scholarly journals Semi-smooth Points in Some Classical Function Spaces

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Beata Derȩgowska ◽  
Beata Gryszka ◽  
Karol Gryszka ◽  
Paweł Wójcik
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
Spaces Of Continuous Functions ◽  
Classical Function ◽  
Necessary And Sufficient ◽  
Vector Valued

AbstractThe investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space $$\mathcal {C}(\Omega )$$ C ( Ω ) . Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions $$\mathcal {C}(\mathcal {T},E)$$ C ( T , E ) , where $$\mathcal {T}$$ T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions $$\mathcal {C}_0(\mathcal {T},E)$$ C 0 ( T , E ) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.

scholarly journals ‎INSERTION OF A CONTRA-CONTINUOUS FUNCTION BETWEEN TWO ‎‎COMPARABLE CONTRA-$\alpha-$CONTINUOUS (CONTRA-$C-$CONTINUOUS) FUNCTIONS

2019 ◽  
pp. 13
Author(s):  
Majid Mirmiran ◽  
Binesh Naderi
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

‎A necessary and sufficient condition in terms of lower cut sets ‎are given for the insertion of a contra-continuous function ‎between two comparable real-valued functions on such topological ‎spaces that kernel of sets are open‎. 

scholarly journals On Weighted Polynomial Approximation With Gaps

2005 ◽  
Vol 178 ◽  
pp. 55-61 ◽  
Author(s):  
Guantie Deng
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Polynomial Approximation ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Weighted Polynomial Approximation ◽  
Necessary And Sufficient Condition ◽  
Weighted Banach Space ◽  
Nonnegative Continuous Function ◽  
Necessary And Sufficient

Let α be a nonnegative continuous function on ℝ. In this paper, the author obtains a necessary and sufficient condition for polynomials with gaps to be dense in Cα, where Cα is the weighted Banach space of complex continuous functions ƒ on ℝ with ƒ(t) exp(−α(t)) vanishing at infinity.

scholarly journals Dunkl harmonic analysis and fundamental sets of continuous functions on the unit sphere

2015 ◽  
Vol 6 (3) ◽  
Author(s):  
Roman A. Veprintsev
Keyword(s):  
Continuous Function ◽  
Harmonic Analysis ◽  
Unit Sphere ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Family ◽  
Necessary And Sufficient

AbstractWe establish a necessary and sufficient condition on a continuous function on [-1,1] under which the family of functions on the unit sphere 𝕊

scholarly journals Random products of maps synchronizing on average

2020 ◽  
Vol 22 (08) ◽  
pp. 1950086
Author(s):  
Edgar Matias ◽  
Ítalo Melo
Keyword(s):  
Metric Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
Random Product ◽  
Random Products ◽  
Necessary And Sufficient

We present a necessary and sufficient condition for a random product of maps on a compact metric space to be (strongly) synchronizing on average.

LIMITS OF HOMOMORPHISMS WITH FINITE-DIMENSIONAL RANGE

2005 ◽  
Vol 16 (07) ◽  
pp. 807-821 ◽  
Author(s):  
SHANWEN HU ◽  
HUAXIN LIN ◽  
YIFENG XUE
Keyword(s):  
Metric Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
C Algebra ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Unital Simple ◽  
Dimensional Range

Let X be a compact metric space and A be a unital simple C*-algebra with TR (A)=0. Suppose that ϕ : C(X) → A is a unital monomorphism. We study the problem when ϕ can be approximated by homomorphisms with finite-dimensional range. We give a K-theoretical necessary and sufficient condition for ϕ being approximated by homomorphisms with finite-dimensional range.

Ideal convergence of nets of functions with values in uniform spaces

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6281-6292
Author(s):  
Athanasios Megaritis
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Uniform Space ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Sets ◽  
Uniform Spaces ◽  
Ideal Convergence ◽  
Necessary And Sufficient

We consider the pointwise, uniform, quasi-uniform, and the almost uniform I-convergence for a net (fd)d?D of functions from a topological space X into a uniform space (Y,U), where I is an ideal on D. The purpose of the present paper is to provide ideal versions of some classical results and to extend these to nets of functions with values in uniform spaces. In particular, we define the notion of I-equicontinuous family of functions on which pointwise and uniform I-convergence coincide on compact sets. Generalizing the theorem of Arzel?, we give a necessary and sufficient condition for a net of continuous functions from a compact space into a uniform space to I-converge pointwise to a continuous function.

scholarly journals XV.—On a Test for Continuity.

1908 ◽  
Vol 28 ◽  
pp. 249-258
Author(s):  
W. H. Young
Keyword(s):  
Uniform Convergence ◽  
Continuous Functions ◽  
Usual Method ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

§ 1. THE usual method of proving that a function defined as the limit of a sequence of continuous functions is continuous is by proving that the convergence is uniform. This method may fail owing to the presence of points at which the convergence is non-uniform although the limiting function is continuous. In such a case it would be necessary to apply a further test, e.g. that of Arzelà (“uniform convergence by segments”).In some cases the continuity may be proved directly by means of a totally different principle, without reference to modes of convergence at all. It is, in fact, a necessary and sufficient condition for the continuity of a function that it should be possible to express it at the same time as the limit of a monotone ascending and of a monotone descending sequence of continuous functions.

On Global Inverse Theorems of Szász and Baskakov Operators

1979 ◽  
Vol 31 (2) ◽  
pp. 255-263 ◽  
Author(s):  
Z. Ditzian
Keyword(s):  
Rate Of Convergence ◽  
Exponential Growth ◽  
Polynomial Growth ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Baskakov Operators ◽  
Inverse Theorems ◽  
Image Position ◽  
Necessary And Sufficient

The Szász and Baskakov approximation operators are given by1.11.2respectively. For continuous functions on [0, ∞) with exponential growth (i.e. ‖ƒ‖A ≡ supx\ƒ(x)e–Ax\ < M) the modulus of continuity is defined by1.3where ƒ ∈ Lip* (∝, A) for some 0 < ∝ ≦ 2 if w2(ƒ, δ, A) ≦ Mδ∝ for all δ < 1. We shall find a necessary and sufficient condition on the rate of convergence of An(ƒ, x) (representing Sn(ƒ, x) or Vn(ƒ, x)) to ƒ(x) for ƒ(x) ∈ Lip* (∝, A). In a recent paper of M. Becker [1] such conditions were found for functions of polynomial growth (where (1 + \x\N)−1 replaced e–Ax in the above). M. Becker explained the difficulties in treating functions of exponential growth.

Non-Extendability of Bounded Continuous Functions

1980 ◽  
Vol 32 (4) ◽  
pp. 867-879
Author(s):  
Ronnie Levy
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Dense Subspace ◽  
Compact Metric Space ◽  
Bounded Continuous Function ◽  
Bounded Functions ◽  
Bounded Continuous Functions ◽  
Completely Metrizable

If X is a dense subspace of Y, much is known about the question of when every bounded continuous real-valued function on X extends to a continuous function on Y. Indeed, this is one of the central topics of [5]. In this paper we are interested in the opposite question: When are there continuous bounded real-valued functions on X which extend to no point of Y – X? (Of course, we cannot hope that every function on X fails to extend since the restrictions to X of continuous functions on Y extend to Y.) In this paper, we show that if Y is a compact metric space and if X is a dense subset of Y, then X admits a bounded continuous function which extends to no point of Y – X if and only if X is completely metrizable. We also show that for certain spaces Y and dense subsets X, the set of bounded functions on X which extend to a point of Y – X form a first category subset of C*(X).

Oscillation and Global Attractivity in a Periodic Delay Equation

1996 ◽  
Vol 39 (3) ◽  
pp. 275-283 ◽  
Author(s):  
J. R. Graef ◽  
C. Qian ◽  
P. W. Spikes
Keyword(s):  
Global Attractivity ◽  
Positive Periodic Solution ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
All Solutions ◽  
Periodic Delay ◽  
Image Position ◽  
Delay Differential ◽  
Necessary And Sufficient

AbstractConsider the delay differential equationwhere α(t) and β(t) are positive, periodic, and continuous functions with period w > 0, and m is a nonnegative integer. We show that this equation has a positive periodic solution x*(t) with period w. We also establish a necessary and sufficient condition for every solution of the equation to oscillate about x*(t) and a sufficient condition for x*(t) to be a global attractor of all solutions of the equation.

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