"Zonal spherical function"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
(→‎메타데이터: 새 문단)
(→‎노트: 새 문단)
44번째 줄: 44번째 줄:
 
* [{'LOWER': 'cohen'}, {'OP': '*'}, {'LEMMA': 'Lenstra'}]
 
* [{'LOWER': 'cohen'}, {'OP': '*'}, {'LEMMA': 'Lenstra'}]
 
* [{'LOWER': 'zonal'}, {'LOWER': 'spherical'}, {'LEMMA': 'function'}]
 
* [{'LOWER': 'zonal'}, {'LOWER': 'spherical'}, {'LEMMA': 'function'}]
 +
 +
== 노트 ==
 +
 +
===말뭉치===
 +
# Previous work characterized certain left coideal subalgebras in the quantized enveloping algebra and established an appropriate framework for quantum zonal spherical functions.<ref name="ref_a3254751">[https://www.sciencedirect.com/science/article/pii/S0001870803003530 Quantum zonal spherical functions and Macdonald polynomials]</ref>
 +
# The zonal spherical functions are a broad extension of the notion of zonal spherical harmonics to allow for a more general symmetry group.<ref name="ref_8b1a5a04">[https://en.wikipedia.org/wiki/Zonal_spherical_harmonics Zonal spherical harmonics]</ref>
 +
# Zonal spherical functions have been explicitly determined for real semisimple groups by Harish-Chandra.<ref name="ref_bbc7de78">[https://en.wikipedia.org/wiki/Zonal_spherical_function Zonal spherical function]</ref>
 +
# The abstract functional analytic theory of zonal spherical functions was first developed by Roger Godement.<ref name="ref_bbc7de78" />
 +
# For semisimple p-adic Lie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake and Ian G. Macdonald.<ref name="ref_bbc7de78" />
 +
# Properties 2, 3 and 4 or properties 3, 4 and 5 characterize zonal spherical functions.<ref name="ref_bbc7de78" />
 +
# I tried to rotate zonal spherical function, projected onto spherical harmonic basis.<ref name="ref_677b2b6d">[https://ask.sagemath.org/question/48775/rotation-of-zonal-spherical-functions/ Rotation of zonal spherical functions [closed]]</ref>
 +
# For zonal spherical functions see Spherical harmonics.<ref name="ref_dd393068">[https://encyclopediaofmath.org/wiki/Spherical_functions Encyclopedia of Mathematics]</ref>
 +
# A more common usage of the phrase "spherical function" is as follows.<ref name="ref_dd393068" />
 +
# These solutions include the spherical functions associated with irreducible unitary representations.<ref name="ref_dd393068" />
 +
# S.  ICHIRO SATAKE Theory of spherical functions on reductive algebraic groups over p-adic elds  Publications mathmatiques de lI.H..<ref name="ref_82532dca">[http://www.numdam.org/article/PMIHES_1963__18__5_0.pdf Publications mathématiques de l’i.h.é.s.]</ref>
 +
# 69INTRODUCTIONThe theory of spherical functions on semi-simple Lie groups has been developedby Gelfand, Neumark, Harish-Chandra, Godement and others.<ref name="ref_82532dca" />
 +
# But to determine the explicitform of zonal spherical functions and the Plancherel measure, it seems necessary toknow the (infinite) matrix of this Fourier transformation more explicitly.<ref name="ref_82532dca" />
 +
# For g,eG, /eJf01, we put(5.7) (T,V)(^=/(^)-Then we have:248THEORY OF SPHERICAL FUNCTIONS 25PROPOSITION 5.1.<ref name="ref_82532dca" />
 +
# Am I simply making mistakes in my computation, or is there something wrong in this approach for spherical functions?<ref name="ref_6f1371b1">[https://math.stackexchange.com/questions/3217854/expressing-spherical-functions-with-zonal-harmonics Expressing spherical functions with zonal harmonics]</ref>
 +
# 3  zonal spherical functions  k(x), k(y) = Gk(x, y),  where Gk are Gegenbauer polynomials, zonal spherical functions associated with Harmk(Sd1).<ref name="ref_3ed88387">[https://www.math.fsu.edu/~vlasiuk/pdf/glazyrin.pdf Mapping to the space of spherical harmonics]</ref>
 +
# 4  zonal spherical functions  k(x), k(y) = Gk(x, y),  where Gk are Gegenbauer polynomials, zonal spherical functions associated with Harmk(Sd1).<ref name="ref_3ed88387" />
 +
# Let us recall rst what is a spherical function for a Gelfand pair.<ref name="ref_62cf8405">[https://webusers.imj-prg.fr/~jacques.faraut/MPI.JF.pdf Asymptotics of spherical functions for large rank,]</ref>
 +
# Z  G  where  is a bounded spherical function (m is a Haar measure on the group G, which is unimodular since (G, K) is a Gelfand pair).<ref name="ref_62cf8405" />
 +
# We will denote the spherical functions of positive type for the Gelfand pair (G, K) (; x) (  , x  G).<ref name="ref_62cf8405" />
 +
# In the same way the spherical dual  is identied with the set of spherical functions of positive type.<ref name="ref_62cf8405" />
 +
# We present a sparse analytic representation for spherical functions, in- cluding those expressed in a spherical harmonic (SH) expansion, that is amenable to fast and accurate rotation on the GPU.<ref name="ref_b659cc36">[http://www.cim.mcgill.ca/~derek/files/SHRot_mine.pdf Sparse zonal harmonic factorization for efficient sh rotation]</ref>
 +
# We present an alternative basis for spherical functions based on rotated zonal harmonic (ZH) lobes.<ref name="ref_b659cc36" />
 +
# Let f () be a spherical function, with  = (x, y, z) = (, )  S2, and  and  are spherical (lat-long) coordinates of point (x, y, z) on the spheres surface, S2.<ref name="ref_b659cc36" />
 +
# The process of representing a spherical function, with an initial SH coefcient vector f , in the RZHB with a new coefcient vector z is called Zonal Harmonic Factorization (ZHF).<ref name="ref_b659cc36" />
 +
===소스===
 +
<references />

2021년 2월 22일 (월) 20:25 판

노트

말뭉치

  1. For zonal spherical functions see Spherical harmonics.[1]
  2. Zonal spherical functions have been explicitly determined for real semisimple groups by Harish-Chandra.[2]
  3. The abstract functional analytic theory of zonal spherical functions was first developed by Roger Godement.[2]
  4. For semisimple p-adic Lie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake and Ian G. Macdonald.[2]
  5. Properties 2, 3 and 4 or properties 3, 4 and 5 characterize zonal spherical functions.[2]

소스

  1. ↑ Encyclopedia of Mathematics
  2. ↑ 2.0 2.1 2.2 2.3 Zonal spherical function

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'cohen'}, {'OP': '*'}, {'LOWER': 'lenstra'}, {'LEMMA': 'heuristic'}]
  • [{'LOWER': 'cohen'}, {'OP': '*'}, {'LEMMA': 'Lenstra'}]
  • [{'LOWER': 'zonal'}, {'LOWER': 'spherical'}, {'LEMMA': 'function'}]

노트

말뭉치

  1. Previous work characterized certain left coideal subalgebras in the quantized enveloping algebra and established an appropriate framework for quantum zonal spherical functions.[1]
  2. The zonal spherical functions are a broad extension of the notion of zonal spherical harmonics to allow for a more general symmetry group.[2]
  3. Zonal spherical functions have been explicitly determined for real semisimple groups by Harish-Chandra.[3]
  4. The abstract functional analytic theory of zonal spherical functions was first developed by Roger Godement.[3]
  5. For semisimple p-adic Lie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake and Ian G. Macdonald.[3]
  6. Properties 2, 3 and 4 or properties 3, 4 and 5 characterize zonal spherical functions.[3]
  7. I tried to rotate zonal spherical function, projected onto spherical harmonic basis.[4]
  8. For zonal spherical functions see Spherical harmonics.[5]
  9. But to determine the explicitform of zonal spherical functions and the Plancherel measure, it seems necessary toknow the (infinite) matrix of this Fourier transformation more explicitly.[6]
  10. 3 zonal spherical functions k(x), k(y) = Gk(x, y), where Gk are Gegenbauer polynomials, zonal spherical functions associated with Harmk(Sd1).[7]
  11. 4 zonal spherical functions k(x), k(y) = Gk(x, y), where Gk are Gegenbauer polynomials, zonal spherical functions associated with Harmk(Sd1).[7]

소스

  1. ↑ Quantum zonal spherical functions and Macdonald polynomials
  2. ↑ Zonal spherical harmonics
  3. ↑ 3.0 3.1 3.2 3.3 Zonal spherical function
  4. ↑ Rotation of zonal spherical functions [closed]
  5. ↑ Encyclopedia of Mathematics
  6. ↑ Publications mathématiques de l’i.h.é.s.
  7. ↑ 7.0 7.1 Mapping to the space of spherical harmonics

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'cohen'}, {'OP': '*'}, {'LOWER': 'lenstra'}, {'LEMMA': 'heuristic'}]
  • [{'LOWER': 'cohen'}, {'OP': '*'}, {'LEMMA': 'Lenstra'}]
  • [{'LOWER': 'zonal'}, {'LOWER': 'spherical'}, {'LEMMA': 'function'}]

노트

말뭉치

  1. Previous work characterized certain left coideal subalgebras in the quantized enveloping algebra and established an appropriate framework for quantum zonal spherical functions.[1]
  2. The zonal spherical functions are a broad extension of the notion of zonal spherical harmonics to allow for a more general symmetry group.[2]
  3. Zonal spherical functions have been explicitly determined for real semisimple groups by Harish-Chandra.[3]
  4. The abstract functional analytic theory of zonal spherical functions was first developed by Roger Godement.[3]
  5. For semisimple p-adic Lie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake and Ian G. Macdonald.[3]
  6. Properties 2, 3 and 4 or properties 3, 4 and 5 characterize zonal spherical functions.[3]
  7. I tried to rotate zonal spherical function, projected onto spherical harmonic basis.[4]
  8. For zonal spherical functions see Spherical harmonics.[5]
  9. A more common usage of the phrase "spherical function" is as follows.[5]
  10. These solutions include the spherical functions associated with irreducible unitary representations.[5]
  11. S. ICHIRO SATAKE Theory of spherical functions on reductive algebraic groups over p-adic elds Publications mathmatiques de lI.H..[6]
  12. 69INTRODUCTIONThe theory of spherical functions on semi-simple Lie groups has been developedby Gelfand, Neumark, Harish-Chandra, Godement and others.[6]
  13. But to determine the explicitform of zonal spherical functions and the Plancherel measure, it seems necessary toknow the (infinite) matrix of this Fourier transformation more explicitly.[6]
  14. For g,eG, /eJf01, we put(5.7) (T,V)(^=/(^)-Then we have:248THEORY OF SPHERICAL FUNCTIONS 25PROPOSITION 5.1.[6]
  15. Am I simply making mistakes in my computation, or is there something wrong in this approach for spherical functions?[7]
  16. 3 zonal spherical functions k(x), k(y) = Gk(x, y), where Gk are Gegenbauer polynomials, zonal spherical functions associated with Harmk(Sd1).[8]
  17. 4 zonal spherical functions k(x), k(y) = Gk(x, y), where Gk are Gegenbauer polynomials, zonal spherical functions associated with Harmk(Sd1).[8]
  18. Let us recall rst what is a spherical function for a Gelfand pair.[9]
  19. Z G where is a bounded spherical function (m is a Haar measure on the group G, which is unimodular since (G, K) is a Gelfand pair).[9]
  20. We will denote the spherical functions of positive type for the Gelfand pair (G, K) (; x) ( , x G).[9]
  21. In the same way the spherical dual is identied with the set of spherical functions of positive type.[9]
  22. We present a sparse analytic representation for spherical functions, in- cluding those expressed in a spherical harmonic (SH) expansion, that is amenable to fast and accurate rotation on the GPU.[10]
  23. We present an alternative basis for spherical functions based on rotated zonal harmonic (ZH) lobes.[10]
  24. Let f () be a spherical function, with = (x, y, z) = (, ) S2, and and are spherical (lat-long) coordinates of point (x, y, z) on the spheres surface, S2.[10]
  25. The process of representing a spherical function, with an initial SH coefcient vector f , in the RZHB with a new coefcient vector z is called Zonal Harmonic Factorization (ZHF).[10]

소스

  1. ↑ Quantum zonal spherical functions and Macdonald polynomials
  2. ↑ Zonal spherical harmonics
  3. ↑ 3.0 3.1 3.2 3.3 Zonal spherical function
  4. ↑ Rotation of zonal spherical functions [closed]
  5. ↑ 5.0 5.1 5.2 Encyclopedia of Mathematics
  6. ↑ 6.0 6.1 6.2 6.3 Publications mathématiques de l’i.h.é.s.
  7. ↑ Expressing spherical functions with zonal harmonics
  8. ↑ 8.0 8.1 Mapping to the space of spherical harmonics
  9. ↑ 9.0 9.1 9.2 9.3 Asymptotics of spherical functions for large rank,
  10. ↑ 10.0 10.1 10.2 10.3 Sparse zonal harmonic factorization for efficient sh rotation
원본 주소 "https://wiki.mathnt.net/index.php?title=Zonal_spherical_function&oldid=52628"