Extremal Properties of Logarithmic Derivatives of Polynomials
We study extremal properties of simple partial fractions ρ n (i.e., the logarithmic derivatives of algebraic polynomials of degree n ) on a segment and on a circle. We prove that for any a > 1 the poles of a fraction ρ n whose sup norm does not exceed ln(1 + a − n ) on [−1, 1] lie in the exterior...
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| Veröffentlicht in: | Journal of mathematical sciences (New York, N.Y.) N.Y.), 2020-10, Vol.250 (1), p.1-9 |
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| container_title | Journal of mathematical sciences (New York, N.Y.) |
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| creator | Komarov, M. A. |
| description | We study extremal properties of simple partial fractions
ρ
n
(i.e., the logarithmic derivatives of algebraic polynomials of degree
n
) on a segment and on a circle. We prove that for any
a
> 1 the poles of a fraction
ρ
n
whose sup norm does not exceed ln(1 +
a
−
n
) on [−1, 1] lie in the exterior of the ellipse with foci ±1 and sum of half-axes
a
. For a real-valued analytic function
f
bounded in the ellipse with
a
=
3
+
2
2
we show that if a real-valued simple partial fraction of order not greater than
n
is least deviating from
f
in the
C
([−1, 1])-metric, then such a fraction is unique and is characterized by an alternance of
n
+ 1 points in the segment [−1, 1]. |
| doi_str_mv | 10.1007/s10958-020-04991-y |
| format | Article |
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ρ
n
(i.e., the logarithmic derivatives of algebraic polynomials of degree
n
) on a segment and on a circle. We prove that for any
a
> 1 the poles of a fraction
ρ
n
whose sup norm does not exceed ln(1 +
a
−
n
) on [−1, 1] lie in the exterior of the ellipse with foci ±1 and sum of half-axes
a
. For a real-valued analytic function
f
bounded in the ellipse with
a
=
3
+
2
2
we show that if a real-valued simple partial fraction of order not greater than
n
is least deviating from
f
in the
C
([−1, 1])-metric, then such a fraction is unique and is characterized by an alternance of
n
+ 1 points in the segment [−1, 1].</description><identifier>ISSN: 1072-3374</identifier><identifier>EISSN: 1573-8795</identifier><identifier>DOI: 10.1007/s10958-020-04991-y</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Analytic functions ; Derivatives ; Mathematical analysis ; Mathematics ; Mathematics and Statistics ; Polynomials</subject><ispartof>Journal of mathematical sciences (New York, N.Y.), 2020-10, Vol.250 (1), p.1-9</ispartof><rights>Springer Science+Business Media, LLC, part of Springer Nature 2020</rights><rights>COPYRIGHT 2020 Springer</rights><rights>Springer Science+Business Media, LLC, part of Springer Nature 2020.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c512y-c11b8c34947c78b27b3af3a24c51f3acd3d126c09d6f17fa48ac56463b80223a3</citedby><cites>FETCH-LOGICAL-c512y-c11b8c34947c78b27b3af3a24c51f3acd3d126c09d6f17fa48ac56463b80223a3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10958-020-04991-y$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10958-020-04991-y$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Komarov, M. A.</creatorcontrib><title>Extremal Properties of Logarithmic Derivatives of Polynomials</title><title>Journal of mathematical sciences (New York, N.Y.)</title><addtitle>J Math Sci</addtitle><description>We study extremal properties of simple partial fractions
ρ
n
(i.e., the logarithmic derivatives of algebraic polynomials of degree
n
) on a segment and on a circle. We prove that for any
a
> 1 the poles of a fraction
ρ
n
whose sup norm does not exceed ln(1 +
a
−
n
) on [−1, 1] lie in the exterior of the ellipse with foci ±1 and sum of half-axes
a
. For a real-valued analytic function
f
bounded in the ellipse with
a
=
3
+
2
2
we show that if a real-valued simple partial fraction of order not greater than
n
is least deviating from
f
in the
C
([−1, 1])-metric, then such a fraction is unique and is characterized by an alternance of
n
+ 1 points in the segment [−1, 1].</description><subject>Analytic functions</subject><subject>Derivatives</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Polynomials</subject><issn>1072-3374</issn><issn>1573-8795</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNqNkktLxDAUhYso-PwDrgZcuYgmuWnSLFyIbxhQfKxDJpPWSNuMSUfsvzdawRkYVLK44d7vnAvJybJ9go8IxuI4EizzAmGKEWZSEtSvZVskF4AKIfP1dMeCIgDBNrPtGF9wEvECtrKTi_cu2EbXo7vgZzZ0zsaRL0djX-nguufGmdG5De5Nd-5tGN35um9943Qdd7ONMhW79113sqfLi8ezazS-vbo5Ox0jkxPaI0PIpDDAJBNGFBMqJqBL0JSlcapmClNCucFyyksiSs0KbXLOOEwKTClo2MkOBt9Z8K9zGzv14uehTSsVZQxTySmBH6rStVWuLX0XtGlcNOpUUM4xQJ7_SvGCSQBJeaLQCqqyrQ269q0tXWovuf6HX_Q_WsGnM7XpyVcu-JdgccPhkiAxnX3vKj2PUd083C-b_8Uu-tKBNcHHGGypZsE1OvSKYPUZRjWEUaUwqq8wqj6JYBDFBLeVDT8f-IvqA51c3DA</recordid><startdate>20201001</startdate><enddate>20201001</enddate><creator>Komarov, M. A.</creator><general>Springer US</general><general>Springer</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>ISR</scope></search><sort><creationdate>20201001</creationdate><title>Extremal Properties of Logarithmic Derivatives of Polynomials</title><author>Komarov, M. A.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c512y-c11b8c34947c78b27b3af3a24c51f3acd3d126c09d6f17fa48ac56463b80223a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Analytic functions</topic><topic>Derivatives</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Polynomials</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Komarov, M. A.</creatorcontrib><collection>CrossRef</collection><collection>Gale In Context: Science</collection><jtitle>Journal of mathematical sciences (New York, N.Y.)</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Komarov, M. A.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Extremal Properties of Logarithmic Derivatives of Polynomials</atitle><jtitle>Journal of mathematical sciences (New York, N.Y.)</jtitle><stitle>J Math Sci</stitle><date>2020-10-01</date><risdate>2020</risdate><volume>250</volume><issue>1</issue><spage>1</spage><epage>9</epage><pages>1-9</pages><issn>1072-3374</issn><eissn>1573-8795</eissn><abstract>We study extremal properties of simple partial fractions
ρ
n
(i.e., the logarithmic derivatives of algebraic polynomials of degree
n
) on a segment and on a circle. We prove that for any
a
> 1 the poles of a fraction
ρ
n
whose sup norm does not exceed ln(1 +
a
−
n
) on [−1, 1] lie in the exterior of the ellipse with foci ±1 and sum of half-axes
a
. For a real-valued analytic function
f
bounded in the ellipse with
a
=
3
+
2
2
we show that if a real-valued simple partial fraction of order not greater than
n
is least deviating from
f
in the
C
([−1, 1])-metric, then such a fraction is unique and is characterized by an alternance of
n
+ 1 points in the segment [−1, 1].</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10958-020-04991-y</doi><tpages>9</tpages></addata></record> |
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| issn | 1072-3374 1573-8795 |
| language | eng |
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| subjects | Analytic functions Derivatives Mathematical analysis Mathematics Mathematics and Statistics Polynomials |
| title | Extremal Properties of Logarithmic Derivatives of Polynomials |
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