Rational approximation near zero sets of functions

Approximation by Rational Functions

Approximation by Rational Functions

Making use of the Hardy-Littlewood maximal function, we give a new proof of the following theorem of Pekarski: If f' is in L log L on a finite interval, then f can be approximated in the uniform norm by rational functions of degree n to an error 0(1/n) on that interval. It is well known that approximation by rational functions of degree n can produce a dramatically smaller error than that for p...

Approximation by Rational Functions in

The denseness of rational functions with prescribed poles in the Hardy space and disk algebra is considered. Notations. C complex plane D unit disk fz : jzj < 1g Tunit circle fz : jzj = 1g H p Hardy space of analytic functions on D kfk 1 := supfjf(z)j : z 2 D g, the H 1 norm A(D) disk algebra of functions analytic on D and continuous on D P n set of polynomials of degree at most n

The best uniform polynomial approximation of two classes of rational functions

In this paper we obtain the explicit form of the best uniform polynomial approximations out of Pn of two classes of rational functions using properties of Chebyshev polynomials. In this way we present some new theorems and lemmas. Some examples will be given to support the results.

Density near zero

Let $S$ be a dense subsemigroup of $(0,+infty)$. In this paper, we state   definition of thick near zero, and also  we will introduce a definition that is equivalent to the definition of piecewise syndetic near zero which presented by Hindman  and Leader in [6].  We define density near zero for subsets of $S$ by a collection of nonempty finite subsets of $S$ and we investigate the conditions un...