Analytical expressions and recurrence relations for the $$P_{n-1}(t) - P_{n + 1}(t)$$ function, derivative and integral
Abstract In this paper, we discuss some methods for the calculation of the Legendre polynomial difference $$P_{n - 1}(t) - P_{n + 1}(t)$$. Such differences frequently appear when calculating the spherical harmonic coefficients of truncated spatial filters. By examining the relation between the Legen...
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| Autor*in: |
Piretzidis, Dimitrios [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2021 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer-Verlag GmbH Germany, part of Springer Nature 2021 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of geodesy - Springer Berlin Heidelberg, 1995, 95(2021), 6 vom: 28. Mai |
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| Übergeordnetes Werk: |
volume:95 ; year:2021 ; number:6 ; day:28 ; month:05 |
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| DOI / URN: |
10.1007/s00190-021-01518-4 |
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| 520 | |a Abstract In this paper, we discuss some methods for the calculation of the Legendre polynomial difference $$P_{n - 1}(t) - P_{n + 1}(t)$$. Such differences frequently appear when calculating the spherical harmonic coefficients of truncated spatial filters. By examining the relation between the Legendre polynomials and other classical orthogonal polynomials (i.e., Gegenbauer and Jacobi polynomials), we provide analytical expressions for $$P_{n - 1}(t) - P_{n + 1}(t)$$ in terms of the latter. We also derive a recurrence relation that avoids the direct evaluation of the Legendre polynomials. Analogous analytical expressions and recurrence relations are provided for the calculation of the $$P_{n - 1}(t) - P_{n + 1}(t)$$ derivative and integral. We show that these recurrence relations can be derived from a more general recurrence relation; thus, the former are special cases of the latter. The expressions given in this study can be useful for the more efficient calculation of truncated kernels in the spherical harmonic domain. We demonstrate the applicability of these new expressions for the Pellinen (i.e., rectangular) filter kernel that is regularly used in regional gravity field modeling. | ||
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| 650 | 4 | |a Orthogonal polynomials | |
| 650 | 4 | |a Recurrence relation | |
| 650 | 4 | |a Truncated filter kernel | |
| 700 | 1 | |a Sideris, Michael G. |4 aut | |
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10.1007/s00190-021-01518-4 doi (DE-627)OLC2125755947 (DE-He213)s00190-021-01518-4-p DE-627 ger DE-627 rakwb eng 550 VZ 14 ssgn Piretzidis, Dimitrios verfasserin (orcid)0000-0003-0148-8063 aut Analytical expressions and recurrence relations for the $$P_{n-1}(t) - P_{n + 1}(t)$$ function, derivative and integral 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2021 Abstract In this paper, we discuss some methods for the calculation of the Legendre polynomial difference $$P_{n - 1}(t) - P_{n + 1}(t)$$. Such differences frequently appear when calculating the spherical harmonic coefficients of truncated spatial filters. By examining the relation between the Legendre polynomials and other classical orthogonal polynomials (i.e., Gegenbauer and Jacobi polynomials), we provide analytical expressions for $$P_{n - 1}(t) - P_{n + 1}(t)$$ in terms of the latter. We also derive a recurrence relation that avoids the direct evaluation of the Legendre polynomials. Analogous analytical expressions and recurrence relations are provided for the calculation of the $$P_{n - 1}(t) - P_{n + 1}(t)$$ derivative and integral. We show that these recurrence relations can be derived from a more general recurrence relation; thus, the former are special cases of the latter. The expressions given in this study can be useful for the more efficient calculation of truncated kernels in the spherical harmonic domain. We demonstrate the applicability of these new expressions for the Pellinen (i.e., rectangular) filter kernel that is regularly used in regional gravity field modeling. Legendre polynomials Orthogonal polynomials Recurrence relation Truncated filter kernel Sideris, Michael G. aut Enthalten in Journal of geodesy Springer Berlin Heidelberg, 1995 95(2021), 6 vom: 28. Mai (DE-627)191686298 (DE-600)1302972-1 (DE-576)051377373 0949-7714 nnns volume:95 year:2021 number:6 day:28 month:05 https://doi.org/10.1007/s00190-021-01518-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-GEO SSG-OPC-GGO SSG-OPC-GEO GBV_ILN_2018 GBV_ILN_4277 AR 95 2021 6 28 05 |
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10.1007/s00190-021-01518-4 doi (DE-627)OLC2125755947 (DE-He213)s00190-021-01518-4-p DE-627 ger DE-627 rakwb eng 550 VZ 14 ssgn Piretzidis, Dimitrios verfasserin (orcid)0000-0003-0148-8063 aut Analytical expressions and recurrence relations for the $$P_{n-1}(t) - P_{n + 1}(t)$$ function, derivative and integral 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2021 Abstract In this paper, we discuss some methods for the calculation of the Legendre polynomial difference $$P_{n - 1}(t) - P_{n + 1}(t)$$. Such differences frequently appear when calculating the spherical harmonic coefficients of truncated spatial filters. By examining the relation between the Legendre polynomials and other classical orthogonal polynomials (i.e., Gegenbauer and Jacobi polynomials), we provide analytical expressions for $$P_{n - 1}(t) - P_{n + 1}(t)$$ in terms of the latter. We also derive a recurrence relation that avoids the direct evaluation of the Legendre polynomials. Analogous analytical expressions and recurrence relations are provided for the calculation of the $$P_{n - 1}(t) - P_{n + 1}(t)$$ derivative and integral. We show that these recurrence relations can be derived from a more general recurrence relation; thus, the former are special cases of the latter. The expressions given in this study can be useful for the more efficient calculation of truncated kernels in the spherical harmonic domain. We demonstrate the applicability of these new expressions for the Pellinen (i.e., rectangular) filter kernel that is regularly used in regional gravity field modeling. Legendre polynomials Orthogonal polynomials Recurrence relation Truncated filter kernel Sideris, Michael G. aut Enthalten in Journal of geodesy Springer Berlin Heidelberg, 1995 95(2021), 6 vom: 28. Mai (DE-627)191686298 (DE-600)1302972-1 (DE-576)051377373 0949-7714 nnns volume:95 year:2021 number:6 day:28 month:05 https://doi.org/10.1007/s00190-021-01518-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-GEO SSG-OPC-GGO SSG-OPC-GEO GBV_ILN_2018 GBV_ILN_4277 AR 95 2021 6 28 05 |
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Analytical expressions and recurrence relations for the $$P_{n-1}(t) - P_{n + 1}(t)$$ function, derivative and integral |
| abstract |
Abstract In this paper, we discuss some methods for the calculation of the Legendre polynomial difference $$P_{n - 1}(t) - P_{n + 1}(t)$$. Such differences frequently appear when calculating the spherical harmonic coefficients of truncated spatial filters. By examining the relation between the Legendre polynomials and other classical orthogonal polynomials (i.e., Gegenbauer and Jacobi polynomials), we provide analytical expressions for $$P_{n - 1}(t) - P_{n + 1}(t)$$ in terms of the latter. We also derive a recurrence relation that avoids the direct evaluation of the Legendre polynomials. Analogous analytical expressions and recurrence relations are provided for the calculation of the $$P_{n - 1}(t) - P_{n + 1}(t)$$ derivative and integral. We show that these recurrence relations can be derived from a more general recurrence relation; thus, the former are special cases of the latter. The expressions given in this study can be useful for the more efficient calculation of truncated kernels in the spherical harmonic domain. We demonstrate the applicability of these new expressions for the Pellinen (i.e., rectangular) filter kernel that is regularly used in regional gravity field modeling. © Springer-Verlag GmbH Germany, part of Springer Nature 2021 |
| abstractGer |
Abstract In this paper, we discuss some methods for the calculation of the Legendre polynomial difference $$P_{n - 1}(t) - P_{n + 1}(t)$$. Such differences frequently appear when calculating the spherical harmonic coefficients of truncated spatial filters. By examining the relation between the Legendre polynomials and other classical orthogonal polynomials (i.e., Gegenbauer and Jacobi polynomials), we provide analytical expressions for $$P_{n - 1}(t) - P_{n + 1}(t)$$ in terms of the latter. We also derive a recurrence relation that avoids the direct evaluation of the Legendre polynomials. Analogous analytical expressions and recurrence relations are provided for the calculation of the $$P_{n - 1}(t) - P_{n + 1}(t)$$ derivative and integral. We show that these recurrence relations can be derived from a more general recurrence relation; thus, the former are special cases of the latter. The expressions given in this study can be useful for the more efficient calculation of truncated kernels in the spherical harmonic domain. We demonstrate the applicability of these new expressions for the Pellinen (i.e., rectangular) filter kernel that is regularly used in regional gravity field modeling. © Springer-Verlag GmbH Germany, part of Springer Nature 2021 |
| abstract_unstemmed |
Abstract In this paper, we discuss some methods for the calculation of the Legendre polynomial difference $$P_{n - 1}(t) - P_{n + 1}(t)$$. Such differences frequently appear when calculating the spherical harmonic coefficients of truncated spatial filters. By examining the relation between the Legendre polynomials and other classical orthogonal polynomials (i.e., Gegenbauer and Jacobi polynomials), we provide analytical expressions for $$P_{n - 1}(t) - P_{n + 1}(t)$$ in terms of the latter. We also derive a recurrence relation that avoids the direct evaluation of the Legendre polynomials. Analogous analytical expressions and recurrence relations are provided for the calculation of the $$P_{n - 1}(t) - P_{n + 1}(t)$$ derivative and integral. We show that these recurrence relations can be derived from a more general recurrence relation; thus, the former are special cases of the latter. The expressions given in this study can be useful for the more efficient calculation of truncated kernels in the spherical harmonic domain. We demonstrate the applicability of these new expressions for the Pellinen (i.e., rectangular) filter kernel that is regularly used in regional gravity field modeling. © Springer-Verlag GmbH Germany, part of Springer Nature 2021 |
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| title_short |
Analytical expressions and recurrence relations for the $$P_{n-1}(t) - P_{n + 1}(t)$$ function, derivative and integral |
| url |
https://doi.org/10.1007/s00190-021-01518-4 |
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| author2 |
Sideris, Michael G. |
| author2Str |
Sideris, Michael G. |
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| doi_str |
10.1007/s00190-021-01518-4 |
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2024-07-04T04:47:01.954Z |
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