On direct and converse theorems in the theory of weighted polynomial approximation

1972 ◽  
Vol 126 (2) ◽  
pp. 123-134 ◽  
Author(s):  
G�za Freud
Keyword(s):  
Polynomial Approximation ◽  
Weighted Polynomial Approximation ◽  
Converse Theorems

Weighted polynomial approximation with Freud weights

1994 ◽  
Vol 10 (3) ◽  
pp. 301-315 ◽  
Author(s):  
Doron S. Lubinsky ◽  
Vilmos Totik
Keyword(s):  
Polynomial Approximation ◽  
Weighted Polynomial Approximation ◽  
Freud Weights

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.

The Cauchy integral and analytic continuation

1985 ◽  
Vol 97 (3) ◽  
pp. 491-498 ◽  
Author(s):  
James. E. Brennan
Keyword(s):  
Analytic Continuation ◽  
Polynomial Approximation ◽  
Rational Functions ◽  
Minimum Problem ◽  
Weighted Polynomial Approximation ◽  
Cauchy Integral ◽  
Typical Problem ◽  
Bernstein Problem ◽  
Open Set ◽  
Simply Connected

One of the most important concepts in the theory of approximation by analytic functions is that of analytic continuation. In a typical problem, for example, there is generally a region Ω, a Banach space B of functions analytic in Ω and a subfamily ℱ ⊂ B, each member of which is analytic in some larger open set, and one might be asked to decide whether or not ℱ is dense in B. It often happens, however, that either ℱ is dense or the only functions which can be so approximated have a natural analytic continuation across ∂Ω. A similar phenomenon is also known to occur even for approximation on sets without interior. In this article we shall describe a method for proving such theorems which can be applied in a variety of settings and, in particular, to: (1)  the Bernštein problem for weighted polynomial approximation on the real line; (2)  the completeness problem for weighted polynomial approximation on bounded simply connected regions; (3) the Shapiro overconvergence problem for sequences of rational functions with sparse poles; (4) the Akutowicz-Carleson minimum problem for interpolating functions. Although we shall present no new results, the method of proof, which is based on an argument of the author [6], seems sufficiently versatile to warrant exposition.

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