Jackson-type Theorems Research Articles

Jackson-Type Theorem on Approximation by Algebraic Polynomials in the Uniform Metric with Laguerre Weight

Jackson-Type Theorem on Approximation by Algebraic Polynomials in the Uniform Metric with Laguerre Weight

Теорема типа Джексона о приближении алгебраическими многочленами в равномерной метрике с весом Лагерра

И. И. Шарапудинов при исследовании аппроксимативных свойств частичных сумм специального ряда по полиномам Лагерра ввел взвешенную величину наилучшего приближения $E_n(f,u_r)$, зависящую от параметра $r$. В настоящей работе для этой величины при $r=1$ доказана теорема типа Джексона. Библиография: 12 названий.

Jackson-type theorem in the weak $L_{1}$-space

The weak $L_{1}$-space meets in many areas of mathematics. For example, the conjugate functions of Lebesgue integrable functions belong to the weak $L_{1}$-space. The difficulty of working with the weak $L_{1}$-space is that the weak $L_{1}$-space is not a normed space. Moreover, infinitely differentiable (even continuous) functions are not dense in this space. Due to this, the theory of approximation was not produced in this space. In the present paper, we introduced the concept of the modulus of continuity of the functions from the weak $L_{1}$-space, studied its properties, found a criterion for convergence to zero of the modulus of continuity of the function from the weak $L_{1}$-space, and proved in this space an analogue of the Jackson-type theorem.

Jackson-type theorem on approximation by non-stationary periodic wavelets

In this paper, sufficient conditions on non-stationary periodic wavelet systems to provide good approximation properties of wavelet expansions are established. The approximation error is estimated in terms of modulus of continuity in [Formula: see text]-spaces.

Spherical Approximation on Unit Sphere

In this paper we introduce a Jackson type theorem for functions in LP spaces on sphere And study on best approximation of functions in spaces defined on unit sphere.
 our central problem is to describe the approximation behavior of functions in spaces for by modulus of smoothness of functions.

Approximation by network operators with logistic activation functions

This paper aims to study the construction and multivariate approximation of a class of network operators with logistic sigmoidal functions. First, a class of even and bell-shaped function with support on R is constructed by using appropriate translation and combination of the logistic function. Then, the constructed function is employed as activation function to construct a kind of so-called Cardaliaguet–Euvrard type network operators. Finally, these network operators are used to approximate bivariate functions in C[-1,1]2, and a Jackson type theorem for the approximation errors is established.

Scattered data quasi‐interpolation on spheres

This paper studies the construction and approximation of quasi‐interpolation for spherical scattered data. First of all, a kind of quasi‐interpolation operator with Gaussian kernel is constructed to approximate the spherical function, and two Jackson type theorems are established. Second, the classical Shepard operator is extended from Euclidean space to the unit sphere, and the error of approximation by the spherical Shepard operator is estimated. Finally, the compact supported kernel is used to construct quasi‐interpolation operator for fitting spherical scattered data, where the spherical modulus of continuity and separation distance of scattered sampling points are employed as the measurements of approximation error. Copyright © 2014 John Wiley & Sons, Ltd.

Exact order of approximation of periodic functions by one nonclassical method of summation of Fourier series

By using an exact estimate for approximation by known trigonometric polynomials, we strengthen a Jackson-type theorem. Moreover, we determine the exact order of approximation of some periodic functions by these polynomials. For this purpose, we introduce a special modulus of smoothness.

Complex rational type interpolation at disturbed fejér points on a curve

The results of accurate order of uniform approximation and simultaneous approximation in the early work “Jackson Type Theorems on Complex Curves” are improved from Fejér points to disturbed Fejér points in this article. Furthermore, the stability of convergence of Tn,∈*(f,z) with disturbed sample values f(zk*)+∈k are also proved in this article.

Learning rates of least-square regularized regression with strongly mixing observation

This paper considers the regularized learning algorithm associated with the least-square loss, strongly mixing observations and reproducing kernel Hilbert spaces. We first give the bound of the sample error with exponentially strongly mixing observations and the rate of approximation by Jackson-type theorem of approximation based on exponentially strongly mixing sequence. Then the generalization error of the least-square regularized regression is obtained by estimating sample error and regularization error.

The estimate for approximation error of spherical neural networks

Compared with planar hyperplane, fitting data on the sphere has been an important and an active issue in geoscience, metrology, brain imaging, and so on. In this paper, with the help of the Jackson-type theorem of polynomial approximation on the sphere, we construct spherical feed-forward neural networks to approximate the continuous function defined on the sphere. As a metric, the modulus of smoothness of spherical function is used to measure the error of the approximation, and a Jackson-type theorem on the approximation is established. Copyright © 2011 John Wiley & Sons, Ltd.

Jackson type theorems on complex curves

Let Γ be a closed smooth Jordan curve in the complex plane. In this paper, with the help of a class of fundamental functions of Hermite interpolation, the author introduces a continuous function interpolation which uniformly approximates to f(z) e C(Γ) with the same order of approximation as that in Jackson Theorem 1 on real interval [−1, 1]. The accuracy of the order of approximation is proved. Using the method different from the early works, the author studies simultaneous approximation to function and its derivatives and the desired results analogues to that in Jackson Theorem 2 on real interval [−1, 1] are obtained.

Jackson-type theorems in homogeneous approximation

The classical Weierstrass theorem states that any function continuous on a compact set K ⊂ R d ( d ≥ 1 ) can be uniformly approximated by algebraic polynomials. In this paper we study the possible extensions of this celebrated result for approximation by homogeneous algebraic polynomials on star-like and convex surfaces in R d such that K = - K . A previous conjecture states that functions continuous on a convex surface can be approximated by a pair of homogeneous polynomials. We verify this conjecture under the mild condition of Dini–Lipschitz smoothness of the convex surface considered. It is also shown that the density fails on surfaces with outer cusps. In addition, we give Jackson-type estimates for the rate of best approximation by homogeneous polynomials on convex surfaces. A new phenomenon here consists of the fact that this rate depends not only on the moduli of continuity of the functions considered, but also on the smoothness properties of convex surfaces.

Continuous wavelet transform and wavelet frames on the sphere using Clifford analysis

In this paper we construct a continuous wavelet transform (CWT) on the sphere $S^{n-1}$ based on the conformal group of the sphere, the Lorentz group Spin$(1,n)$. For this purpose, we present a short survey on the existing techniques of continuous wavelet transform and of conformal transformations on the unit sphere. We decompose the conformal group into the maximal compact subgroup of rotations Spin$(n)$ and the set of Mobius transformations of the form $\varphi_a(x) = (x-a)(1+ax)^{-1}$, where $a \in B^n$ and $B^n$ denotes the unit ball in $\mathbb R^n$. Based on a study of the influence of the parameter $a$ arising in the definition of dilations/contractions on the sphere we define a class of local conformal dilation operators and consequently a family of continuous wavelet transforms for the Hilbert space of square integrable functions on the sphere $L_2(S^{n-1})$ and the Hardy space $H^2$. In the end we construct Banach frames for our wavelets and prove Jackson-type theorems for the best $n$-point approximation.

Monogenic frames for an integral transform on the unit sphere

In this article we construct Banach frames of monogenic functions and prove Jackson-type theorems for the best n-point approximation.

Jackson-type theorems on some transcendental curves in [formula omitted

“Efficient curves” in the sense of the best rate of multivariate polynomial approximation to contractive functions on these curves, were first introduced by D.J. Newman and L. Raymon in 1969. They proved that algebraic curves are efficient, but claimed that the exponential curve γ : = { ( t , e t ) : 0 ⩽ t ⩽ 1 } is not. We prove to the contrary that this exponential curve and its generalization to higher dimensions are indeed efficient. We also investigate helical curves in R d and show that they too are efficient. Transcendental curves of the form { ( t , t λ ) : δ ⩽ t ⩽ 1 } are shown to be efficient for δ > 0 , contradicting another claim of Newman and Raymon.

Modulus of smoothness and theorems concerning approximation on compact groups

We consider the generalized shift operator defined by (Shuf)(g) = ∫Gf(tut−1g)dt on a compact group G, and by using this operator, we define spherical modulus of smoothness. So, we prove Stechkin and Jackson‐type theorems.

Jackson’s theorem for compact groups

In this article we consider the generalized shift operator defined by $$(Sh_\alpha f)(g) = \int_G {f(tut^{ - 1} g)dt} $$ on compact group G and by help of this operator we define “Spherical” modulus of continuity. So we prove Stechkin and Jackson type theorems.

Approximation with interpolatory constraints

Given a triangular array of points on [ − 1 , 1 ] [-1,1] satisfying certain minimal separation conditions, a classical theorem of Szabados asserts the existence of polynomial operators that provide interpolation at these points as well as a near-optimal degree of approximation for arbitrary continuous functions on the interval. This paper provides a simple, functional-analytic proof of this fact. This abstract technique also leads to similar results in general situations where an analogue of the classical Jackson-type theorem holds. In particular, it allows one to obtain simultaneous interpolation and a near-optimal degree of approximation by neural networks on a cube, radial-basis functions on a torus, and Gaussian networks on Euclidean space. These ideas are illustrated by a discussion of simultaneous approximation and interpolation by polynomials and also by zonal-function networks on the unit sphere in Euclidean space.

JACKSON TYPE THEOREMS FOR COMPACT RANK 1 SYMMETRIC SPACES

Since recently various problems of approximation theory have been actively studied for functions on the n-dimensional sphere Sn (see [1–3] and the bibliography therein). A broader but natural class of spaces for which we can pose such problems is that of all compact rank 1 symmetric spaces (CROSP in the terminology of the monograph [4]). Some results are already available for these spaces (see [5–9]); however, the main problems still remain open. This article is devoted to proving the direct Jackson-type theorems for an arbitrary CROSP M . The complete classification of all CROSPs is well known. It comprises the four series: the spheres Sn (n = 1, 2, . . . ) and the real, complex, and quaternionic projective spaces (Pn(R), n = 2, 3, . . . ; Pn(C), n = 4, 6, . . . ; and Pn(H), n = 8, 12, 16, . . . ) (everywhere the superscript n denotes the dimension of the space) and one special space, the Cayley elliptic plane P 16(Cay). Since the problems of harmonic analysis on Pn(R) are easily reduced to the corresponding problems on Sn, we assume that M 6= Pn(R). A CROSP M is always a Riemannian manifold. Let ∆ be the Laplace–Beltrami operator on M . The spectrum of ∆ is discrete, real, and nonpositive. We arrange it in decreasing order (0 = λ0 > λ1 > λ2 > . . . ) and denote the eigensubspace of ∆ corresponding to the eigenvalue λk by Hk (it is always finite-dimensional). Put PN (M) := H0 + H1 + · · · + HN . The functions in PN (M) are referred to as spherical polynomials on M of degree m (for M = Sn they coincide with the ordinary spherical polynomials). Given a set X with a measure dσ, we as usual denote by Lp(X, dσ) the Banach space of complexvalued measurable functions f(x) on X with the finite norm