Jackson-type inequality on the sphere

In a recent paper, I introduced new moduli of smoothness for functions on the sphere which did not use averages and, as a result, had some interesting properties. The direct, Jackson-type, estimate of the best approximation by spherical harmonics using the new moduli will be proved here. Equivalence with the appropriate K-functionals will be given. Relations with the moduli used earlier will be shown and used to prove new results for these moduli.     

Approximation by polynomials in Bergman spaces of slice regular functions in the unit ball

In this paper, we show that the set of quaternionic polynomials is dense in the Bergman spaces of slice regular functions in the unit ball, both of the first and of the second kind. Several proofs are presented, including constructive methods based on the Taylor expansion and on the convolution polynomials. In the last case, quantitative estimates in terms of higher‐order moduli of smoothness and of best approximation quantity are obtained.     

On approximation of functions by algebraic polynomials in Hölder spaces

We study approximation of functions by algebraic polynomials in the Hölder spaces corresponding to the generalized Jacobi translation and the Ditzian–Totik moduli of smoothness. By using modifications of the classical moduli of smoothness, we give improvements of the direct and inverse theorems of approximation and prove the criteria of the precise order of decrease of the best approximation in these spaces. Moreover, we obtain strong converse inequalities for some methods of approximation of functions. As an example, we consider approximation by the Durrmeyer–Bernstein polynomial operators.     

Jackson inequality for Banach spaces on the sphere

The best rate of approximation of functions on the sphere by spherical polynomials is majorized by recently introduced moduli of smoothness. The treatment applies to a wide class of Banach spaces of functions.      

Polynomial approximation in Sobolev spaces on the unit sphere and the unit ball

This work is a continuation of the recent study by the authors on approximation theory over the sphere and the ball. The main results define new Sobolev spaces on these domains and study polynomial approximations for functions in these spaces, including simultaneous approximation by polynomials and the relation between the best approximation of a function and its derivatives.     

A modulus of smoothness on the unit sphere

For functions onS ^d−1 (the unit sphere inR ^d) and, in particular, forf∈L _p(S ^d−1), we define new simple moduli of smoothness. We relate different orders of these moduli, and we also relate these moduli to best approximation by spherical harmonics of order smaller thann. Our new moduli lead to sharper results than those now available for the known moduli onL _p(S ^d−1). Supported by NSERC Grant A4816 of Canada.     

On a Sequence of Positive Linear Operators Associated with a Continuous Selection of Borel Measures

In this paper we introduce and study a new sequence of positive linear operators acting on the space of Lebesgue-integrable functions on the unit interval. These operators are defined by means of continuous selections of Borel measures and generalize the Kantorovich operators. We investigate their approximation properties by presenting several estimates of the rate of convergence by means of suitable moduli of smoothness. Some shape preserving properties are also shown. Dedicated to the memory of Professor Aldo Cossu     

New Moduli of Smoothness on the Unit Ball and Other Domains,Introduction and Main Properties

A new set of moduli of smoothness on a large variety of Banach spaces of functions on the unit ball is introduced. These measures of smoothness utilize uniformly bounded holomorphic semigroups on the Banach space in question. The new moduli are “correct” in the sense that they satisfy direct (Jackson) and weak converse inequalities. The method used also applies to spaces of functions on the simplex and the unit sphere, and while the main goal is the investigation of properties and relations concerning the unit ball, many of the results will be given for other domains and situations. The classic properties, including equivalence with appropriate \(K\) -functionals or realization functionals, will be established. Bernstein- and Kolmogorov-type inequalities are proved.     

Evaluating polynomials over the unit disk and the unit ball

We investigate the use of orthonormal polynomials over the unit disk ??² in ?² and the unit ball ??³ in ?³. An efficient evaluation of an orthonormal polynomial basis is given, and it is used in evaluating general polynomials over ??² and ??³. The least squares approximation of a function f on the unit disk by polynomials of a given degree is investigated, including how to write a polynomial using the orthonormal basis. Matlab codes are given.     

Polynomial approximation with an exponential weight on the real semiaxis

We consider the polynomial approximation on (0,+∞), with the weight $u(x)= x^{\gamma}e^{-x^{-\alpha}-x^{\beta}}$ , α>0, β>1 and γ≧0. We introduce new moduli of smoothness and related K-functionals for functions defined on the real semiaxis, which can grow exponentially both at 0 and at +∞. Then we prove the Jackson theorem, also in its weaker form, and the Stechkin inequality. Moreover, we study the behavior of the derivatives of polynomials of best approximation.     

Smoothness and Best Approximation with Respect to Freud-Type Weights,a New Approach

A new measure of smoothness is defined and related to best approximation by polynomials with respect to weighted L _p (R) with Freud-type weights. Other related norms are also discussed. Comparisons with the known measure of smoothness on weighted L _p spaces are obtained. Related K-functionals and realization functionals are introduced. The new measure of smoothness allows us to consider a more general class of function spaces, to achieve Marchaud, Jackson and Bernstein-type inequalities, and to relate it to expressions involving the coefficients of the expansion by orthogonal polynomials with respect to Freud-type weights. Some of the results are new for approximation by Hermite polynomials in the weighted L _p space with the weight \({e^{-x^{2}}}\) .     

球面带形平移网络逼近的Jackson定理

研究了球面带型平移网络逼近阶用球面调和多项式的最佳逼近及光滑模的刻画问题．借助于球调和多项式的最佳逼近多项式和Riesz平均构造出了单位球面Sq上的带形平移网络,并建立了球面带形平移网络对Lp(Sq)中函数一致逼近的Jackson型定理．所得结果表明球面带形平移网络可以达到球调和多项式的逼近阶．     

球面上Peetre K－模和最佳逼近

本文研究了球面上三种ＰｅｅｔｒｅＫ－模与最佳逼近的关系，建立起它们之间的若干强型和弱型不等式．此外，还讨论了Ｋ－模与光滑模的等价性．     

Padé approximation and Apostol-Bernoulli and Apostol-Euler polynomials

Using the Padé approximation of the exponential function, we obtain recurrence relations between Apostol-Bernoulli and between Apostol-Euler polynomials. As applications, we derive some new lacunary recurrence relations for Bernoulli and Euler polynomials with gap of length 4 and lacunary relations for Bernoulli and Euler numbers with gap of length 6.     

球面S~(d-1)上的广义Ul'yanov型不等式

本文对于单位球面上的经典连续模,给出了一个非常有用的广义Ul'yanov型不等式.该不等式在球面多项式逼近、球面嵌入理论以及球面上函数空间的插值理论等领域有着非常重要的应用.我们的证明基于球面调和多项式展开的新的估计,这些估计本身也具有独立的意义.     

球面Jackson多项式逼近的正逆定理

研究了球面Jackson多项式J_(v,s)f的逼近阶,建立了该多项式逼近的强型正向与逆向不等式.利用球面光滑模较好地刻画了Jackson多项式的逼近性能,证明了存在与v和f无关的常数C_1和C_2,使得对于定义在球面上任意p-幂勒贝格可积或连续函数f成立C_1ω(f,1/v)_p≤‖J_(v,s)f-f‖_p≤C_2ω(f,1/v)_p,其中ω(f,t)p是f的光滑模.     

Weak Type Inequalities for Best Simultaneous Approximation in Banach Spaces

For arbitrary Banach spaces Butzer and Scherer in 1968 showed that the approximation order of best approximation can characterized by the order of certain K-functionals. This general theorem has many applications such as the characterization of the best approximation of algebraic polynomials by moduli of smoothness involving the Legendre, Chebyshev, or more general the Jacobi transform. In this paper we introduce a family of seminorms on the underlying approximation space which leads to a generalization of the Butzer–Scherer theorems. Now the characterization of the weighted best algebraic approximation in terms of the so-called main part modulus of Ditzian and Totik is included in our frame as another particular application. The goal of the paper is to show that for the characterization of the orders of best approximation, simultaneous approximation (in different spaces), reduction theorems, and K-functionals one has (essentially) only to verify three types of inequalities, namely inequalities of Jackson-, Bernstein-type and an equivalence condition which guarantees the equivalence of the seminorm and the underlying norm on certain subspaces. All the results are given in weak-type estimates for almost arbitrary approximation orders, the proofs use only functional analytic methods.     

The uniform norm of hyperinterpolation on the unit sphere in an arbitrary number of dimensions

In this paper we study the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere S ^r−1 ⊂ R^r. The hyperinterpolation approximation L _n ƒ, where ƒ ∈ C(S ^r −1), is derived from the exact L ² orthogonal projection Π ƒ onto the space P _n ^r (S ^r −1) of spherical polynomials of degree n or less, with the Fourier coefficients approximated by a positive weight quadrature rule that integrates exactly all polynomials of degree ≤ 2n. We extend to arbitrary r the recent r = 3 result of Sloan and Womersley [9], by proving that under an additional “quadrature regularity” assumption on the quadrature rule, the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere is O(n ^r /2−1), which is the same as that of the orthogonal projection Π_n, and best possible among all linear projections onto P _n ^r (S ^r −1).     

Approximation inL 2 Sobolev spaces on the 2-sphere by quasi-interpolation

In this article we consider a simple method of radial quasi-interpolation by polynomials on the unit sphere in ℝ³, and present rates of covergence for this method in Sobolev spaces of square integrable functions. We write the discrete Fourier series as a quasi-interpolant and hence obtain convergence rates, in the aforementioned Sobolev spaces, for the discrete Fourier projection. We also discuss some typical practical examples used in the context of spherical wavelets.