Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations

Abstract We study linear transformations %$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]%$ of the form %$T[x^n]=P_n(x)%$ where %$\{P_n(x)\}%$ is a real orthogonal polynomial system. With %$T=\sum \tfrac{Q_k(x)}{k!}D^k%$, we seek to understand the behavior of the transformation T by studying the roots...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Cardon, David A. [verfasserIn] Sorensen, Evan L. White, Jason C.

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2022

Schlagwörter:	Transformations preserving real-rootedness Hyperbolic polynomials Stable polynomials Orthogonal polynomials Interlacing

Anmerkung:	© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022

Übergeordnetes Werk:	Enthalten in: Constructive approximation - New York, NY : Springer, 1985, 57(2022), 1 vom: 18. Juli, Seite 235-253
Übergeordnetes Werk:	volume:57 ; year:2022 ; number:1 ; day:18 ; month:07 ; pages:235-253

Links:	Volltext

DOI / URN:	10.1007/s00365-022-09581-6

Katalog-ID:	SPR049206664

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100	1		\|a Cardon, David A. \|e verfasserin \|4 aut
245	1	0	\|a Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations
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520			\|a Abstract We study linear transformations %$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]%$ of the form %$T[x^n]=P_n(x)%$ where %$\{P_n(x)\}%$ is a real orthogonal polynomial system. With %$T=\sum \tfrac{Q_k(x)}{k!}D^k%$, we seek to understand the behavior of the transformation T by studying the roots of the %$Q_k(x)%$. We prove four main things. First, we show that the only case where the %$Q_k(x)%$ are constant and %$\{P_n(x)\}%$ is an orthogonal system is when the %$P_n(x)%$ form a shifted set of generalized probabilist Hermite polynomials. Second, we show that the coefficient polynomials %$Q_k(x)%$ have real roots when the %$P_n(x)%$ are the physicist Hermite polynomials or the Laguerre polynomials. Next, we show that in these cases, the roots of successive polynomials strictly interlace, a property that has not yet been studied for coefficient polynomials. We conclude by discussing the Chebyshev and Legendre polynomials, proving a conjecture of Chasse, and presenting several open problems.
650		4	\|a Transformations preserving real-rootedness \|7 (dpeaa)DE-He213
650		4	\|a Hyperbolic polynomials \|7 (dpeaa)DE-He213
650		4	\|a Stable polynomials \|7 (dpeaa)DE-He213
650		4	\|a Orthogonal polynomials \|7 (dpeaa)DE-He213
650		4	\|a Interlacing \|7 (dpeaa)DE-He213
700	1		\|a Sorensen, Evan L. \|4 aut
700	1		\|a White, Jason C. \|4 aut
773	0	8	\|i Enthalten in \|t Constructive approximation \|d New York, NY : Springer, 1985 \|g 57(2022), 1 vom: 18. Juli, Seite 235-253 \|w (DE-627)265505518 \|w (DE-600)1463844-7 \|x 1432-0940 \|7 nnns
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allfields	10.1007/s00365-022-09581-6 doi (DE-627)SPR049206664 (SPR)s00365-022-09581-6-e DE-627 ger DE-627 rakwb eng Cardon, David A. verfasserin aut Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022 Abstract We study linear transformations %$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]%$ of the form %$T[x^n]=P_n(x)%$ where %$\{P_n(x)\}%$ is a real orthogonal polynomial system. With %$T=\sum \tfrac{Q_k(x)}{k!}D^k%$, we seek to understand the behavior of the transformation T by studying the roots of the %$Q_k(x)%$. We prove four main things. First, we show that the only case where the %$Q_k(x)%$ are constant and %$\{P_n(x)\}%$ is an orthogonal system is when the %$P_n(x)%$ form a shifted set of generalized probabilist Hermite polynomials. Second, we show that the coefficient polynomials %$Q_k(x)%$ have real roots when the %$P_n(x)%$ are the physicist Hermite polynomials or the Laguerre polynomials. Next, we show that in these cases, the roots of successive polynomials strictly interlace, a property that has not yet been studied for coefficient polynomials. We conclude by discussing the Chebyshev and Legendre polynomials, proving a conjecture of Chasse, and presenting several open problems. Transformations preserving real-rootedness (dpeaa)DE-He213 Hyperbolic polynomials (dpeaa)DE-He213 Stable polynomials (dpeaa)DE-He213 Orthogonal polynomials (dpeaa)DE-He213 Interlacing (dpeaa)DE-He213 Sorensen, Evan L. aut White, Jason C. aut Enthalten in Constructive approximation New York, NY : Springer, 1985 57(2022), 1 vom: 18. Juli, Seite 235-253 (DE-627)265505518 (DE-600)1463844-7 1432-0940 nnns volume:57 year:2022 number:1 day:18 month:07 pages:235-253 https://dx.doi.org/10.1007/s00365-022-09581-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 57 2022 1 18 07 235-253
spelling	10.1007/s00365-022-09581-6 doi (DE-627)SPR049206664 (SPR)s00365-022-09581-6-e DE-627 ger DE-627 rakwb eng Cardon, David A. verfasserin aut Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022 Abstract We study linear transformations %$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]%$ of the form %$T[x^n]=P_n(x)%$ where %$\{P_n(x)\}%$ is a real orthogonal polynomial system. With %$T=\sum \tfrac{Q_k(x)}{k!}D^k%$, we seek to understand the behavior of the transformation T by studying the roots of the %$Q_k(x)%$. We prove four main things. First, we show that the only case where the %$Q_k(x)%$ are constant and %$\{P_n(x)\}%$ is an orthogonal system is when the %$P_n(x)%$ form a shifted set of generalized probabilist Hermite polynomials. Second, we show that the coefficient polynomials %$Q_k(x)%$ have real roots when the %$P_n(x)%$ are the physicist Hermite polynomials or the Laguerre polynomials. Next, we show that in these cases, the roots of successive polynomials strictly interlace, a property that has not yet been studied for coefficient polynomials. We conclude by discussing the Chebyshev and Legendre polynomials, proving a conjecture of Chasse, and presenting several open problems. Transformations preserving real-rootedness (dpeaa)DE-He213 Hyperbolic polynomials (dpeaa)DE-He213 Stable polynomials (dpeaa)DE-He213 Orthogonal polynomials (dpeaa)DE-He213 Interlacing (dpeaa)DE-He213 Sorensen, Evan L. aut White, Jason C. aut Enthalten in Constructive approximation New York, NY : Springer, 1985 57(2022), 1 vom: 18. Juli, Seite 235-253 (DE-627)265505518 (DE-600)1463844-7 1432-0940 nnns volume:57 year:2022 number:1 day:18 month:07 pages:235-253 https://dx.doi.org/10.1007/s00365-022-09581-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 57 2022 1 18 07 235-253
allfields_unstemmed	10.1007/s00365-022-09581-6 doi (DE-627)SPR049206664 (SPR)s00365-022-09581-6-e DE-627 ger DE-627 rakwb eng Cardon, David A. verfasserin aut Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022 Abstract We study linear transformations %$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]%$ of the form %$T[x^n]=P_n(x)%$ where %$\{P_n(x)\}%$ is a real orthogonal polynomial system. With %$T=\sum \tfrac{Q_k(x)}{k!}D^k%$, we seek to understand the behavior of the transformation T by studying the roots of the %$Q_k(x)%$. We prove four main things. First, we show that the only case where the %$Q_k(x)%$ are constant and %$\{P_n(x)\}%$ is an orthogonal system is when the %$P_n(x)%$ form a shifted set of generalized probabilist Hermite polynomials. Second, we show that the coefficient polynomials %$Q_k(x)%$ have real roots when the %$P_n(x)%$ are the physicist Hermite polynomials or the Laguerre polynomials. Next, we show that in these cases, the roots of successive polynomials strictly interlace, a property that has not yet been studied for coefficient polynomials. We conclude by discussing the Chebyshev and Legendre polynomials, proving a conjecture of Chasse, and presenting several open problems. Transformations preserving real-rootedness (dpeaa)DE-He213 Hyperbolic polynomials (dpeaa)DE-He213 Stable polynomials (dpeaa)DE-He213 Orthogonal polynomials (dpeaa)DE-He213 Interlacing (dpeaa)DE-He213 Sorensen, Evan L. aut White, Jason C. aut Enthalten in Constructive approximation New York, NY : Springer, 1985 57(2022), 1 vom: 18. Juli, Seite 235-253 (DE-627)265505518 (DE-600)1463844-7 1432-0940 nnns volume:57 year:2022 number:1 day:18 month:07 pages:235-253 https://dx.doi.org/10.1007/s00365-022-09581-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 57 2022 1 18 07 235-253
allfieldsGer	10.1007/s00365-022-09581-6 doi (DE-627)SPR049206664 (SPR)s00365-022-09581-6-e DE-627 ger DE-627 rakwb eng Cardon, David A. verfasserin aut Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022 Abstract We study linear transformations %$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]%$ of the form %$T[x^n]=P_n(x)%$ where %$\{P_n(x)\}%$ is a real orthogonal polynomial system. With %$T=\sum \tfrac{Q_k(x)}{k!}D^k%$, we seek to understand the behavior of the transformation T by studying the roots of the %$Q_k(x)%$. We prove four main things. First, we show that the only case where the %$Q_k(x)%$ are constant and %$\{P_n(x)\}%$ is an orthogonal system is when the %$P_n(x)%$ form a shifted set of generalized probabilist Hermite polynomials. Second, we show that the coefficient polynomials %$Q_k(x)%$ have real roots when the %$P_n(x)%$ are the physicist Hermite polynomials or the Laguerre polynomials. Next, we show that in these cases, the roots of successive polynomials strictly interlace, a property that has not yet been studied for coefficient polynomials. We conclude by discussing the Chebyshev and Legendre polynomials, proving a conjecture of Chasse, and presenting several open problems. Transformations preserving real-rootedness (dpeaa)DE-He213 Hyperbolic polynomials (dpeaa)DE-He213 Stable polynomials (dpeaa)DE-He213 Orthogonal polynomials (dpeaa)DE-He213 Interlacing (dpeaa)DE-He213 Sorensen, Evan L. aut White, Jason C. aut Enthalten in Constructive approximation New York, NY : Springer, 1985 57(2022), 1 vom: 18. Juli, Seite 235-253 (DE-627)265505518 (DE-600)1463844-7 1432-0940 nnns volume:57 year:2022 number:1 day:18 month:07 pages:235-253 https://dx.doi.org/10.1007/s00365-022-09581-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 57 2022 1 18 07 235-253
allfieldsSound	10.1007/s00365-022-09581-6 doi (DE-627)SPR049206664 (SPR)s00365-022-09581-6-e DE-627 ger DE-627 rakwb eng Cardon, David A. verfasserin aut Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022 Abstract We study linear transformations %$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]%$ of the form %$T[x^n]=P_n(x)%$ where %$\{P_n(x)\}%$ is a real orthogonal polynomial system. With %$T=\sum \tfrac{Q_k(x)}{k!}D^k%$, we seek to understand the behavior of the transformation T by studying the roots of the %$Q_k(x)%$. We prove four main things. First, we show that the only case where the %$Q_k(x)%$ are constant and %$\{P_n(x)\}%$ is an orthogonal system is when the %$P_n(x)%$ form a shifted set of generalized probabilist Hermite polynomials. Second, we show that the coefficient polynomials %$Q_k(x)%$ have real roots when the %$P_n(x)%$ are the physicist Hermite polynomials or the Laguerre polynomials. Next, we show that in these cases, the roots of successive polynomials strictly interlace, a property that has not yet been studied for coefficient polynomials. We conclude by discussing the Chebyshev and Legendre polynomials, proving a conjecture of Chasse, and presenting several open problems. Transformations preserving real-rootedness (dpeaa)DE-He213 Hyperbolic polynomials (dpeaa)DE-He213 Stable polynomials (dpeaa)DE-He213 Orthogonal polynomials (dpeaa)DE-He213 Interlacing (dpeaa)DE-He213 Sorensen, Evan L. aut White, Jason C. aut Enthalten in Constructive approximation New York, NY : Springer, 1985 57(2022), 1 vom: 18. Juli, Seite 235-253 (DE-627)265505518 (DE-600)1463844-7 1432-0940 nnns volume:57 year:2022 number:1 day:18 month:07 pages:235-253 https://dx.doi.org/10.1007/s00365-022-09581-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 57 2022 1 18 07 235-253
language	English
source	Enthalten in Constructive approximation 57(2022), 1 vom: 18. Juli, Seite 235-253 volume:57 year:2022 number:1 day:18 month:07 pages:235-253
sourceStr	Enthalten in Constructive approximation 57(2022), 1 vom: 18. Juli, Seite 235-253 volume:57 year:2022 number:1 day:18 month:07 pages:235-253
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Transformations preserving real-rootedness Hyperbolic polynomials Stable polynomials Orthogonal polynomials Interlacing
isfreeaccess_bool	false
container_title	Constructive approximation
authorswithroles_txt_mv	Cardon, David A. @@aut@@ Sorensen, Evan L. @@aut@@ White, Jason C. @@aut@@
publishDateDaySort_date	2022-07-18T00:00:00Z
hierarchy_top_id	265505518
id	SPR049206664
language_de	englisch
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author	Cardon, David A.
spellingShingle	Cardon, David A. misc Transformations preserving real-rootedness misc Hyperbolic polynomials misc Stable polynomials misc Orthogonal polynomials misc Interlacing Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations
authorStr	Cardon, David A.
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topic_title	Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations Transformations preserving real-rootedness (dpeaa)DE-He213 Hyperbolic polynomials (dpeaa)DE-He213 Stable polynomials (dpeaa)DE-He213 Orthogonal polynomials (dpeaa)DE-He213 Interlacing (dpeaa)DE-He213
topic	misc Transformations preserving real-rootedness misc Hyperbolic polynomials misc Stable polynomials misc Orthogonal polynomials misc Interlacing
topic_unstemmed	misc Transformations preserving real-rootedness misc Hyperbolic polynomials misc Stable polynomials misc Orthogonal polynomials misc Interlacing
topic_browse	misc Transformations preserving real-rootedness misc Hyperbolic polynomials misc Stable polynomials misc Orthogonal polynomials misc Interlacing
format_facet	Elektronische Aufsätze Aufsätze Elektronische Ressource
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title	Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations
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title_full	Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations
author_sort	Cardon, David A.
journal	Constructive approximation
journalStr	Constructive approximation
lang_code	eng
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container_start_page	235
author_browse	Cardon, David A. Sorensen, Evan L. White, Jason C.
container_volume	57
format_se	Elektronische Aufsätze
author-letter	Cardon, David A.
doi_str_mv	10.1007/s00365-022-09581-6
title_sort	interlacing properties of coefficient polynomials in differential operator representations of real-root preserving linear transformations
title_auth	Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations
abstract	Abstract We study linear transformations %$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]%$ of the form %$T[x^n]=P_n(x)%$ where %$\{P_n(x)\}%$ is a real orthogonal polynomial system. With %$T=\sum \tfrac{Q_k(x)}{k!}D^k%$, we seek to understand the behavior of the transformation T by studying the roots of the %$Q_k(x)%$. We prove four main things. First, we show that the only case where the %$Q_k(x)%$ are constant and %$\{P_n(x)\}%$ is an orthogonal system is when the %$P_n(x)%$ form a shifted set of generalized probabilist Hermite polynomials. Second, we show that the coefficient polynomials %$Q_k(x)%$ have real roots when the %$P_n(x)%$ are the physicist Hermite polynomials or the Laguerre polynomials. Next, we show that in these cases, the roots of successive polynomials strictly interlace, a property that has not yet been studied for coefficient polynomials. We conclude by discussing the Chebyshev and Legendre polynomials, proving a conjecture of Chasse, and presenting several open problems. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022
abstractGer	Abstract We study linear transformations %$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]%$ of the form %$T[x^n]=P_n(x)%$ where %$\{P_n(x)\}%$ is a real orthogonal polynomial system. With %$T=\sum \tfrac{Q_k(x)}{k!}D^k%$, we seek to understand the behavior of the transformation T by studying the roots of the %$Q_k(x)%$. We prove four main things. First, we show that the only case where the %$Q_k(x)%$ are constant and %$\{P_n(x)\}%$ is an orthogonal system is when the %$P_n(x)%$ form a shifted set of generalized probabilist Hermite polynomials. Second, we show that the coefficient polynomials %$Q_k(x)%$ have real roots when the %$P_n(x)%$ are the physicist Hermite polynomials or the Laguerre polynomials. Next, we show that in these cases, the roots of successive polynomials strictly interlace, a property that has not yet been studied for coefficient polynomials. We conclude by discussing the Chebyshev and Legendre polynomials, proving a conjecture of Chasse, and presenting several open problems. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022
abstract_unstemmed	Abstract We study linear transformations %$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]%$ of the form %$T[x^n]=P_n(x)%$ where %$\{P_n(x)\}%$ is a real orthogonal polynomial system. With %$T=\sum \tfrac{Q_k(x)}{k!}D^k%$, we seek to understand the behavior of the transformation T by studying the roots of the %$Q_k(x)%$. We prove four main things. First, we show that the only case where the %$Q_k(x)%$ are constant and %$\{P_n(x)\}%$ is an orthogonal system is when the %$P_n(x)%$ form a shifted set of generalized probabilist Hermite polynomials. Second, we show that the coefficient polynomials %$Q_k(x)%$ have real roots when the %$P_n(x)%$ are the physicist Hermite polynomials or the Laguerre polynomials. Next, we show that in these cases, the roots of successive polynomials strictly interlace, a property that has not yet been studied for coefficient polynomials. We conclude by discussing the Chebyshev and Legendre polynomials, proving a conjecture of Chasse, and presenting several open problems. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022
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title_short	Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations
url	https://dx.doi.org/10.1007/s00365-022-09581-6
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author2	Sorensen, Evan L. White, Jason C.
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doi_str	10.1007/s00365-022-09581-6
up_date	2024-07-03T23:50:07.458Z
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fullrecord_marcxml	<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">SPR049206664</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230510061839.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">230202s2022 xx \|\|\|\|\|o 00\| \|\|eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s00365-022-09581-6</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR049206664</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s00365-022-09581-6-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Cardon, David A.</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2022</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract We study linear transformations %$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]%$ of the form %$T[x^n]=P_n(x)%$ where %$\{P_n(x)\}%$ is a real orthogonal polynomial system. With %$T=\sum \tfrac{Q_k(x)}{k!}D^k%$, we seek to understand the behavior of the transformation T by studying the roots of the %$Q_k(x)%$. We prove four main things. First, we show that the only case where the %$Q_k(x)%$ are constant and %$\{P_n(x)\}%$ is an orthogonal system is when the %$P_n(x)%$ form a shifted set of generalized probabilist Hermite polynomials. Second, we show that the coefficient polynomials %$Q_k(x)%$ have real roots when the %$P_n(x)%$ are the physicist Hermite polynomials or the Laguerre polynomials. Next, we show that in these cases, the roots of successive polynomials strictly interlace, a property that has not yet been studied for coefficient polynomials. 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Interlacing Properties of Coefficient Polynomials in Differential Operator Representations of Real-Root Preserving Linear Transformations

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