Hermite-Poulain Theorems for Linear Finite Difference Operators

Abstract We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form Δθ,h(f)(z...
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Autor*in:	Katkova, Olga [verfasserIn] Tyaglov, Mikhail [verfasserIn] Vishnyakova, Anna [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2020

Schlagwörter:	Hyperbolic polynomials Laguerre-Pólya class Finite difference operators Hyperbolicity preserving linear operators Mesh of polynomial

Übergeordnetes Werk:	Enthalten in: Constructive approximation - New York, NY : Springer, 1985, 52(2020), 3 vom: 12. Mai, Seite 357-393
Übergeordnetes Werk:	volume:52 ; year:2020 ; number:3 ; day:12 ; month:05 ; pages:357-393

Links:	Volltext

DOI / URN:	10.1007/s00365-020-09507-0

Katalog-ID:	SPR042194334

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520			\|a Abstract We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form Δθ,h(f)(z)=eiθf(z+ih)-e-iθf(z-ih),θ∈[0,π),h∈C\{0},%$\begin{aligned} \Delta _{\theta , h}(f)(z)=e^{i\theta }f(z+ih)-e^{-i\theta }f(z-ih), \quad \theta \in [0,\pi ),\ \ h\in \mathbb {C}{\setminus }\{0\}, \end{aligned}%$where f is a polynomial or an entire function of a certain kind, and prove that the roots of %$\Delta _{\theta , h}(f)%$ are simple under some conditions. Moreover, we prove that the operator %$\Delta _{\theta , h}%$ does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of %$\Delta _{\theta , h}(p)%$ as %$\|h\|\rightarrow \infty %$ are found for any complex polynomial p. Some other interesting roots preserving properties of the operator %$\Delta _{\theta , h}%$ are also studied, and a few examples are presented.
650		4	\|a Hyperbolic polynomials \|7 (dpeaa)DE-He213
650		4	\|a Laguerre-Pólya class \|7 (dpeaa)DE-He213
650		4	\|a Finite difference operators \|7 (dpeaa)DE-He213
650		4	\|a Hyperbolicity preserving linear operators \|7 (dpeaa)DE-He213
650		4	\|a Mesh of polynomial \|7 (dpeaa)DE-He213
700	1		\|a Tyaglov, Mikhail \|e verfasserin \|4 aut
700	1		\|a Vishnyakova, Anna \|e verfasserin \|4 aut
773	0	8	\|i Enthalten in \|t Constructive approximation \|d New York, NY : Springer, 1985 \|g 52(2020), 3 vom: 12. Mai, Seite 357-393 \|w (DE-627)265505518 \|w (DE-600)1463844-7 \|x 1432-0940 \|7 nnns
773	1	8	\|g volume:52 \|g year:2020 \|g number:3 \|g day:12 \|g month:05 \|g pages:357-393
856	4	0	\|u https://dx.doi.org/10.1007/s00365-020-09507-0 \|z lizenzpflichtig \|3 Volltext
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912			\|a GBV_ILN_69
912			\|a GBV_ILN_70
912			\|a GBV_ILN_73
912			\|a GBV_ILN_74
912			\|a GBV_ILN_90
912			\|a GBV_ILN_95
912			\|a GBV_ILN_100
912			\|a GBV_ILN_105
912			\|a GBV_ILN_110
912			\|a GBV_ILN_120
912			\|a GBV_ILN_138
912			\|a GBV_ILN_150
912			\|a GBV_ILN_151
912			\|a GBV_ILN_152
912			\|a GBV_ILN_161
912			\|a GBV_ILN_170
912			\|a GBV_ILN_171
912			\|a GBV_ILN_187
912			\|a GBV_ILN_206
912			\|a GBV_ILN_213
912			\|a GBV_ILN_224
912			\|a GBV_ILN_230
912			\|a GBV_ILN_250
912			\|a GBV_ILN_267
912			\|a GBV_ILN_281
912			\|a GBV_ILN_285
912			\|a GBV_ILN_293
912			\|a GBV_ILN_370
912			\|a GBV_ILN_602
912			\|a GBV_ILN_636
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2001
912			\|a GBV_ILN_2003
912			\|a GBV_ILN_2004
912			\|a GBV_ILN_2005
912			\|a GBV_ILN_2006
912			\|a GBV_ILN_2007
912			\|a GBV_ILN_2008
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912			\|a GBV_ILN_2011
912			\|a GBV_ILN_2014
912			\|a GBV_ILN_2015
912			\|a GBV_ILN_2020
912			\|a GBV_ILN_2021
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912			\|a GBV_ILN_2044
912			\|a GBV_ILN_2048
912			\|a GBV_ILN_2049
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912			\|a GBV_ILN_2055
912			\|a GBV_ILN_2056
912			\|a GBV_ILN_2057
912			\|a GBV_ILN_2059
912			\|a GBV_ILN_2061
912			\|a GBV_ILN_2064
912			\|a GBV_ILN_2065
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912			\|a GBV_ILN_2088
912			\|a GBV_ILN_2093
912			\|a GBV_ILN_2106
912			\|a GBV_ILN_2107
912			\|a GBV_ILN_2108
912			\|a GBV_ILN_2110
912			\|a GBV_ILN_2111
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912			\|a GBV_ILN_2113
912			\|a GBV_ILN_2118
912			\|a GBV_ILN_2122
912			\|a GBV_ILN_2129
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912			\|a GBV_ILN_4251
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author_variant	o k ok m t mt a v av
matchkey_str	article:14320940:2020----::emtpuanhoesolnafntdfe
hierarchy_sort_str	2020
bklnumber	31.49
publishDate	2020
allfields	10.1007/s00365-020-09507-0 doi (DE-627)SPR042194334 (SPR)s00365-020-09507-0-e DE-627 ger DE-627 rakwb eng 510 ASE 510 ASE 31.49 bkl Katkova, Olga verfasserin aut Hermite-Poulain Theorems for Linear Finite Difference Operators 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form Δθ,h(f)(z)=eiθf(z+ih)-e-iθf(z-ih),θ∈[0,π),h∈C\{0},%$\begin{aligned} \Delta _{\theta , h}(f)(z)=e^{i\theta }f(z+ih)-e^{-i\theta }f(z-ih), \quad \theta \in [0,\pi ),\ \ h\in \mathbb {C}{\setminus }\{0\}, \end{aligned}%$where f is a polynomial or an entire function of a certain kind, and prove that the roots of %$\Delta _{\theta , h}(f)%$ are simple under some conditions. Moreover, we prove that the operator %$\Delta _{\theta , h}%$ does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of %$\Delta _{\theta , h}(p)%$ as %$\|h\|\rightarrow \infty %$ are found for any complex polynomial p. Some other interesting roots preserving properties of the operator %$\Delta _{\theta , h}%$ are also studied, and a few examples are presented. Hyperbolic polynomials (dpeaa)DE-He213 Laguerre-Pólya class (dpeaa)DE-He213 Finite difference operators (dpeaa)DE-He213 Hyperbolicity preserving linear operators (dpeaa)DE-He213 Mesh of polynomial (dpeaa)DE-He213 Tyaglov, Mikhail verfasserin aut Vishnyakova, Anna verfasserin aut Enthalten in Constructive approximation New York, NY : Springer, 1985 52(2020), 3 vom: 12. Mai, Seite 357-393 (DE-627)265505518 (DE-600)1463844-7 1432-0940 nnns volume:52 year:2020 number:3 day:12 month:05 pages:357-393 https://dx.doi.org/10.1007/s00365-020-09507-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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 31.49 ASE AR 52 2020 3 12 05 357-393
spelling	10.1007/s00365-020-09507-0 doi (DE-627)SPR042194334 (SPR)s00365-020-09507-0-e DE-627 ger DE-627 rakwb eng 510 ASE 510 ASE 31.49 bkl Katkova, Olga verfasserin aut Hermite-Poulain Theorems for Linear Finite Difference Operators 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form Δθ,h(f)(z)=eiθf(z+ih)-e-iθf(z-ih),θ∈[0,π),h∈C\{0},%$\begin{aligned} \Delta _{\theta , h}(f)(z)=e^{i\theta }f(z+ih)-e^{-i\theta }f(z-ih), \quad \theta \in [0,\pi ),\ \ h\in \mathbb {C}{\setminus }\{0\}, \end{aligned}%$where f is a polynomial or an entire function of a certain kind, and prove that the roots of %$\Delta _{\theta , h}(f)%$ are simple under some conditions. Moreover, we prove that the operator %$\Delta _{\theta , h}%$ does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of %$\Delta _{\theta , h}(p)%$ as %$\|h\|\rightarrow \infty %$ are found for any complex polynomial p. Some other interesting roots preserving properties of the operator %$\Delta _{\theta , h}%$ are also studied, and a few examples are presented. Hyperbolic polynomials (dpeaa)DE-He213 Laguerre-Pólya class (dpeaa)DE-He213 Finite difference operators (dpeaa)DE-He213 Hyperbolicity preserving linear operators (dpeaa)DE-He213 Mesh of polynomial (dpeaa)DE-He213 Tyaglov, Mikhail verfasserin aut Vishnyakova, Anna verfasserin aut Enthalten in Constructive approximation New York, NY : Springer, 1985 52(2020), 3 vom: 12. Mai, Seite 357-393 (DE-627)265505518 (DE-600)1463844-7 1432-0940 nnns volume:52 year:2020 number:3 day:12 month:05 pages:357-393 https://dx.doi.org/10.1007/s00365-020-09507-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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 31.49 ASE AR 52 2020 3 12 05 357-393
allfields_unstemmed	10.1007/s00365-020-09507-0 doi (DE-627)SPR042194334 (SPR)s00365-020-09507-0-e DE-627 ger DE-627 rakwb eng 510 ASE 510 ASE 31.49 bkl Katkova, Olga verfasserin aut Hermite-Poulain Theorems for Linear Finite Difference Operators 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form Δθ,h(f)(z)=eiθf(z+ih)-e-iθf(z-ih),θ∈[0,π),h∈C\{0},%$\begin{aligned} \Delta _{\theta , h}(f)(z)=e^{i\theta }f(z+ih)-e^{-i\theta }f(z-ih), \quad \theta \in [0,\pi ),\ \ h\in \mathbb {C}{\setminus }\{0\}, \end{aligned}%$where f is a polynomial or an entire function of a certain kind, and prove that the roots of %$\Delta _{\theta , h}(f)%$ are simple under some conditions. Moreover, we prove that the operator %$\Delta _{\theta , h}%$ does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of %$\Delta _{\theta , h}(p)%$ as %$\|h\|\rightarrow \infty %$ are found for any complex polynomial p. Some other interesting roots preserving properties of the operator %$\Delta _{\theta , h}%$ are also studied, and a few examples are presented. Hyperbolic polynomials (dpeaa)DE-He213 Laguerre-Pólya class (dpeaa)DE-He213 Finite difference operators (dpeaa)DE-He213 Hyperbolicity preserving linear operators (dpeaa)DE-He213 Mesh of polynomial (dpeaa)DE-He213 Tyaglov, Mikhail verfasserin aut Vishnyakova, Anna verfasserin aut Enthalten in Constructive approximation New York, NY : Springer, 1985 52(2020), 3 vom: 12. Mai, Seite 357-393 (DE-627)265505518 (DE-600)1463844-7 1432-0940 nnns volume:52 year:2020 number:3 day:12 month:05 pages:357-393 https://dx.doi.org/10.1007/s00365-020-09507-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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 31.49 ASE AR 52 2020 3 12 05 357-393
allfieldsGer	10.1007/s00365-020-09507-0 doi (DE-627)SPR042194334 (SPR)s00365-020-09507-0-e DE-627 ger DE-627 rakwb eng 510 ASE 510 ASE 31.49 bkl Katkova, Olga verfasserin aut Hermite-Poulain Theorems for Linear Finite Difference Operators 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form Δθ,h(f)(z)=eiθf(z+ih)-e-iθf(z-ih),θ∈[0,π),h∈C\{0},%$\begin{aligned} \Delta _{\theta , h}(f)(z)=e^{i\theta }f(z+ih)-e^{-i\theta }f(z-ih), \quad \theta \in [0,\pi ),\ \ h\in \mathbb {C}{\setminus }\{0\}, \end{aligned}%$where f is a polynomial or an entire function of a certain kind, and prove that the roots of %$\Delta _{\theta , h}(f)%$ are simple under some conditions. Moreover, we prove that the operator %$\Delta _{\theta , h}%$ does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of %$\Delta _{\theta , h}(p)%$ as %$\|h\|\rightarrow \infty %$ are found for any complex polynomial p. Some other interesting roots preserving properties of the operator %$\Delta _{\theta , h}%$ are also studied, and a few examples are presented. Hyperbolic polynomials (dpeaa)DE-He213 Laguerre-Pólya class (dpeaa)DE-He213 Finite difference operators (dpeaa)DE-He213 Hyperbolicity preserving linear operators (dpeaa)DE-He213 Mesh of polynomial (dpeaa)DE-He213 Tyaglov, Mikhail verfasserin aut Vishnyakova, Anna verfasserin aut Enthalten in Constructive approximation New York, NY : Springer, 1985 52(2020), 3 vom: 12. Mai, Seite 357-393 (DE-627)265505518 (DE-600)1463844-7 1432-0940 nnns volume:52 year:2020 number:3 day:12 month:05 pages:357-393 https://dx.doi.org/10.1007/s00365-020-09507-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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 31.49 ASE AR 52 2020 3 12 05 357-393
allfieldsSound	10.1007/s00365-020-09507-0 doi (DE-627)SPR042194334 (SPR)s00365-020-09507-0-e DE-627 ger DE-627 rakwb eng 510 ASE 510 ASE 31.49 bkl Katkova, Olga verfasserin aut Hermite-Poulain Theorems for Linear Finite Difference Operators 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form Δθ,h(f)(z)=eiθf(z+ih)-e-iθf(z-ih),θ∈[0,π),h∈C\{0},%$\begin{aligned} \Delta _{\theta , h}(f)(z)=e^{i\theta }f(z+ih)-e^{-i\theta }f(z-ih), \quad \theta \in [0,\pi ),\ \ h\in \mathbb {C}{\setminus }\{0\}, \end{aligned}%$where f is a polynomial or an entire function of a certain kind, and prove that the roots of %$\Delta _{\theta , h}(f)%$ are simple under some conditions. Moreover, we prove that the operator %$\Delta _{\theta , h}%$ does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of %$\Delta _{\theta , h}(p)%$ as %$\|h\|\rightarrow \infty %$ are found for any complex polynomial p. Some other interesting roots preserving properties of the operator %$\Delta _{\theta , h}%$ are also studied, and a few examples are presented. Hyperbolic polynomials (dpeaa)DE-He213 Laguerre-Pólya class (dpeaa)DE-He213 Finite difference operators (dpeaa)DE-He213 Hyperbolicity preserving linear operators (dpeaa)DE-He213 Mesh of polynomial (dpeaa)DE-He213 Tyaglov, Mikhail verfasserin aut Vishnyakova, Anna verfasserin aut Enthalten in Constructive approximation New York, NY : Springer, 1985 52(2020), 3 vom: 12. Mai, Seite 357-393 (DE-627)265505518 (DE-600)1463844-7 1432-0940 nnns volume:52 year:2020 number:3 day:12 month:05 pages:357-393 https://dx.doi.org/10.1007/s00365-020-09507-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE 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 31.49 ASE AR 52 2020 3 12 05 357-393
language	English
source	Enthalten in Constructive approximation 52(2020), 3 vom: 12. Mai, Seite 357-393 volume:52 year:2020 number:3 day:12 month:05 pages:357-393
sourceStr	Enthalten in Constructive approximation 52(2020), 3 vom: 12. Mai, Seite 357-393 volume:52 year:2020 number:3 day:12 month:05 pages:357-393
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Hyperbolic polynomials Laguerre-Pólya class Finite difference operators Hyperbolicity preserving linear operators Mesh of polynomial
dewey-raw	510
isfreeaccess_bool	false
container_title	Constructive approximation
authorswithroles_txt_mv	Katkova, Olga @@aut@@ Tyaglov, Mikhail @@aut@@ Vishnyakova, Anna @@aut@@
publishDateDaySort_date	2020-05-12T00:00:00Z
hierarchy_top_id	265505518
dewey-sort	3510
id	SPR042194334
language_de	englisch
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author	Katkova, Olga
spellingShingle	Katkova, Olga ddc 510 bkl 31.49 misc Hyperbolic polynomials misc Laguerre-Pólya class misc Finite difference operators misc Hyperbolicity preserving linear operators misc Mesh of polynomial Hermite-Poulain Theorems for Linear Finite Difference Operators
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topic_title	510 ASE 31.49 bkl Hermite-Poulain Theorems for Linear Finite Difference Operators Hyperbolic polynomials (dpeaa)DE-He213 Laguerre-Pólya class (dpeaa)DE-He213 Finite difference operators (dpeaa)DE-He213 Hyperbolicity preserving linear operators (dpeaa)DE-He213 Mesh of polynomial (dpeaa)DE-He213
topic	ddc 510 bkl 31.49 misc Hyperbolic polynomials misc Laguerre-Pólya class misc Finite difference operators misc Hyperbolicity preserving linear operators misc Mesh of polynomial
topic_unstemmed	ddc 510 bkl 31.49 misc Hyperbolic polynomials misc Laguerre-Pólya class misc Finite difference operators misc Hyperbolicity preserving linear operators misc Mesh of polynomial
topic_browse	ddc 510 bkl 31.49 misc Hyperbolic polynomials misc Laguerre-Pólya class misc Finite difference operators misc Hyperbolicity preserving linear operators misc Mesh of polynomial
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title	Hermite-Poulain Theorems for Linear Finite Difference Operators
ctrlnum	(DE-627)SPR042194334 (SPR)s00365-020-09507-0-e
title_full	Hermite-Poulain Theorems for Linear Finite Difference Operators
author_sort	Katkova, Olga
journal	Constructive approximation
journalStr	Constructive approximation
lang_code	eng
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author_browse	Katkova, Olga Tyaglov, Mikhail Vishnyakova, Anna
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class	510 ASE 31.49 bkl
format_se	Elektronische Aufsätze
author-letter	Katkova, Olga
doi_str_mv	10.1007/s00365-020-09507-0
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title_sort	hermite-poulain theorems for linear finite difference operators
title_auth	Hermite-Poulain Theorems for Linear Finite Difference Operators
abstract	Abstract We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form Δθ,h(f)(z)=eiθf(z+ih)-e-iθf(z-ih),θ∈[0,π),h∈C\{0},%$\begin{aligned} \Delta _{\theta , h}(f)(z)=e^{i\theta }f(z+ih)-e^{-i\theta }f(z-ih), \quad \theta \in [0,\pi ),\ \ h\in \mathbb {C}{\setminus }\{0\}, \end{aligned}%$where f is a polynomial or an entire function of a certain kind, and prove that the roots of %$\Delta _{\theta , h}(f)%$ are simple under some conditions. Moreover, we prove that the operator %$\Delta _{\theta , h}%$ does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of %$\Delta _{\theta , h}(p)%$ as %$\|h\|\rightarrow \infty %$ are found for any complex polynomial p. Some other interesting roots preserving properties of the operator %$\Delta _{\theta , h}%$ are also studied, and a few examples are presented.
abstractGer	Abstract We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form Δθ,h(f)(z)=eiθf(z+ih)-e-iθf(z-ih),θ∈[0,π),h∈C\{0},%$\begin{aligned} \Delta _{\theta , h}(f)(z)=e^{i\theta }f(z+ih)-e^{-i\theta }f(z-ih), \quad \theta \in [0,\pi ),\ \ h\in \mathbb {C}{\setminus }\{0\}, \end{aligned}%$where f is a polynomial or an entire function of a certain kind, and prove that the roots of %$\Delta _{\theta , h}(f)%$ are simple under some conditions. Moreover, we prove that the operator %$\Delta _{\theta , h}%$ does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of %$\Delta _{\theta , h}(p)%$ as %$\|h\|\rightarrow \infty %$ are found for any complex polynomial p. Some other interesting roots preserving properties of the operator %$\Delta _{\theta , h}%$ are also studied, and a few examples are presented.
abstract_unstemmed	Abstract We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form Δθ,h(f)(z)=eiθf(z+ih)-e-iθf(z-ih),θ∈[0,π),h∈C\{0},%$\begin{aligned} \Delta _{\theta , h}(f)(z)=e^{i\theta }f(z+ih)-e^{-i\theta }f(z-ih), \quad \theta \in [0,\pi ),\ \ h\in \mathbb {C}{\setminus }\{0\}, \end{aligned}%$where f is a polynomial or an entire function of a certain kind, and prove that the roots of %$\Delta _{\theta , h}(f)%$ are simple under some conditions. Moreover, we prove that the operator %$\Delta _{\theta , h}%$ does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of %$\Delta _{\theta , h}(p)%$ as %$\|h\|\rightarrow \infty %$ are found for any complex polynomial p. Some other interesting roots preserving properties of the operator %$\Delta _{\theta , h}%$ are also studied, and a few examples are presented.
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title_short	Hermite-Poulain Theorems for Linear Finite Difference Operators
url	https://dx.doi.org/10.1007/s00365-020-09507-0
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author2	Tyaglov, Mikhail Vishnyakova, Anna
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doi_str	10.1007/s00365-020-09507-0
up_date	2024-07-04T01:12:49.517Z
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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">SPR042194334</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20220110174403.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">201201s2020 xx \|\|\|\|\|o 00\| \|\|eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s00365-020-09507-0</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR042194334</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s00365-020-09507-0-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="082" ind1="0" ind2="4"><subfield code="a">510</subfield><subfield code="q">ASE</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">510</subfield><subfield code="q">ASE</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">31.49</subfield><subfield code="2">bkl</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Katkova, Olga</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Hermite-Poulain Theorems for Linear Finite Difference Operators</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2020</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="520" ind1=" " ind2=" "><subfield code="a">Abstract We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form Δθ,h(f)(z)=eiθf(z+ih)-e-iθf(z-ih),θ∈[0,π),h∈C\{0},%$\begin{aligned} \Delta _{\theta , h}(f)(z)=e^{i\theta }f(z+ih)-e^{-i\theta }f(z-ih), \quad \theta \in [0,\pi ),\ \ h\in \mathbb {C}{\setminus }\{0\}, \end{aligned}%$where f is a polynomial or an entire function of a certain kind, and prove that the roots of %$\Delta _{\theta , h}(f)%$ are simple under some conditions. Moreover, we prove that the operator %$\Delta _{\theta , h}%$ does not decrease the mesh on the set of polynomials with roots on a line and find the minimal mesh. The asymptotics of the roots of %$\Delta _{\theta , h}(p)%$ as %$\|h\|\rightarrow \infty %$ are found for any complex polynomial p. Some other interesting roots preserving properties of the operator %$\Delta _{\theta , h}%$ are also studied, and a few examples are presented.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Hyperbolic polynomials</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Laguerre-Pólya class</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Finite difference operators</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Hyperbolicity preserving linear operators</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Mesh of polynomial</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Tyaglov, Mikhail</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Vishnyakova, Anna</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Constructive approximation</subfield><subfield code="d">New York, NY : Springer, 1985</subfield><subfield code="g">52(2020), 3 vom: 12. Mai, Seite 357-393</subfield><subfield code="w">(DE-627)265505518</subfield><subfield code="w">(DE-600)1463844-7</subfield><subfield code="x">1432-0940</subfield><subfield code="7">nnns</subfield></datafield><datafield tag="773" ind1="1" ind2="8"><subfield code="g">volume:52</subfield><subfield code="g">year:2020</subfield><subfield code="g">number:3</subfield><subfield code="g">day:12</subfield><subfield code="g">month:05</subfield><subfield code="g">pages:357-393</subfield></datafield><datafield tag="856" ind1="4" ind2="0"><subfield code="u">https://dx.doi.org/10.1007/s00365-020-09507-0</subfield><subfield code="z">lizenzpflichtig</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_USEFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SYSFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_SPRINGER</subfield></datafield><datafield tag="912" 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