On harmonic entire mappings II
Abstract In this paper, we investigate properties of harmonic entire mappings. First, we study lower order of harmonic entire mappings. For a harmonic entire mappings f of order %$\rho %$, we also discuss the case %$\rho =\infty %$ by introducing the quantities %$\rho (k)%$, %$\tau (k)%$, %$\lambda...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Deng, Hua [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2023 |
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| Schlagwörter: |
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| Anmerkung: |
© The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| Übergeordnetes Werk: |
Enthalten in: Monatshefte für Mathematik - Wien [u.a.] : Springer, 1890, 201(2023), 4 vom: 12. Mai, Seite 1059-1092 |
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| Übergeordnetes Werk: |
volume:201 ; year:2023 ; number:4 ; day:12 ; month:05 ; pages:1059-1092 |
| Links: |
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| DOI / URN: |
10.1007/s00605-023-01866-7 |
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| Katalog-ID: |
SPR051870606 |
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| 520 | |a Abstract In this paper, we investigate properties of harmonic entire mappings. First, we study lower order of harmonic entire mappings. For a harmonic entire mappings f of order %$\rho %$, we also discuss the case %$\rho =\infty %$ by introducing the quantities %$\rho (k)%$, %$\tau (k)%$, %$\lambda (k)%$, %$\omega (k)%$, and also the case %$\rho =0%$ by studying logarithmic order %$\rho _l%$, logarithmic type %$\tau _l%$, logarithmic lower order %$\lambda _l%$, and logarithmic lower type %$\omega _l%$. Secondly, we investigate approximation by harmonic polynomials of harmonic entire mappings. For a real valued continuous function f on %$[-1,1]%$, let En(f)=infpn∈πn‖f-pn‖,n=0,1,2,⋯,%$\begin{aligned} E_n(f)=\inf _{p_n\in \pi _n}\Vert f-p_n\Vert , \;\; n=0,1,2,\cdots , \end{aligned}%$where the norm is the maximum norm on %$[-1,1]%$ and %$\pi _n%$ denotes the class of all harmonic polynomials with real coefficients of degree at most n. It is known that limn→∞En1/n(f)=0%$\begin{aligned} \lim _{n\rightarrow \infty }E_n^{1/n}(f)=0 \end{aligned}%$if and only if f is the restriction to %$[-1,1]%$ of an entire function (cf. [5, Theorem 7, p. 76]). We prove that this result continues to hold for harmonic entire mappings. We also study the relationship of %$\rho (k)%$ and %$\lambda (k)%$ with the rate growth of %$E_n^{1/n}(f)%$ and investigate the relationship of %$\rho _l%$, %$\tau _l%$, %$\lambda _l%$, %$\omega _l%$ with the asymptotic behaviour of %$E_n^{1/n}(f)%$. | ||
| 650 | 4 | |a Harmonic entire mapping |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Order |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Type |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Lower order |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Approximation |7 (dpeaa)DE-He213 | |
| 700 | 1 | |a Ponnusamy, Saminathan |0 (orcid)0000-0002-3699-2713 |4 aut | |
| 700 | 1 | |a Qiao, Jinjing |4 aut | |
| 700 | 1 | |a Tian, Yue |4 aut | |
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10.1007/s00605-023-01866-7 doi (DE-627)SPR051870606 (SPR)s00605-023-01866-7-e DE-627 ger DE-627 rakwb eng Deng, Hua verfasserin aut On harmonic entire mappings II 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract In this paper, we investigate properties of harmonic entire mappings. First, we study lower order of harmonic entire mappings. For a harmonic entire mappings f of order %$\rho %$, we also discuss the case %$\rho =\infty %$ by introducing the quantities %$\rho (k)%$, %$\tau (k)%$, %$\lambda (k)%$, %$\omega (k)%$, and also the case %$\rho =0%$ by studying logarithmic order %$\rho _l%$, logarithmic type %$\tau _l%$, logarithmic lower order %$\lambda _l%$, and logarithmic lower type %$\omega _l%$. Secondly, we investigate approximation by harmonic polynomials of harmonic entire mappings. For a real valued continuous function f on %$[-1,1]%$, let En(f)=infpn∈πn‖f-pn‖,n=0,1,2,⋯,%$\begin{aligned} E_n(f)=\inf _{p_n\in \pi _n}\Vert f-p_n\Vert , \;\; n=0,1,2,\cdots , \end{aligned}%$where the norm is the maximum norm on %$[-1,1]%$ and %$\pi _n%$ denotes the class of all harmonic polynomials with real coefficients of degree at most n. It is known that limn→∞En1/n(f)=0%$\begin{aligned} \lim _{n\rightarrow \infty }E_n^{1/n}(f)=0 \end{aligned}%$if and only if f is the restriction to %$[-1,1]%$ of an entire function (cf. [5, Theorem 7, p. 76]). We prove that this result continues to hold for harmonic entire mappings. We also study the relationship of %$\rho (k)%$ and %$\lambda (k)%$ with the rate growth of %$E_n^{1/n}(f)%$ and investigate the relationship of %$\rho _l%$, %$\tau _l%$, %$\lambda _l%$, %$\omega _l%$ with the asymptotic behaviour of %$E_n^{1/n}(f)%$. Harmonic entire mapping (dpeaa)DE-He213 Order (dpeaa)DE-He213 Type (dpeaa)DE-He213 Lower order (dpeaa)DE-He213 Approximation (dpeaa)DE-He213 Ponnusamy, Saminathan (orcid)0000-0002-3699-2713 aut Qiao, Jinjing aut Tian, Yue aut Enthalten in Monatshefte für Mathematik Wien [u.a.] : Springer, 1890 201(2023), 4 vom: 12. Mai, Seite 1059-1092 (DE-627)254638058 (DE-600)1462913-6 1436-5081 nnns volume:201 year:2023 number:4 day:12 month:05 pages:1059-1092 https://dx.doi.org/10.1007/s00605-023-01866-7 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_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 201 2023 4 12 05 1059-1092 |
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10.1007/s00605-023-01866-7 doi (DE-627)SPR051870606 (SPR)s00605-023-01866-7-e DE-627 ger DE-627 rakwb eng Deng, Hua verfasserin aut On harmonic entire mappings II 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract In this paper, we investigate properties of harmonic entire mappings. First, we study lower order of harmonic entire mappings. For a harmonic entire mappings f of order %$\rho %$, we also discuss the case %$\rho =\infty %$ by introducing the quantities %$\rho (k)%$, %$\tau (k)%$, %$\lambda (k)%$, %$\omega (k)%$, and also the case %$\rho =0%$ by studying logarithmic order %$\rho _l%$, logarithmic type %$\tau _l%$, logarithmic lower order %$\lambda _l%$, and logarithmic lower type %$\omega _l%$. Secondly, we investigate approximation by harmonic polynomials of harmonic entire mappings. For a real valued continuous function f on %$[-1,1]%$, let En(f)=infpn∈πn‖f-pn‖,n=0,1,2,⋯,%$\begin{aligned} E_n(f)=\inf _{p_n\in \pi _n}\Vert f-p_n\Vert , \;\; n=0,1,2,\cdots , \end{aligned}%$where the norm is the maximum norm on %$[-1,1]%$ and %$\pi _n%$ denotes the class of all harmonic polynomials with real coefficients of degree at most n. It is known that limn→∞En1/n(f)=0%$\begin{aligned} \lim _{n\rightarrow \infty }E_n^{1/n}(f)=0 \end{aligned}%$if and only if f is the restriction to %$[-1,1]%$ of an entire function (cf. [5, Theorem 7, p. 76]). We prove that this result continues to hold for harmonic entire mappings. We also study the relationship of %$\rho (k)%$ and %$\lambda (k)%$ with the rate growth of %$E_n^{1/n}(f)%$ and investigate the relationship of %$\rho _l%$, %$\tau _l%$, %$\lambda _l%$, %$\omega _l%$ with the asymptotic behaviour of %$E_n^{1/n}(f)%$. Harmonic entire mapping (dpeaa)DE-He213 Order (dpeaa)DE-He213 Type (dpeaa)DE-He213 Lower order (dpeaa)DE-He213 Approximation (dpeaa)DE-He213 Ponnusamy, Saminathan (orcid)0000-0002-3699-2713 aut Qiao, Jinjing aut Tian, Yue aut Enthalten in Monatshefte für Mathematik Wien [u.a.] : Springer, 1890 201(2023), 4 vom: 12. Mai, Seite 1059-1092 (DE-627)254638058 (DE-600)1462913-6 1436-5081 nnns volume:201 year:2023 number:4 day:12 month:05 pages:1059-1092 https://dx.doi.org/10.1007/s00605-023-01866-7 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_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 201 2023 4 12 05 1059-1092 |
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10.1007/s00605-023-01866-7 doi (DE-627)SPR051870606 (SPR)s00605-023-01866-7-e DE-627 ger DE-627 rakwb eng Deng, Hua verfasserin aut On harmonic entire mappings II 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract In this paper, we investigate properties of harmonic entire mappings. First, we study lower order of harmonic entire mappings. For a harmonic entire mappings f of order %$\rho %$, we also discuss the case %$\rho =\infty %$ by introducing the quantities %$\rho (k)%$, %$\tau (k)%$, %$\lambda (k)%$, %$\omega (k)%$, and also the case %$\rho =0%$ by studying logarithmic order %$\rho _l%$, logarithmic type %$\tau _l%$, logarithmic lower order %$\lambda _l%$, and logarithmic lower type %$\omega _l%$. Secondly, we investigate approximation by harmonic polynomials of harmonic entire mappings. For a real valued continuous function f on %$[-1,1]%$, let En(f)=infpn∈πn‖f-pn‖,n=0,1,2,⋯,%$\begin{aligned} E_n(f)=\inf _{p_n\in \pi _n}\Vert f-p_n\Vert , \;\; n=0,1,2,\cdots , \end{aligned}%$where the norm is the maximum norm on %$[-1,1]%$ and %$\pi _n%$ denotes the class of all harmonic polynomials with real coefficients of degree at most n. It is known that limn→∞En1/n(f)=0%$\begin{aligned} \lim _{n\rightarrow \infty }E_n^{1/n}(f)=0 \end{aligned}%$if and only if f is the restriction to %$[-1,1]%$ of an entire function (cf. [5, Theorem 7, p. 76]). We prove that this result continues to hold for harmonic entire mappings. We also study the relationship of %$\rho (k)%$ and %$\lambda (k)%$ with the rate growth of %$E_n^{1/n}(f)%$ and investigate the relationship of %$\rho _l%$, %$\tau _l%$, %$\lambda _l%$, %$\omega _l%$ with the asymptotic behaviour of %$E_n^{1/n}(f)%$. Harmonic entire mapping (dpeaa)DE-He213 Order (dpeaa)DE-He213 Type (dpeaa)DE-He213 Lower order (dpeaa)DE-He213 Approximation (dpeaa)DE-He213 Ponnusamy, Saminathan (orcid)0000-0002-3699-2713 aut Qiao, Jinjing aut Tian, Yue aut Enthalten in Monatshefte für Mathematik Wien [u.a.] : Springer, 1890 201(2023), 4 vom: 12. Mai, Seite 1059-1092 (DE-627)254638058 (DE-600)1462913-6 1436-5081 nnns volume:201 year:2023 number:4 day:12 month:05 pages:1059-1092 https://dx.doi.org/10.1007/s00605-023-01866-7 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_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 201 2023 4 12 05 1059-1092 |
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10.1007/s00605-023-01866-7 doi (DE-627)SPR051870606 (SPR)s00605-023-01866-7-e DE-627 ger DE-627 rakwb eng Deng, Hua verfasserin aut On harmonic entire mappings II 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract In this paper, we investigate properties of harmonic entire mappings. First, we study lower order of harmonic entire mappings. For a harmonic entire mappings f of order %$\rho %$, we also discuss the case %$\rho =\infty %$ by introducing the quantities %$\rho (k)%$, %$\tau (k)%$, %$\lambda (k)%$, %$\omega (k)%$, and also the case %$\rho =0%$ by studying logarithmic order %$\rho _l%$, logarithmic type %$\tau _l%$, logarithmic lower order %$\lambda _l%$, and logarithmic lower type %$\omega _l%$. Secondly, we investigate approximation by harmonic polynomials of harmonic entire mappings. For a real valued continuous function f on %$[-1,1]%$, let En(f)=infpn∈πn‖f-pn‖,n=0,1,2,⋯,%$\begin{aligned} E_n(f)=\inf _{p_n\in \pi _n}\Vert f-p_n\Vert , \;\; n=0,1,2,\cdots , \end{aligned}%$where the norm is the maximum norm on %$[-1,1]%$ and %$\pi _n%$ denotes the class of all harmonic polynomials with real coefficients of degree at most n. It is known that limn→∞En1/n(f)=0%$\begin{aligned} \lim _{n\rightarrow \infty }E_n^{1/n}(f)=0 \end{aligned}%$if and only if f is the restriction to %$[-1,1]%$ of an entire function (cf. [5, Theorem 7, p. 76]). We prove that this result continues to hold for harmonic entire mappings. We also study the relationship of %$\rho (k)%$ and %$\lambda (k)%$ with the rate growth of %$E_n^{1/n}(f)%$ and investigate the relationship of %$\rho _l%$, %$\tau _l%$, %$\lambda _l%$, %$\omega _l%$ with the asymptotic behaviour of %$E_n^{1/n}(f)%$. Harmonic entire mapping (dpeaa)DE-He213 Order (dpeaa)DE-He213 Type (dpeaa)DE-He213 Lower order (dpeaa)DE-He213 Approximation (dpeaa)DE-He213 Ponnusamy, Saminathan (orcid)0000-0002-3699-2713 aut Qiao, Jinjing aut Tian, Yue aut Enthalten in Monatshefte für Mathematik Wien [u.a.] : Springer, 1890 201(2023), 4 vom: 12. Mai, Seite 1059-1092 (DE-627)254638058 (DE-600)1462913-6 1436-5081 nnns volume:201 year:2023 number:4 day:12 month:05 pages:1059-1092 https://dx.doi.org/10.1007/s00605-023-01866-7 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_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 201 2023 4 12 05 1059-1092 |
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10.1007/s00605-023-01866-7 doi (DE-627)SPR051870606 (SPR)s00605-023-01866-7-e DE-627 ger DE-627 rakwb eng Deng, Hua verfasserin aut On harmonic entire mappings II 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract In this paper, we investigate properties of harmonic entire mappings. First, we study lower order of harmonic entire mappings. For a harmonic entire mappings f of order %$\rho %$, we also discuss the case %$\rho =\infty %$ by introducing the quantities %$\rho (k)%$, %$\tau (k)%$, %$\lambda (k)%$, %$\omega (k)%$, and also the case %$\rho =0%$ by studying logarithmic order %$\rho _l%$, logarithmic type %$\tau _l%$, logarithmic lower order %$\lambda _l%$, and logarithmic lower type %$\omega _l%$. Secondly, we investigate approximation by harmonic polynomials of harmonic entire mappings. For a real valued continuous function f on %$[-1,1]%$, let En(f)=infpn∈πn‖f-pn‖,n=0,1,2,⋯,%$\begin{aligned} E_n(f)=\inf _{p_n\in \pi _n}\Vert f-p_n\Vert , \;\; n=0,1,2,\cdots , \end{aligned}%$where the norm is the maximum norm on %$[-1,1]%$ and %$\pi _n%$ denotes the class of all harmonic polynomials with real coefficients of degree at most n. It is known that limn→∞En1/n(f)=0%$\begin{aligned} \lim _{n\rightarrow \infty }E_n^{1/n}(f)=0 \end{aligned}%$if and only if f is the restriction to %$[-1,1]%$ of an entire function (cf. [5, Theorem 7, p. 76]). We prove that this result continues to hold for harmonic entire mappings. We also study the relationship of %$\rho (k)%$ and %$\lambda (k)%$ with the rate growth of %$E_n^{1/n}(f)%$ and investigate the relationship of %$\rho _l%$, %$\tau _l%$, %$\lambda _l%$, %$\omega _l%$ with the asymptotic behaviour of %$E_n^{1/n}(f)%$. Harmonic entire mapping (dpeaa)DE-He213 Order (dpeaa)DE-He213 Type (dpeaa)DE-He213 Lower order (dpeaa)DE-He213 Approximation (dpeaa)DE-He213 Ponnusamy, Saminathan (orcid)0000-0002-3699-2713 aut Qiao, Jinjing aut Tian, Yue aut Enthalten in Monatshefte für Mathematik Wien [u.a.] : Springer, 1890 201(2023), 4 vom: 12. Mai, Seite 1059-1092 (DE-627)254638058 (DE-600)1462913-6 1436-5081 nnns volume:201 year:2023 number:4 day:12 month:05 pages:1059-1092 https://dx.doi.org/10.1007/s00605-023-01866-7 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_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 201 2023 4 12 05 1059-1092 |
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Enthalten in Monatshefte für Mathematik 201(2023), 4 vom: 12. Mai, Seite 1059-1092 volume:201 year:2023 number:4 day:12 month:05 pages:1059-1092 |
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Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract In this paper, we investigate properties of harmonic entire mappings. First, we study lower order of harmonic entire mappings. For a harmonic entire mappings f of order %$\rho %$, we also discuss the case %$\rho =\infty %$ by introducing the quantities %$\rho (k)%$, %$\tau (k)%$, %$\lambda (k)%$, %$\omega (k)%$, and also the case %$\rho =0%$ by studying logarithmic order %$\rho _l%$, logarithmic type %$\tau _l%$, logarithmic lower order %$\lambda _l%$, and logarithmic lower type %$\omega _l%$. Secondly, we investigate approximation by harmonic polynomials of harmonic entire mappings. For a real valued continuous function f on %$[-1,1]%$, let En(f)=infpn∈πn‖f-pn‖,n=0,1,2,⋯,%$\begin{aligned} E_n(f)=\inf _{p_n\in \pi _n}\Vert f-p_n\Vert , \;\; n=0,1,2,\cdots , \end{aligned}%$where the norm is the maximum norm on %$[-1,1]%$ and %$\pi _n%$ denotes the class of all harmonic polynomials with real coefficients of degree at most n. It is known that limn→∞En1/n(f)=0%$\begin{aligned} \lim _{n\rightarrow \infty }E_n^{1/n}(f)=0 \end{aligned}%$if and only if f is the restriction to %$[-1,1]%$ of an entire function (cf. [5, Theorem 7, p. 76]). We prove that this result continues to hold for harmonic entire mappings. We also study the relationship of %$\rho (k)%$ and %$\lambda (k)%$ with the rate growth of %$E_n^{1/n}(f)%$ and investigate the relationship of %$\rho _l%$, %$\tau _l%$, %$\lambda _l%$, %$\omega _l%$ with the asymptotic behaviour of %$E_n^{1/n}(f)%$.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Harmonic entire mapping</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Order</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Type</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Lower order</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Approximation</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Ponnusamy, Saminathan</subfield><subfield code="0">(orcid)0000-0002-3699-2713</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Qiao, Jinjing</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Tian, Yue</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Monatshefte für Mathematik</subfield><subfield code="d">Wien [u.a.] : Springer, 1890</subfield><subfield code="g">201(2023), 4 vom: 12. 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On harmonic entire mappings II Harmonic entire mapping (dpeaa)DE-He213 Order (dpeaa)DE-He213 Type (dpeaa)DE-He213 Lower order (dpeaa)DE-He213 Approximation (dpeaa)DE-He213 |
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on harmonic entire mappings ii |
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On harmonic entire mappings II |
| abstract |
Abstract In this paper, we investigate properties of harmonic entire mappings. First, we study lower order of harmonic entire mappings. For a harmonic entire mappings f of order %$\rho %$, we also discuss the case %$\rho =\infty %$ by introducing the quantities %$\rho (k)%$, %$\tau (k)%$, %$\lambda (k)%$, %$\omega (k)%$, and also the case %$\rho =0%$ by studying logarithmic order %$\rho _l%$, logarithmic type %$\tau _l%$, logarithmic lower order %$\lambda _l%$, and logarithmic lower type %$\omega _l%$. Secondly, we investigate approximation by harmonic polynomials of harmonic entire mappings. For a real valued continuous function f on %$[-1,1]%$, let En(f)=infpn∈πn‖f-pn‖,n=0,1,2,⋯,%$\begin{aligned} E_n(f)=\inf _{p_n\in \pi _n}\Vert f-p_n\Vert , \;\; n=0,1,2,\cdots , \end{aligned}%$where the norm is the maximum norm on %$[-1,1]%$ and %$\pi _n%$ denotes the class of all harmonic polynomials with real coefficients of degree at most n. It is known that limn→∞En1/n(f)=0%$\begin{aligned} \lim _{n\rightarrow \infty }E_n^{1/n}(f)=0 \end{aligned}%$if and only if f is the restriction to %$[-1,1]%$ of an entire function (cf. [5, Theorem 7, p. 76]). We prove that this result continues to hold for harmonic entire mappings. We also study the relationship of %$\rho (k)%$ and %$\lambda (k)%$ with the rate growth of %$E_n^{1/n}(f)%$ and investigate the relationship of %$\rho _l%$, %$\tau _l%$, %$\lambda _l%$, %$\omega _l%$ with the asymptotic behaviour of %$E_n^{1/n}(f)%$. © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
| abstractGer |
Abstract In this paper, we investigate properties of harmonic entire mappings. First, we study lower order of harmonic entire mappings. For a harmonic entire mappings f of order %$\rho %$, we also discuss the case %$\rho =\infty %$ by introducing the quantities %$\rho (k)%$, %$\tau (k)%$, %$\lambda (k)%$, %$\omega (k)%$, and also the case %$\rho =0%$ by studying logarithmic order %$\rho _l%$, logarithmic type %$\tau _l%$, logarithmic lower order %$\lambda _l%$, and logarithmic lower type %$\omega _l%$. Secondly, we investigate approximation by harmonic polynomials of harmonic entire mappings. For a real valued continuous function f on %$[-1,1]%$, let En(f)=infpn∈πn‖f-pn‖,n=0,1,2,⋯,%$\begin{aligned} E_n(f)=\inf _{p_n\in \pi _n}\Vert f-p_n\Vert , \;\; n=0,1,2,\cdots , \end{aligned}%$where the norm is the maximum norm on %$[-1,1]%$ and %$\pi _n%$ denotes the class of all harmonic polynomials with real coefficients of degree at most n. It is known that limn→∞En1/n(f)=0%$\begin{aligned} \lim _{n\rightarrow \infty }E_n^{1/n}(f)=0 \end{aligned}%$if and only if f is the restriction to %$[-1,1]%$ of an entire function (cf. [5, Theorem 7, p. 76]). We prove that this result continues to hold for harmonic entire mappings. We also study the relationship of %$\rho (k)%$ and %$\lambda (k)%$ with the rate growth of %$E_n^{1/n}(f)%$ and investigate the relationship of %$\rho _l%$, %$\tau _l%$, %$\lambda _l%$, %$\omega _l%$ with the asymptotic behaviour of %$E_n^{1/n}(f)%$. © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
| abstract_unstemmed |
Abstract In this paper, we investigate properties of harmonic entire mappings. First, we study lower order of harmonic entire mappings. For a harmonic entire mappings f of order %$\rho %$, we also discuss the case %$\rho =\infty %$ by introducing the quantities %$\rho (k)%$, %$\tau (k)%$, %$\lambda (k)%$, %$\omega (k)%$, and also the case %$\rho =0%$ by studying logarithmic order %$\rho _l%$, logarithmic type %$\tau _l%$, logarithmic lower order %$\lambda _l%$, and logarithmic lower type %$\omega _l%$. Secondly, we investigate approximation by harmonic polynomials of harmonic entire mappings. For a real valued continuous function f on %$[-1,1]%$, let En(f)=infpn∈πn‖f-pn‖,n=0,1,2,⋯,%$\begin{aligned} E_n(f)=\inf _{p_n\in \pi _n}\Vert f-p_n\Vert , \;\; n=0,1,2,\cdots , \end{aligned}%$where the norm is the maximum norm on %$[-1,1]%$ and %$\pi _n%$ denotes the class of all harmonic polynomials with real coefficients of degree at most n. It is known that limn→∞En1/n(f)=0%$\begin{aligned} \lim _{n\rightarrow \infty }E_n^{1/n}(f)=0 \end{aligned}%$if and only if f is the restriction to %$[-1,1]%$ of an entire function (cf. [5, Theorem 7, p. 76]). We prove that this result continues to hold for harmonic entire mappings. We also study the relationship of %$\rho (k)%$ and %$\lambda (k)%$ with the rate growth of %$E_n^{1/n}(f)%$ and investigate the relationship of %$\rho _l%$, %$\tau _l%$, %$\lambda _l%$, %$\omega _l%$ with the asymptotic behaviour of %$E_n^{1/n}(f)%$. © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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On harmonic entire mappings II |
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Ponnusamy, Saminathan Qiao, Jinjing Tian, Yue |
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| score |
7.3995275 |