How close is the approximation by Bernstein polynomials?

2020 ◽  
Vol 104 (561) ◽  
pp. 482-494
Author(s):  
G. J. O. Jameson
Keyword(s):  
Continuous Function ◽  
Bernstein Polynomials ◽  
Real Interval ◽  
Famous Theorem

A famous theorem of Weierstrass, dating from 1885, states that any continuous function can be uniformly approximated by polynomials on a bounded, closed real interval.

The r-monotonicity of generalized Bernstein polynomials

2012 ◽  
Vol 55 (3) ◽  
pp. 797-807
Author(s):  
Laiyi Zhu ◽  
Zhiyong Huang
Keyword(s):  
Continuous Function ◽  
Bernstein Polynomials ◽  
Sufficient Condition ◽  
Generalized Bernstein Polynomials

AbstractLet f ∊ C[0, 1] and let the Bn(f, q; x) be generalized Bernstein polynomials based on the q-integers that were introduced by Phillips. We prove that if f is r-monotone, then Bn(f, q; x) is r-monotone, generalizing well-known results when q = 1 and the results when r = 1 and r = 2 by Goodman et al. We also prove a sufficient condition for a continuous function to be r-monotone.

scholarly journals Lusin-type Theorems for Cheeger Derivatives on Metric Measure Spaces

2015 ◽  
Vol 3 (1) ◽  
Keyword(s):  
Continuous Function ◽  
Large Class ◽  
Borel Function ◽  
Metric Measure Spaces ◽  
Famous Theorem ◽  
Poincaré Inequalities ◽  
Almost Everywhere ◽  
Poincare Inequalities ◽  
Measure Spaces ◽  
Doubling Measures

Abstract A theorem of Lusin states that every Borel function onRis equal almost everywhere to the derivative of a continuous function. This result was later generalized to Rn in works of Alberti and Moonens-Pfeffer. In this note, we prove direct analogs of these results on a large class of metric measure spaces, those with doubling measures and Poincaré inequalities, which admit a form of differentiation by a famous theorem of Cheeger.

Invariant measures for maps of the interval that do not have points of some period

1994 ◽  
Vol 14 (1) ◽  
pp. 9-21 ◽  
Author(s):  
Jozef Bobok ◽  
Milan Kuchta
Keyword(s):  
Fixed Point ◽  
Continuous Function ◽  
Invariant Measures ◽  
Periodic Points ◽  
Real Interval

AbstractWe study invariant measures for a continuous function which maps a real interval into itself. We show that the ratio of the measures of the two subintervals into which it is divided by a fixed point is constrained by the the set of periods of periodic points. As a consequence of this we give a new forcing relation between periodic points.

scholarly journals The minimal number of periodic orbits of periods guaranteed in Sharkovskii's theorem

1985 ◽  
Vol 31 (1) ◽  
pp. 89-103 ◽  
Author(s):  
Bau-Sen Du
Keyword(s):  
Lower Bound ◽  
Continuous Function ◽  
Positive Integer ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Real Interval ◽  
Minimal Period ◽  
Complete Answer ◽  
Integral Power

Let f(x) be a continuous function from a compact real interval into itself with a periodic orbit of minimal period m, where m is not an integral power of 2. Then, by Sharkovskii's theorem, for every positive integer n with m → n in the Sharkovskii ordering defined below, a lower bound on the number of periodic orbits of f(x) with minimal period n is 1. Could we improve this lower bound from 1 to some larger number? In this paper, we give a complete answer to this question.

scholarly journals On the Sets of Convergence for Sequences of the -Bernstein Polynomials with

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Sofiya Ostrovska ◽  
Ahmet Yaşar Özban
Keyword(s):  
Continuous Function ◽  
Uniform Approximation ◽  
Time Scale ◽  
Bernstein Polynomials ◽  
Numerical Examples

The aim of this paper is to present new results related to the convergence of the sequence of the -Bernstein polynomials in the case , where is a continuous function on . It is shown that the polynomials converge to uniformly on the time scale , and that this result is sharp in the sense that the sequence may be divergent for all . Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper, the results are illustrated by numerical examples.

Decision making based on intuitionistic fuzzy preference relations with additive approximate consistency

2020 ◽  
Vol 39 (3) ◽  
pp. 4041-4058
Author(s):  
Fang Liu ◽  
Xu Tan ◽  
Hui Yang ◽  
Hui Zhao
Keyword(s):  
Decision Making ◽  
Decision Makers ◽  
Real Interval ◽  
Preference Relations ◽  
Intuitionistic Fuzzy ◽  
Proposed Model ◽  
Fuzzy Preference Relations ◽  
Novel Concept ◽  
Fuzzy Weights ◽  
Fuzzy Preference

Intuitionistic fuzzy preference relations (IFPRs) have the natural ability to reflect the positive, the negative and the non-determinative judgements of decision makers. A decision making model is proposed by considering the inherent property of IFPRs in this study, where the main novelty comes with the introduction of the concept of additive approximate consistency. First, the consistency definitions of IFPRs are reviewed and the underlying ideas are analyzed. Second, by considering the allocation of the non-determinacy degree of decision makers’ opinions, the novel concept of approximate consistency for IFPRs is proposed. Then the additive approximate consistency of IFPRs is defined and the properties are studied. Third, the priorities of alternatives are derived from IFPRs with additive approximate consistency by considering the effects of the permutations of alternatives and the allocation of the non-determinacy degree. The rankings of alternatives based on real, interval and intuitionistic fuzzy weights are investigated, respectively. Finally, some comparisons are reported by carrying out numerical examples to show the novelty and advantage of the proposed model. It is found that the proposed model can offer various decision schemes due to the allocation of the non-determinacy degree of IFPRs.

Generalized ʎ-Closed Sets and (ʎ-ʏ)*-Continuous Function

2017 ◽  
Vol 4 (ICBS Conference) ◽  
pp. 1-17 ◽  
Author(s):  
Alias Khalaf ◽  
Sarhad Nami
Keyword(s):  
Continuous Function ◽  
Closed Sets

Almost Somewhat Near Continuity and Near Regularity

2021 ◽  
Vol 7 (1) ◽  
pp. 88-99
Author(s):  
Zanyar A. Ameen
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
One To One ◽  
Nearly Continuous

AbstractThe notions of almost somewhat near continuity of functions and near regularity of spaces are introduced. Some properties of almost somewhat nearly continuous functions and their connections are studied. At the end, it is shown that a one-to-one almost somewhat nearly continuous function f from a space X onto a space Y is somewhat nearly continuous if and only if the range of f is nearly regular.

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