The polynomial approximation of continuous functions defined on $(-\infty ,\infty )$

Leetsch C. Hsu

Displaying similar documents to “The polynomial approximation of continuous functions defined on $(- \infty, \infty)$ ”

$L_{\infty}$ -Convergence of Finite Element Approximation

J. A. Nitsche (1975)

Publications mathématiques et informatique de Rennes

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Lebesgue type points in strong (C,α) approximation of Fourier series

Włodzimierz Łenski, Bogdan Roszak (2011)

Banach Center Publications

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We present an estimation of the $H_{k ₀, k_{r}}^{q, α} f$ and $H_{λ, u}^{ϕ, α} f$ means as approximation versions of the Totik type generalization (see [5], [6]) of the result of G. H. Hardy, J. E. Littlewood. Some corollaries on the norm approximation are also given.

Some duality results on bounded approximation properties of pairs

Eve Oja, Silja Treialt (2013)

Studia Mathematica

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The main result is as follows. Let X be a Banach space and let Y be a closed subspace of X. Assume that the pair $(X *, Y^{⊥})$ has the λ-bounded approximation property. Then there exists a net $(S_{α})$ of finite-rank operators on X such that $S_{α} (Y) \subset Y$ and $| | S_{α} | | \leq λ$ for all α, and $(S_{α})$ and $(S *_{α})$ converge pointwise to the identity operators on X and X*, respectively. This means that the pair (X,Y) has the λ-bounded duality approximation property.

On subextension and approximation of plurisubharmonic functions with given boundary values

Hichame Amal (2014)

Annales Polonici Mathematici

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Our aim in this article is the study of subextension and approximation of plurisubharmonic functions in $_{χ} (Ω, H)$ , the class of functions with finite χ-energy and given boundary values. We show that, under certain conditions, one can approximate any function in $_{χ} (Ω, H)$ by an increasing sequence of plurisubharmonic functions defined on strictly larger domains.

Routh-type $L_{2}$ model reduction revisited

Wiesław Krajewski, Umberto Viaro (2018)

Kybernetika

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A computationally simple method for generating reduced-order models that minimise the $L_{2}$ norm of the approximation error while preserving a number of second-order information indices as well as the steady-state value of the step response, is presented. The method exploits the energy-conservation property peculiar to the Routh reduction method and the interpolation property of the $L_{2}$ -optimal approximation. Two examples taken from the relevant literature show that the suggested techniques...

On the approximation of real continuous functions by series of solutions of a single system of partial differential equations

Carsten Elsner (2006)

Colloquium Mathematicae

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We prove the existence of an effectively computable integer polynomial P(x,t₀,...,t₅) having the following property. Every continuous function $f : ℝ^{s} \to ℝ$ can be approximated with arbitrary accuracy by an infinite sum $\sum_{r = 1}^{\infty} H_{r} (x ₁, . . ., x_{s}) \in C^{\infty} (ℝ^{s})$ of analytic functions $H_{r}$ , each solving the same system of universal partial differential equations, namely $P (x_{σ}; H_{r}, \partial H_{r} / \partial x_{σ}, . . ., \partial ⁵ H_{r} / \partial x ⁵_{σ} ⁵) = 0$ (σ = 1,..., s).

Functions in $L_{p}$ -space and their approximations

P. Chandra (1988)

Matematički Vesnik

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Whitney type inequality, pointwise version

Yu. A. Brudnyi, I. E. Gopengauz (2013)

Studia Mathematica

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The main result of the paper estimates the asymptotic behavior of local polynomial approximation for $L_{p}$ functions at a point via the behavior of μ-differences, a generalization of the kth difference. The result is applied to prove several new and extend classical results on pointwise differentiability of $L_{p}$ functions including Marcinkiewicz-Zygmund’s and M. Weiss’ theorems. In particular, we present a solution of the problem posed in the 30s by Marcinkiewicz and Zygmund.

Approximation of functions from $L^{p} {(ω ̃)}_{β}$ by linear operators of conjugate Fourier series

WŁodzimierz Łenski, Bogdan Szal (2011)

Banach Center Publications

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We show the results corresponding to some theorems of S. Lal and H. K. Nigam [Int. J. Math. Math. Sci. 27 (2001), 555-563] on the norm and pointwise approximation of conjugate functions and to the results of the authors [Acta Comment. Univ. Tartu. Math. 13 (2009), 11-24] also on such approximations.

Rational approximation to real points on conics

Damien Roy (2013)

Annales de l’institut Fourier

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A point $(ξ_{1}, ξ_{2})$ with coordinates in a subfield of $ℝ$ of transcendence degree one over $ℚ$ , with $1, ξ_{1}, ξ_{2}$ linearly independent over $ℚ$ , may have a uniform exponent of approximation by elements of $ℚ^{2}$ that is strictly larger than the lower bound $1 / 2$ given by Dirichlet’s box principle. This appeared as a surprise, in connection to work of Davenport and Schmidt, for points of the parabola ${(ξ, ξ^{2}); ξ \in ℝ}$ . The goal of this paper is to show that this phenomenon extends to all real conics defined over $ℚ$ , and that the largest...

Convergence of greedy approximation II. The trigonometric system

S. V. Konyagin, V. N. Temlyakov (2003)

Studia Mathematica

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We study the following nonlinear method of approximation by trigonometric polynomials. For a periodic function f we take as an approximant a trigonometric polynomial of the form $G ₘ (f) : = \sum_{k \in Λ} f ̂ (k) e^{i (k, x)}$ , where $Λ \subset ℤ^{d}$ is a set of cardinality m containing the indices of the m largest (in absolute value) Fourier coefficients f̂(k) of the function f. Note that Gₘ(f) gives the best m-term approximant in the L₂-norm, and therefore, for each f ∈ L₂, ||f-Gₘ(f)||₂ → 0 as m → ∞. It is known from previous results that in...

Around the Littlewood conjecture in Diophantine approximation

Yann Bugeaud (2014)

Publications mathématiques de Besançon

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The Littlewood conjecture in Diophantine approximation claims that $inf_{q \geq 1} q \cdot ∥ q α ∥ \cdot ∥ q β ∥ = 0$ holds for all real numbers $α$ and $β$ , where $∥ \cdot ∥$ denotes the distance to the nearest integer. Its $p$ -adic analogue, formulated by de Mathan and Teulié in 2004, asserts that $inf_{q \geq 1} q \cdot ∥ q α ∥ \cdot {| q |}_{p} = 0$ holds for every real number $α$ and every prime number $p$ , where ${| \cdot |}_{p}$ denotes the $p$ -adic absolute value normalized by ${| p |}_{p} = p^{- 1}$ . We survey the known results on these conjectures and highlight recent developments. ...

On the multiples of a badly approximable vector

Yann Bugeaud (2015)

Acta Arithmetica

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Let d be a positive integer and α a real algebraic number of degree d + 1. Set $α ̲ : = (α, α ², . . ., α^{d})$ . It is well-known that $c (α ̲) : = l i m i n f_{q \to \infty} q^{1 / d} \cdot | | q α ̲ | | > 0$ , where ||·|| denotes the distance to the nearest integer. Furthermore, $c (α ̲) n^{- 1 / d} \leq c (n α ̲) \leq n c (α ̲)$ for any integer n ≥ 1. Our main result asserts that there exists a real number C, depending only on α, such that $c (n α ̲) \leq C n^{- 1 / d}$ for any integer n ≥ 1.

Approximation properties for modified $(p, q)$ -Bernstein-Durrmeyer operators

Mohammad Mursaleen, Ahmed A. H. Alabied (2018)

Mathematica Bohemica

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We introduce modified $(p, q)$ -Bernstein-Durrmeyer operators. We discuss approximation properties for these operators based on Korovkin type approximation theorem and compute the order of convergence using usual modulus of continuity. We also study the local approximation property of the sequence of positive linear operators $D_{n, p, q}^{*}$ and compute the rate of convergence for the function $f$ belonging to the class ${Lip}_{M} (γ)$ .

A remark on the approximation theorems of Whitney and Carleman-Scheinberg

Michal Johanis (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show that a $C^{k}$ -smooth mapping on an open subset of $ℝ^{n}$ , $k \in ℕ \cup {0, \infty}$ , can be approximated in a fine topology and together with its derivatives by a restriction of a holomorphic mapping with explicitly described domain. As a corollary we obtain a generalisation of the Carleman-Scheinberg theorem on approximation by entire functions.

Displaying similar documents to “The polynomial approximation of continuous functions defined on ( - ∞ , ∞ ) ”

Displaying similar documents to “The polynomial approximation of continuous functions defined on $(- \infty, \infty)$ ”