OPUSCULA MATHEMATICA
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Roczniki, numery i artykuły czasopisma Opuscula Mathematica
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http://www.opuscula.agh.edu.pl/
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- ArtykułOtwarty dostępGeometric properties of the lattice of polynomials with integer coefficients(Wydawnictwa AGH, 2024) Lipnicki, Artur; Śmietański, Marek J.This paper is related to the classic but still being examined issue of approximation of functions by polynomials with integer coefficients. Let $r$, $n$ be positive integers with $n \ge 6r$. Let $\boldsymbol{P}_n \cap \boldsymbol{M}_r$ be the space of polynomials of degree at most $n$ on $[0,1]$ with integer coefficients such that $P^{(k)}(0)/k!$ and $P^{(k)}(1)/k!$ are integers for $k=0,\dots,r-1$ and let $\boldsymbol{P}_n^\mathbb{Z} \cap \boldsymbol{M}_r$ be the additive group of polynomials with integer coefficients. We explore the problem of estimating the minimal distance of elements of $\boldsymbol{P}_n^\mathbb{Z} \cap \boldsymbol{M}_r$ from $\boldsymbol{P}_n \cap \boldsymbol{M}_r$ in $L_2(0,1)$. We give rather precise quantitative estimations for successive minima of $\boldsymbol{P}_n^\mathbb{Z}$ in certain specific cases. At the end, we study properties of the shortest polynomials in some hyperplane in $\boldsymbol{P}_n \cap \boldsymbol{M}_r$.
- ArtykułOtwarty dostępUniform approximation by polynomials with integer coefficients(2016) Lipnicki, ArturLet r, n be positive integers with n≥6r. Let P be a polynomial of degree at most n on [0,1] with real coefficients, such that [formula] and [formula] are integers for k= 0,. . . , r − 1. It is proved that there is a polynomial Q of degree at most n with integer coefficients such that [formula] for x ∈ [0,1], where C1, C2 are some numerical constants. The result is the best possible up to the constants.