An elementary proof of the Birkhoff-Hopf theorem

1995 ◽  
Vol 117 (1) ◽  
pp. 31-55 ◽  
Author(s):  
Simon P. Eveson ◽  
Roger D. Nussbaum
Keyword(s):  
Elementary Proof ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Important Work ◽  
Large Classes ◽  
Contraction Mappings ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Image Position ◽  
Hopf Theorem

In important work some thirty years ago, G. Birkhoff[2, 3] and E. Hopf [16, 17] showed that large classes of positive linear operators behave like contraction mappings with respect to certain ‘almost’ metrics. Hopf worked in a space of measurable functions and took as his ‘almost’ metric the oscillation ω(y/x) of functions y and x with x(t) > 0 almost everywhere, defined by

On a Class of Positive Linear Operators

1973 ◽  
Vol 16 (4) ◽  
pp. 557-559 ◽  
Author(s):  
J. Swetits ◽  
B. Wood
Keyword(s):  
Linear Operators ◽  
Positive Linear Operators ◽  
Complex Numbers ◽  
Image Position

In a recent paper [3] Meir and Sharma introduced a generalization of the Sα- method of summability. The elements of their matrix, (ank), are defined by(1)where is a sequence of complex numbers. if 0 < αj < l for each j = 0, 1, 2,… then ank≥0 for each n = 0, 1, 2,… and k = 0,1,2,…

scholarly journals Positive linear operators and summability

1970 ◽  
Vol 11 (3) ◽  
pp. 281-290 ◽  
Author(s):  
J. P. King ◽  
J. J. Swetits
Keyword(s):  
Linear Operators ◽  
Positive Linear Operators ◽  
Convergence Properties ◽  
Summability Matrices ◽  
Image Position

Let {Ln} be a sequence of positive linear operators defined on C[a, b] of the form where xnk ∈ [a, b] for each k = 0, 1,…, n = 1, 2,…. The convergence properties of the sequences {Ln(f)} to for each f ∈ C[a, b] have been the object of much recent research (see e.g. [4], [8], [11], [13]). In many cases positive linear operators of the form (1) give rise to interesting summability matrices A = (ank(x)) and vice- versa.

Saturation results for a class of linear operators

1983 ◽  
Vol 94 (1) ◽  
pp. 133-148 ◽  
Author(s):  
B. Kuttner ◽  
R. N. Mohapatra ◽  
B. N. Sahney
Keyword(s):  
Fourier Series ◽  
Linear Operators ◽  
Infinite Matrix ◽  
Measurable Functions ◽  
Image Position

Let B denote the space of bounded measurable functions with period 2π. We will suppose throughout that f(x) ∈ B. All norms considered are essential sup norms. Let the Fourier series of f(x) be given byLet D = (dnk) (n, k = 0, 1, …) be an infinite matrix.Let Ln(f; x) be the D transform of the Fourier series of f(t) at t = x, i.e.where Sk(x) = A0(x) + A1(x) + … + Ak(x). Let us write

scholarly journals (Uniform) convergence of twisted ergodic averages

2015 ◽  
Vol 36 (7) ◽  
pp. 2172-2202 ◽  
Author(s):  
TANJA EISNER ◽  
BEN KRAUSE
Keyword(s):  
Uniform Convergence ◽  
Probability Space ◽  
Elementary Proof ◽  
Ergodic Measure ◽  
Ergodic Averages ◽  
Almost Everywhere ◽  
Measure Preserving Transformation ◽  
Image Position ◽  
Badly Approximable

Let$T$be an ergodic measure-preserving transformation on a non-atomic probability space$(X,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$. We prove uniform extensions of the Wiener–Wintner theorem in two settings: for averages involving weights coming from Hardy field functions $p$,$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(p(n))T^{n}f(x)\bigg\}; & & \displaystyle \nonumber\end{eqnarray}$$and for ‘twisted’ polynomial ergodic averages,$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(n\unicode[STIX]{x1D703})T^{P(n)}f(x)\bigg\} & & \displaystyle \nonumber\end{eqnarray}$$for certain classes of badly approximable$\unicode[STIX]{x1D703}\in [0,1]$. We also give an elementary proof that the above twisted polynomial averages converge pointwise$\unicode[STIX]{x1D707}$-almost everywhere for$f\in L^{p}(X),p>1,$and arbitrary$\unicode[STIX]{x1D703}\in [0,1]$.

scholarly journals Sharp estimates of approximation by some positive linear operators

1984 ◽  
Vol 29 (1) ◽  
pp. 13-18 ◽  
Author(s):  
Ashok Sahai ◽  
Govind Prasad
Keyword(s):  
Modulus Of Continuity ◽  
Bernstein Polynomials ◽  
Higher Order ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Higher Order Moments ◽  
Sharp Estimates ◽  
Quantitative Estimates ◽  
Image Position

Recently, Varshney and Singh [Rend. Mat. (6) 2 (1982), 219–225] have given sharper quantitative estimates of convergence for Bernstein polynomials, Szasz and Meyer-Konig-Zeller operators. We have achieved improvement over these estimates by taking moments of higher order. For example, in case of the Meyer-Konig-Zeller operator, they gave the following estimatewherein ∥·∥ stands for sup norm. We have improved this result toWe may remark here that for this modulus of continuity ) our result cannot be sharpened further by taking higher order moments.

The Carathéodory Pseudodistance and Positive Linear Operators

1997 ◽  
Vol 08 (06) ◽  
pp. 809-824 ◽  
Author(s):  
Ralf Meyer
Keyword(s):  
Elementary Proof ◽  
General Setting ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Convex Domains ◽  
Product Property ◽  
Uniform Algebras ◽  
Lempert Function ◽  
Kobayashi Pseudodistance

We give a new elementary proof of Lempert's theorem, which states that for convex domains the Carathéodory pseudodistance coincides with the Lempert function and thus with the Kobayashi pseudodistance. Moreover, we prove the product property of the Carathéodory pseudodistance. Our methods are functional analytic and work also in the more general setting of uniform algebras.

Approximation in statistical sense to B -continuous functions by positive linear operators

2010 ◽  
Vol 47 (3) ◽  
pp. 289-298 ◽  
Author(s):  
Fadime Dirik ◽  
Oktay Duman ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Classical Version ◽  
Statistical Sense

In the present work, using the concept of A -statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued B -continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.

On αβ-statistical convergence for sequences of fuzzy mappings and Korovkin type approximation theorem

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3749-3760 ◽  
Author(s):  
Ali Karaisa ◽  
Uğur Kadak
Keyword(s):  
Statistical Convergence ◽  
Approximation Theorem ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Bernstein Operator ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Inclusion Relations ◽  
Prior Investigation

Upon prior investigation on statistical convergence of fuzzy sequences, we study the notion of pointwise ??-statistical convergence of fuzzy mappings of order ?. Also, we establish the concept of strongly ??-summable sequences of fuzzy mappings and investigate some inclusion relations. Further, we get an analogue of Korovkin-type approximation theorem for fuzzy positive linear operators with respect to ??-statistical convergence. Lastly, we apply fuzzy Bernstein operator to construct an example in support of our result.

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