Opuscula Math. 36, no. 4 (2016), 489-498
http://dx.doi.org/10.7494/OpMath.2016.36.4.489

Opuscula Mathematica

Uniform approximation by polynomials with integer coefficients

Artur Lipnicki

Abstract. Let $$r$$, $$n$$ be positive integers with $$n\ge 6r$$. Let $$P$$ be a polynomial of degree at most $$n$$ on $$[0,1]$$ with real coefficients, such that $$P^{(k)}(0)/k!$$ and $$P^{(k)}(1)/k!$$ are integers for $$k=0,\dots,r-1$$. It is proved that there is a polynomial $$Q$$ of degree at most $$n$$ with integer coefficients such that $$|P(x)-Q(x)|\le C_1C_2^r r^{2r+1/2}n^{-2r}$$ for $$x\in[0,1]$$, where $$C_1$$, $$C_2$$ are some numerical constants. The result is the best possible up to the constants.

Keywords: approximation by polynomials with integer coefficients, lattice, covering radius.

Mathematics Subject Classification: 41A10, 52C07.

Full text (pdf)

Artur Lipnicki, Uniform approximation by polynomials with integer coefficients, Opuscula Math. 36, no. 4 (2016), 489-498, http://dx.doi.org/10.7494/OpMath.2016.36.4.489

a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.