scholarly journals Badly approximable functions and interpolation by Blaschke products

1976 ◽  
Vol 20 (2) ◽  
pp. 159-161 ◽  
Author(s):  
L. A. Rubel ◽  
A. L. Shields
Keyword(s):  
Continuous Function ◽  
Supremum Norm ◽  
Sufficient Condition ◽  
Blaschke Products ◽  
Winding Number ◽  
Necessary And Sufficient Condition ◽  
Uniform Limit ◽  
Disc Algebra ◽  
Necessary And Sufficient ◽  
Badly Approximable

A continuous function φ on the unit circle is called badly approximable if ‖ φ − p ‖∞ ≧ ‖ φ |∞ for all polynomials p, where ‖ |∞ is the essential supremum norm. In (4), Poreda asked whether every continuous φ may be written φ = φW+φB, where φW is the uniform limit of polynomials (i.e. φW belongs to the disc algebra A) and φB is badly approximable. We call such a function φ decomposable. In (4), he characterised the badly approximable functions as those of constant non-zero modulus and negative winding number around the origin, i.e. ind (φ)<0. (See (3) for two new proofs of this result.) We show that the answer to Poreda's question is no in general, but give a necessary and sufficient condition for a given φ to have such a decomposition. Then we apply this criterion to solve an interpolation problem.

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 Cesàro kernels on classical groups

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.

scholarly journals The complex geometry of Blaschke products of degree 3 and associated ellipses

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 61-68
Author(s):  
Masayo Fujimuraa
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Complex Geometry ◽  
Sufficient Condition ◽  
Blaschke Products ◽  
Necessary And Sufficient Condition ◽  
Geometrical Properties ◽  
Natural Extensions ◽  
Necessary And Sufficient

A bicentric polygon is a polygon which has both an inscribed circle and a circumscribed one. For given two circles, the necessary and sufficient condition for existence of bicentric triangle for these two circles is known as Chapple?s formula or Euler?s theorem. As one of natural extensions of this formula, we characterize the inscribed ellipses of a triangle which is inscribed in the unit circle. We also discuss the condition for the ?circumscribed? ellipse of a triangle which is circumscribed about the unit circle. For the proof of these results, we use some geometrical properties of Blaschke products on the unit disk.

scholarly journals Insertion of a Contra-continuous Function between Two Comparable Real-valued Functions

2019 ◽  
pp. 21-35 ◽  
Author(s):  
Majid Mirmiran ◽  
Binesh Naderi
Keyword(s):  
Continuous Function ◽  
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 Existence of blowup solutions for nonlinear problems with a gradient term

2006 ◽  
Vol 2006 ◽  
pp. 1-11 ◽  
Author(s):  
Faten Toumi
Keyword(s):  
Continuous Function ◽  
Positive Solution ◽  
Nonlinear Problems ◽  
Gradient Term ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Whole Space ◽  
Necessary And Sufficient

We prove the existence of positive explosive solutions for the equationΔu+λ(|x|)|∇u(x)|=ϕ(x,u(x))in the whole spaceℝN(N≥3), whereλ:[0,∞)→[0,∞)is a continuous function andϕ:ℝN×[0,∞)→[0,∞)is required to satisfy some hypotheses detailed below. More precisely, we will give a necessary and sufficient condition for the existence of a positive solution that blows up at infinity.

scholarly journals Insertion of a Contra-α-continuous Function

2019 ◽  
pp. 383-393 ◽  
Author(s):  
Majid Mirmiran ◽  
Binesh Naderi
Keyword(s):  
Continuous Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

A necessary and sufficient condition in terms of lower cut sets is given for the insertion of a contra-α-continuous function between two comparable real-valued functions.

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