The uniform closure of non-dense rational spaces on the unit interval (original) (raw)

Some dense subset of real numbers and an application

2011

In this paper we first give a collection of subsets which are dense in the set of real numbers. Then, as an application, we show that: for a continuous function f on R\{0}, the integrals Fp,f (x) = ∫ px x f(t)dt and Fq,f (x) = ∫ qx x f(t)dt (where ln p ln q / ∈ Q) are constant functions of x if and only if f = c x , c = f(1) ∈ R.

Uniform approximation by polynomials with integral coefficients. II

Pacific Journal of Mathematics, 1968

Let A be a discrete subring of C of rank 2. Let X be a compact subset of C with transfinite diameter not less than unity or with transfinite diameter less than unity, void interior, and connected complement. In an earlier paper we characterized the complex valued functions on X which can be uniformly approximated by elements from the ring of polynomials A[z], In this paper the same problem is studied where X is a compact subset of C with transfinite diameter d(X) less than unity and with nonvoid interior. It is also studied for certain compact subsets of C n where n is any positive integer. These subsets will have the property that every continuous function holomorphic on the interior is uniformly approximable by complex polynomials. A large class of sets of this type is shown to exist.

The best approximation of some rational functions in uniform norm

Applied Numerical Mathematics, 2005

Here we are concerned with the best approximation by polynomials to rational functions in the uniform norm. We give some new theorems about the best approximation of 1/(1 + x) and 1/(x − a) where a > 1. Finally we extend this problem to that of computing the best approximation of the Chebyshev expansion in uniform norm and give some results and conjectures about this.

On the density principle for rational functions

Numerical Algorithms, 2000

Let E be a subspace of C(X) and define R(E):={g/h: g,heE;h>0}. We prove that R(E) is dense in C(X) if for every X0?X there exists xeX0 such that E contains an approximation to a d-function at the point x on the set X0. We use this principle to study the density of Müntz rationals in two variables.

On norm attaining polynomials

Publications of the Research Institute for Mathematical Sciences, 2003

We show that for every Banach space X the set of 2-homogeneous continuous polynomials whose canonical extension to X * * attain their norm is a dense subset of the space of all 2-homogeneous continuous polynomials P( 2 X).

Approximation by polynomials with bounded coefficients

Journal of Number Theory, 2007

Let θ be a real number satisfying 1 < θ < 2, and let A(θ) be the set of polynomials with coefficients in {0, 1}, evaluated at θ. Using a result of Bugeaud, we prove by elementary methods that θ is a Pisot number when the set (A(θ) − A(θ) − A(θ)) is discrete; the problem whether Pisot numbers are the only numbers θ such that 0 is not a limit point of (A(θ) − A(θ)) is still unsolved. We also determine the three greatest limit points of the quantities inf{c, c > 0, c ∈ C(θ)}, where C(θ) is the set of polynomials with coefficients in {−1, 1}, evaluated at θ , and we find in particular infinitely many Perron numbers θ such that the sets C(θ) are discrete.

Rational combinations of xλk, λk ⩾ 0 are always dense in C[0, 1]

Journal of Approximation Theory, 1978

Several years ago, the second author conjectured that the set of ratios of finite linear combinations of given, distinct monomials ~~1, x%,... is dense in C[O, 11, assuming all X, are 20. This rather bold conjecture was proven true by Somorjai [l] with the assumption that X, ---f 00 as k + co. By an elementary argument, the same result can be extended to any sequence of distinct monomials as long as X, , X, ,... are positive and bounded away from zero. For if {hle}& has a positive limit point, the set of linear combinations of xAk is dense in C [O, 11. (See [2].) However, the above arguments fail if X, -+ 0 as k --f co. The purpose of this note is to show that the original conjecture is true in all cases. Toward that end, we will prove: THEOREM.