Uniform approximation by polynomials with integer coefficients
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 \(n\)...
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| Format: | Article |
| Language: | English |
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AGH Univeristy of Science and Technology Press
2016-01-01
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| Series: | Opuscula Mathematica |
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| Online Access: | http://www.opuscula.agh.edu.pl/vol36/4/art/opuscula_math_3628.pdf |
| _version_ | 1818465945220284416 |
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| author | Artur Lipnicki |
| author_facet | Artur Lipnicki |
| author_sort | Artur Lipnicki |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-13T20:52:45Z |
| format | Article |
| id | doaj.art-82f14a8d0f9343f18144a37127c8e91f |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-04-13T20:52:45Z |
| publishDate | 2016-01-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-82f14a8d0f9343f18144a37127c8e91f2022-12-22T02:30:27ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742016-01-01364489498http://dx.doi.org/10.7494/OpMath.2016.36.4.4893628Uniform approximation by polynomials with integer coefficientsArtur Lipnicki0University of Łódź, Faculty of Mathematics and Computer Science, ul. Banacha 22, 90-238 Łódź, PolandLet \(r\), \(n\) 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.http://www.opuscula.agh.edu.pl/vol36/4/art/opuscula_math_3628.pdfapproximation by polynomials with integer coefficientslatticecovering radius |
| spellingShingle | Artur Lipnicki Uniform approximation by polynomials with integer coefficients Opuscula Mathematica approximation by polynomials with integer coefficients lattice covering radius |
| title | Uniform approximation by polynomials with integer coefficients |
| title_full | Uniform approximation by polynomials with integer coefficients |
| title_fullStr | Uniform approximation by polynomials with integer coefficients |
| title_full_unstemmed | Uniform approximation by polynomials with integer coefficients |
| title_short | Uniform approximation by polynomials with integer coefficients |
| title_sort | uniform approximation by polynomials with integer coefficients |
| topic | approximation by polynomials with integer coefficients lattice covering radius |
| url | http://www.opuscula.agh.edu.pl/vol36/4/art/opuscula_math_3628.pdf |
| work_keys_str_mv | AT arturlipnicki uniformapproximationbypolynomialswithintegercoefficients |