the optimal one provided for the alternating projection method by Kayalar and Weinert (1988). Moreover, we relate our result to the alternating projection formalization of Pierra (1984) in a product space. Finally, we adjust our results to closed affine subspaces and put them in context with recent dichotomy theorems., Accepted for publication in the Journal of Approximation Theory
- Published
- 2017
35. Approximation properties of combination of multivariate averages on Hardy spaces
- Author
-
Fayou Zhao and Dashan Fan
- Subjects
Numerical Analysis ,Multivariate statistics ,Pure mathematics ,Relation (database) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Extension (predicate logic) ,Hardy space ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Rate of approximation ,symbols ,0101 mathematics ,D'Alembert operator ,Analysis ,Mathematics - Abstract
In this paper, we study the rate of approximation of the combination of some generalized multivariate average on Hardy spaces and obtain its equivalent relation to the K -functionals. The result is an extension of a result in Dai and Ditzian (2004). We also extend and improve Theorem 6.2 in Belinsky et al. (2003).
- Published
- 2017
36. Approximation on variable exponent spaces by linear integral operators
- Author
-
Bo-Lu He, Bing-Zheng Li, and Ding-Xuan Zhou
- Subjects
Numerical Analysis ,Smoothness (probability theory) ,Euclidean space ,Applied Mathematics ,General Mathematics ,Open problem ,010102 general mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Function (mathematics) ,Operator theory ,01 natural sciences ,Fourier integral operator ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Exponent ,Applied mathematics ,0101 mathematics ,Linear combination ,Analysis ,Mathematics - Abstract
This paper aims at approximation of functions by linear integral operators on variable exponent spaces associated with a general exponent function on a domain of a Euclidean space. Under a log-Holder continuity assumption of the exponent function, we present quantitative estimates for the approximation and solve an open problem raised in our earlier work. As applications of our key estimates, we provide high orders of approximation by quasi-interpolation type and linear combinations of Bernstein type integral operators on variable exponent spaces. We also introduce K -functionals and moduli of smoothness on variable exponent spaces and discuss their relationships and applications.
- Published
- 2017
37. The growth of polynomials outside of a compact set—The Bernstein–Walsh inequality revisited
- Author
-
Klaus Schiefermayr
- Subjects
Kantorovich inequality ,Hölder's inequality ,Numerical Analysis ,Pure mathematics ,Bernoulli's inequality ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Ky Fan inequality ,MathematicsofComputing_NUMERICALANALYSIS ,010103 numerical & computational mathematics ,Minkowski inequality ,01 natural sciences ,Algebra ,Linear inequality ,Log sum inequality ,Rearrangement inequality ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this paper, we present a new and simple proof of the classical Bernstein–Walsh inequality. Based on this proof, we give some improvements for this inequality in the case that the corresponding compact set is real.
- Published
- 2017
38. Spaceability and strong divergence of the Shannon sampling series and applications
- Author
-
Ezra Tampubolon, Ullrich J. Monich, and Holger Boche
- Subjects
Bandlimiting ,Numerical Analysis ,Class (set theory) ,Integrable system ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Structure (category theory) ,020206 networking & telecommunications ,02 engineering and technology ,Space (mathematics) ,01 natural sciences ,symbols.namesake ,Fourier transform ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,0101 mathematics ,Divergence (statistics) ,Analysis ,Subspace topology ,Mathematics - Abstract
In this paper the structure of the set of functions for which the peak value of the Shannon sampling series is strongly divergent is analyzed. Strong divergence is closely linked to the non-existence of adaptive reconstruction methods. Signals in the Paley–Wiener space PW π 1 of bandlimited functions with absolutely integrable Fourier transform are considered, and it is shown that the set of strong divergence is spaceable and dense-lineable, i.e., that there exist an infinite dimensional closed subspace and an infinite dimensional dense subspace, such that we have strong divergence of the peak value of the Shannon sampling series for all functions from these sets, except the zero function. Further, it is proved that this result is not restricted to the Shannon sampling series, but rather holds for an entire class of reconstruction processes.
- Published
- 2017
39. Kolmogorov widths of weighted Sobolev classes on a multi-dimensional domain with conditions on the derivatives of order r and zero
- Author
-
A. A. Vasil’eva
- Subjects
Mathematics::Functional Analysis ,Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Domain (mathematical analysis) ,Sobolev space ,Norm (mathematics) ,Multi dimensional ,Order (group theory) ,Standard probability space ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this paper order estimates of the Kolmogorov widths for some weighted Sobolev classes in a weighted Lebesgue space are obtained. The weighted Sobolev classes are defined by a restriction on the weighted L p 1 -norm of the highest order derivative and the weighted L p 0 -norm of the function.
- Published
- 2021
40. Sharp Lp Bernstein type inequality for cuspidal domains in Rd
- Author
-
András Kroó
- Subjects
Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Boundary (topology) ,010103 numerical & computational mathematics ,Type inequality ,01 natural sciences ,Measure (mathematics) ,Domain (ring theory) ,Graph (abstract data type) ,0101 mathematics ,Mathematics::Representation Theory ,Analysis ,Mathematics ,Algebraic polynomial - Abstract
In this paper we verify a sharp Bernstein type inequality for multivariate algebraic polynomials in L p norm on general cuspidal domains which is based on a proper measure of the distance to the boundary of the domain. In particular, in case of Lip γ cuspidal graph domains the exact rate is given by n 2 γ − 1 .
- Published
- 2021
41. Estimations of singular functions of kernel cross-covariance operators
- Author
-
Yao Zhao, Heng Chen, and Di-Rong Chen
- Subjects
Numerical Analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,Estimator ,010103 numerical & computational mathematics ,Covariance ,01 natural sciences ,symbols.namesake ,Operator (computer programming) ,Kernel method ,Singular function ,Kernel (statistics) ,symbols ,Applied mathematics ,Cross-covariance ,0101 mathematics ,Analysis ,Mathematics - Abstract
The constrained covariance (COCO) has been proposed for measuring dependence between random vectors. Kernel cross-covariance operators on reproducing kernel Hilbert spaces, as one of kernel methods which could extract nonlinear dependence, have attracted considerable attention. This paper establishes learning rates of some estimators associated with kernel cross-covariance. For kernel cross-covariance operators, we bound a weighted summation of squared estimation errors of empirical singular functions by 16 times of the estimation error of empirical cross-covariance. Our method actually applies in general setting, so that a new bound is obtained for perturbation of singular functions of Hilbert–Schmidt operators. It is much tighter than the classical result as the latter only bounds each error of singular function individually. This is interest in its own right. For normalized cross-covariance operator, we propose an estimator and obtain a learning rate.
- Published
- 2021
42. Norm estimates for Chebyshev polynomials, I
- Author
-
Klaus Schiefermayr and Maxim Zinchenko
- Subjects
Numerical Analysis ,Chebyshev polynomials ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Monotonic function ,010103 numerical & computational mathematics ,01 natural sciences ,Infimum and supremum ,Upper and lower bounds ,Combinatorics ,Limit (mathematics) ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this paper, we extend the sharp upper bound of Christiansen et al. (2017) and the sharp lower bound of Schiefermayr (2008) to the case of weighted Chebyshev polynomials on subsets of [ − 1 , 1 ] for the weight w ( x ) = 1 − x 2 . We then analyse the norm of Chebyshev polynomials on a circular arc, prove monotonicity of the corresponding Widom factors, find exact values of their supremum and infimum, and obtain a new proof for their limit.
- Published
- 2021
43. A note on K-functional, Modulus of smoothness, Jackson theorem and Bernstein–Nikolskii–Stechkin inequality on Damek–Ricci spaces
- Author
-
Vishvesh Kumar and Michael Ruzhansky
- Subjects
Numerical Analysis ,Pure mathematics ,Inequality ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,Operator (physics) ,010102 general mathematics ,Spectrum (functional analysis) ,010103 numerical & computational mathematics ,Space (mathematics) ,01 natural sciences ,Spherical mean ,symbols.namesake ,Fourier transform ,Bounded function ,symbols ,Mathematics::Differential Geometry ,0101 mathematics ,Equivalence (measure theory) ,Analysis ,media_common ,Mathematics - Abstract
In this paper we study approximation theorems for L 2 -space on Damek–Ricci spaces. We prove direct Jackson theorem of approximations for the modulus of smoothness defined using spherical mean operator on Damek–Ricci spaces. We also prove Bernstein–Nikolskii–Stechkin inequality. To prove these inequalities we use functions of bounded spectrum as a tool of approximation. Finally, as an application we prove equivalence of the K -functional and modulus of smoothness for Damek–Ricci spaces.
- Published
- 2021
44. An intersection representation for a class of anisotropic vector-valued function spaces
- Author
-
Nick Lindemulder
- Subjects
Mathematics::Functional Analysis ,Numerical Analysis ,Pure mathematics ,Class (set theory) ,Function space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,010103 numerical & computational mathematics ,01 natural sciences ,Intersection ,Fubini's theorem ,Maximal function ,Boundary value problem ,0101 mathematics ,Representation (mathematics) ,Vector-valued function ,Analysis ,Mathematics - Abstract
The main result of this paper is an intersection representation for a class of anisotropic vector-valued function spaces in an axiomatic setting a la Hedberg and Netrusov (2007), which includes weighted anisotropic mixed-norm Besov and Lizorkin–Triebel spaces. In the special case of the classical Lizorkin–Triebel spaces, the intersection representation gives an improvement of the well-known Fubini property. The main result has applications in the weighted L q - L p -maximal regularity problem for parabolic boundary value problems, where weighted anisotropic mixed-norm Lizorkin–Triebel spaces occur as spaces of boundary data.
- Published
- 2021
45. s-numbers of compact embeddings of some sequence and function spaces
- Author
-
Alicja Dota
- Subjects
Discrete mathematics ,Mathematics::Functional Analysis ,Numerical Analysis ,Sequence ,Function space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,010103 numerical & computational mathematics ,01 natural sciences ,Sobolev space ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this paper we prove asymptotic formulas for the behavior of approximation, Gelfand, Kolmogorov and Weyl numbers of embeddings between some weighted sequence spaces. The formulas are applied to the estimates of asymptotic behavior of these s -numbers of Sobolev embeddings between Besov and Triebel–Lizorkin spaces defined on quasi-bounded domains.
- Published
- 2017
46. On the probability of positive-definiteness in the gGUE via semi-classical Laguerre polynomials
- Author
-
Alfredo Deaño and Nick Simm
- Subjects
Pure mathematics ,Laguerre's method ,General Mathematics ,Gaussian ,FOS: Physical sciences ,Positive-definite matrix ,01 natural sciences ,Combinatorics ,Classical orthogonal polynomials ,Matrix (mathematics) ,symbols.namesake ,0103 physical sciences ,QA351 ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,QA299 ,0101 mathematics ,Mathematical Physics ,60B20, 33C45, 34E05 ,Mathematics ,Numerical Analysis ,010308 nuclear & particles physics ,Applied Mathematics ,010102 general mathematics ,Mathematical Physics (math-ph) ,Positive definiteness ,Mathematics - Classical Analysis and ODEs ,Laguerre polynomials ,symbols ,Gradient descent ,Analysis - Abstract
In this paper, we compute the probability that an $N \times N$ matrix from the generalised Gaussian Unitary Ensemble (gGUE) is positive definite, extending a previous result of Dean and Majumdar \cite{DM}. For this purpose, we work out the large degree asymptotics of semi-classical Laguerre polynomials and their recurrence coefficients, using the steepest descent analysis of the corresponding Riemann--Hilbert problem., 21 pages, 1 figure. Revised version, minor changes and references added
- Published
- 2017
47. Error bounds for multiquadrics without added constants
- Author
-
Martin D. Buhmann and Oleg Davydov
- Subjects
Numerical Analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Expression (computer science) ,01 natural sciences ,Pontryagin's minimum principle ,Radial function ,Radial basis function ,Pairwise comparison ,0101 mathematics ,Constant (mathematics) ,Analysis ,Mathematics ,Interpolation - Abstract
While it was noted by R. Hardy and proved in a famous paper by C. A. Micchelli that radial basis function interpolants s(x)=j(xxj) exist uniquely for the multiquadric radial function (r)=r2+c2 as soon as the (at least two) centres are pairwise distinct, the error bounds for this interpolation problem always demanded an added constant to s. By using Pontryagin native spaces, we obtain error bounds that no longer require this additional constant expression.
- Published
- 2017
48. Representations of hypergeometric functions for arbitrary parameter values and their use
- Author
-
Dmitrii Karp and José L. López
- Subjects
Numerical Analysis ,Factorial ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Gauss ,010103 numerical & computational mathematics ,Function (mathematics) ,Positive-definite matrix ,Generalized hypergeometric function ,01 natural sciences ,symbols.namesake ,symbols ,0101 mathematics ,Hypergeometric function ,Series expansion ,Analysis ,Bessel function ,Mathematics - Abstract
Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and show that the extended representations can be interpreted as examples of regularizations of integrals containing Meijer’s G function. Second, we give new applications of both, known and extended representations. These include: inverse factorial series expansion for the Gauss type function, new information about zeros of the Bessel and Kummer type functions, connection with radial positive definite functions and generalizations of Luke’s inequalities for the Kummer and Gauss type functions.
- Published
- 2017
49. Nonlinear piecewise polynomial approximation and multivariate BV spaces of a Wiener–L. Young type. I
- Author
-
Yu. A. Brudnyi
- Subjects
Discrete mathematics ,Numerical Analysis ,Smoothness (probability theory) ,Degree (graph theory) ,Applied Mathematics ,General Mathematics ,Dyadic cubes ,010102 general mathematics ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Sobolev space ,Order (group theory) ,Interpolation space ,Birnbaum–Orlicz space ,0101 mathematics ,Analysis ,Mathematics - Abstract
The named space denoted by V p q k consists of L q functions on [ 0 , 1 ) d of bounded p -variation of order k ∈ N . It generalizes the classical spaces V p ( 0 , 1 ) ( = V p ∞ 1 ) and B V ( = [ 0 , 1 ) d ) ( V 1 q 1 where q ≔ d d − 1 ) and is closely related to several important smoothness spaces, e.g., to Sobolev spaces over L p , B V and B M O and to Besov spaces. The main approximation result concerns the space V p q k of smoothness s ≔ d 1 p − 1 q ∈ ( 0 , k ] . It asserts the following: Let f ∈ V p q k be of smoothness s ∈ ( 0 , k ] , 1 ≤ p q ∞ and N ∈ N . There exist a family Δ N of N dyadic subcubes of [ 0 , 1 ) d and a piecewise polynomial g N over Δ N of degree k − 1 such that ‖ f − g N ‖ q ⩽ C N − s ∕ d | f | V p q k . This implies similar results for the above mentioned smoothness spaces, in particular, solves the going back to the 1967 Birman–Solomyak paper (Birman and Solomyak, 1967) problem of approximation of functions from W p k ( [ 0 , 1 ) d ) in L q ( [ 0 , 1 ) d ) whenever k d = 1 p − 1 q and q ∞ .
- Published
- 2017
50. Needlet approximation for isotropic random fields on the sphere
- Author
-
Quoc Thong Le Gia, Ian H. Sloan, Robert S. Womersley, and Yu Guang Wang
- Subjects
Pointwise ,Statistics::Theory ,Numerical Analysis ,Random field ,Applied Mathematics ,General Mathematics ,Gaussian ,010102 general mathematics ,Isotropy ,Mathematical analysis ,Mathematics::Classical Analysis and ODEs ,Astrophysics::Cosmology and Extragalactic Astrophysics ,010103 numerical & computational mathematics ,01 natural sciences ,symbols.namesake ,Wavelet ,Approximation error ,Convergence (routing) ,symbols ,0101 mathematics ,Fourier series ,Analysis ,Mathematics - Abstract
In this paper we establish a multiscale approximation for random fields on the sphere using spherical needlets—a class of spherical wavelets. We prove that the semidiscrete needlet decomposition converges in mean and pointwise senses for weakly isotropic random fields on S d , d ≥ 2 . For numerical implementation, we construct a fully discrete needlet approximation of a smooth 2 -weakly isotropic random field on S d and prove that the approximation error for fully discrete needlets has the same convergence order as that for semidiscrete needlets. Numerical examples are carried out for fully discrete needlet approximations of Gaussian random fields and compared to a discrete version of the truncated Fourier expansion.
- Published
- 2017
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