Inequalities between height and deviation of polynomials

In this paper, for polynomials with real coefficients P,QP,Q satisfying ∣P(x)∣≤∣Q(x)∣| P\left(x)| \le | Q\left(x)| for each xx in a real interval II, we prove the bound L(P)≤cL(Q)L\left(P)\le cL\left(Q) between the lengths of PP and QQ with a constant cc, which is exponential in the degree dd of PP...

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Autor principal:	Dubickas Artūras
Formato:	article
Lenguaje:	EN
Publicado:	De Gruyter 2021
Materias:	height of a polynomial chebyshev polynomials 11c08 12d10 26c10 41a50 Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/9b3a9d2c544f4164a840c186b77435a0
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id	oai:doaj.org-article:9b3a9d2c544f4164a840c186b77435a0
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spelling	oai:doaj.org-article:9b3a9d2c544f4164a840c186b77435a02021-12-05T14:10:53ZInequalities between height and deviation of polynomials2391-545510.1515/math-2021-0055https://doaj.org/article/9b3a9d2c544f4164a840c186b77435a02021-07-01T00:00:00Zhttps://doi.org/10.1515/math-2021-0055https://doaj.org/toc/2391-5455In this paper, for polynomials with real coefficients P,QP,Q satisfying ∣P(x)∣≤∣Q(x)∣\| P\left(x)\| \le \| Q\left(x)\| for each xx in a real interval II, we prove the bound L(P)≤cL(Q)L\left(P)\le cL\left(Q) between the lengths of PP and QQ with a constant cc, which is exponential in the degree dd of PP. An example showing that the constant cc in this bound should be at least exponential in dd is also given. Similar inequalities are obtained for the heights of PP and QQ when the interval II is infinite and P,QP,Q are both of degree dd. In the proofs and in the constructions of examples, we use some translations of Chebyshev polynomials.Dubickas ArtūrasDe Gruyterarticleheight of a polynomialchebyshev polynomials11c0812d1026c1041a50MathematicsQA1-939ENOpen Mathematics, Vol 19, Iss 1, Pp 540-550 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	height of a polynomial chebyshev polynomials 11c08 12d10 26c10 41a50 Mathematics QA1-939
spellingShingle	height of a polynomial chebyshev polynomials 11c08 12d10 26c10 41a50 Mathematics QA1-939 Dubickas Artūras Inequalities between height and deviation of polynomials
description	In this paper, for polynomials with real coefficients P,QP,Q satisfying ∣P(x)∣≤∣Q(x)∣\| P\left(x)\| \le \| Q\left(x)\| for each xx in a real interval II, we prove the bound L(P)≤cL(Q)L\left(P)\le cL\left(Q) between the lengths of PP and QQ with a constant cc, which is exponential in the degree dd of PP. An example showing that the constant cc in this bound should be at least exponential in dd is also given. Similar inequalities are obtained for the heights of PP and QQ when the interval II is infinite and P,QP,Q are both of degree dd. In the proofs and in the constructions of examples, we use some translations of Chebyshev polynomials.
format	article
author	Dubickas Artūras
author_facet	Dubickas Artūras
author_sort	Dubickas Artūras
title	Inequalities between height and deviation of polynomials
title_short	Inequalities between height and deviation of polynomials
title_full	Inequalities between height and deviation of polynomials
title_fullStr	Inequalities between height and deviation of polynomials
title_full_unstemmed	Inequalities between height and deviation of polynomials
title_sort	inequalities between height and deviation of polynomials
publisher	De Gruyter
publishDate	2021
url	https://doaj.org/article/9b3a9d2c544f4164a840c186b77435a0
work_keys_str_mv	AT dubickasarturas inequalitiesbetweenheightanddeviationofpolynomials
_version_	1718371618814689280

Inequalities between height and deviation of polynomials

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