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

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.

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

Facta Universitatis Series Mathematics and Informatics ◽

10.22190/fumi1901013m ◽

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.

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On Weighted Polynomial Approximation With Gaps

Nagoya Mathematical Journal ◽

10.1017/s0027763000009119 ◽

2005 ◽

Vol 178 ◽

pp. 55-61 ◽

Cited By ~ 10

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.

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Cesàro kernels on classical groups

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700000951 ◽

1997 ◽

Vol 63 (3) ◽

pp. 364-389

Author(s):

Dashan Fan

Keyword(s):

Continuous Function ◽

Harmonic Analysis ◽

Classical Group ◽

Sufficient Condition ◽

Classical Groups ◽

Cesàro Operator ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient ◽

Cesaro Operator

AbstractWe study the Cesàro operator on the classical group G and give a necessary and sufficient condition on the index α = α(G) for which the operator is convergent to f(U) for any continuous function f as N → ∞. The result in this paper solves a question posed by Gong in the book Harmonic analysis on classical groups.

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Ideal convergence of nets of functions with values in uniform spaces

Filomat ◽

10.2298/fil1720281m ◽

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.

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Binomial subsampling

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2005.1622 ◽

2006 ◽

Vol 462 (2068) ◽

pp. 1181-1195 ◽

Cited By ~ 12

Author(s):

Carsten Wiuf ◽

Michael P.H Stumpf

Keyword(s):

Mathematical Biology ◽

Power Series ◽

Generating Functions ◽

Binomial Distribution ◽

Random Variable ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

The Family ◽

Necessary And Sufficient ◽

Finite Moments

In this paper, we discuss statistical families with the property that if the distribution of a random variable X is in , then so is the distribution of Z ∼Bi( X , p ) for 0≤ p ≤1. (Here we take Z ∼Bi( X , p ) to mean that given X = x , Z is a draw from the binomial distribution Bi( x , p ).) It is said that the family is closed under binomial subsampling. We characterize such families in terms of probability generating functions and for families with finite moments of all orders we give a necessary and sufficient condition for the family to be closed under binomial subsampling. The results are illustrated with power series and other examples, and related to examples from mathematical biology. Finally, some issues concerning inference are discussed.

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XV.—On a Test for Continuity.

Proceedings of the Royal Society of Edinburgh ◽

10.1017/s0370164600011676 ◽

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.

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On Global Inverse Theorems of Szász and Baskakov Operators

Canadian Journal of Mathematics ◽

10.4153/cjm-1979-027-2 ◽

1979 ◽

Vol 31 (2) ◽

pp. 255-263 ◽

Cited By ~ 15

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.

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Oscillation and Global Attractivity in a Periodic Delay Equation

Canadian Mathematical Bulletin ◽

10.4153/cmb-1996-035-9 ◽

1996 ◽

Vol 39 (3) ◽

pp. 275-283 ◽

Cited By ~ 23

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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Proximinal subspaces ofA(K)of finite codimension

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171203212308 ◽

2003 ◽

Vol 2003 (39) ◽

pp. 2501-2505

Author(s):

T. S. S. R. K. Rao

Keyword(s):

Compact Subset ◽

Convex Set ◽

Compact Convex ◽

Continuous Functions ◽

Finite Codimension ◽

Sufficient Condition ◽

Choquet Simplex ◽

Necessary And Sufficient Condition ◽

Compact Convex Set ◽

Necessary And Sufficient

We study an analogue of Garkavi's result on proximinal subspaces ofC(X)of finite codimension in the context of the spaceA(K)of affine continuous functions on a compact convex setK. We give an example to show that a simple-minded analogue of Garkavi's result fails for these spaces. WhenKis a metrizable Choquet simplex, we give a necessary and sufficient condition for a boundary measure to attain its norm onA(K). We also exhibit proximinal subspaces of finite codimension ofA(K)when the measures are supported on a compact subset of the extreme boundary.

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