975:
698:
740:
1257:
1617:
491:
1728:
535:
1413:
1133:
1107:
1493:
970:{\displaystyle {\frac {1}{|\mathbf {x} -\mathbf {y} |^{n-2}}}=\sum _{k=0}^{\infty }c_{n,k}{\frac {|\mathbf {x} |^{k}}{|\mathbf {y} |^{n+k-2}}}Z_{\mathbf {x} /|\mathbf {x} |}^{(k)}(\mathbf {y} /|\mathbf {y} |)}
132:
375:
202:
1634:
261:
1324:
315:
356:
1310:
731:
1009:
693:{\displaystyle {\frac {1}{\omega _{n-1}}}{\frac {1-r^{2}}{|\mathbf {x} -r\mathbf {y} |^{n}}}=\sum _{k=0}^{\infty }r^{k}Z_{\mathbf {x} }^{(k)}(\mathbf {y} ),}
61:
1784:
54:
On the two-dimensional sphere, the unique zonal spherical harmonic of degree ℓ invariant under rotations fixing the north pole is represented in
1760:
1252:{\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )={\frac {n+2\ell -2}{n-2}}C_{\ell }^{(\alpha )}(\mathbf {x} \cdot \mathbf {y} )}
324:
156:
1612:{\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )=\sum _{k=1}^{d}Y_{k}(\mathbf {x} ){\overline {Y_{k}(\mathbf {y} )}}.}
1779:
318:
1116:
486:{\displaystyle Y(\mathbf {x} )=\int _{S^{n-1}}Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )Y(\mathbf {y} )\,d\Omega (y)}
215:
280:
44:
1277:
55:
1723:{\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {x} )=\omega _{n-1}^{-1}\dim \mathbf {H} _{\ell }.}
1408:{\displaystyle Z_{R\mathbf {x} }^{(\ell )}(R\mathbf {y} )=Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )}
703:
146:
40:
32:
369:
212:
is the variable of the function. This can be obtained by rotation of the basic zonal harmonic
1756:
1476:
47:
are a broad extension of the notion of zonal spherical harmonics to allow for a more general
1749:
517:
48:
1773:
1112:
734:
359:
269:-dimensional Euclidean space, zonal spherical harmonics are defined as follows. Let
1452:, and that satisfies this invariance property, is a constant multiple of the degree
733:
is the surface area of the (n-1)-dimensional sphere. They are also related to the
1744:
1740:
28:
17:
508:. The integral is taken with respect to the invariant probability measure.
43:
that are invariant under the rotation through a particular fixed axis. The
1102:{\displaystyle c_{n,k}={\frac {1}{\omega _{n-1}}}{\frac {2k+n-2}{(n-2)}}.}
1119:. Thus, the zonal spherical harmonics can be expressed as follows. If
1321:
The zonal spherical harmonics are rotationally invariant, meaning that
1115:
of the Newton kernel (with suitable normalization) are precisely the
368:
of spherical harmonics of degree ℓ. In other words, the following
127:{\displaystyle Z^{(\ell )}(\theta ,\phi )=P_{\ell }(\cos \theta )}
153:. The general zonal spherical harmonic of degree ℓ is denoted by
516:
The zonal harmonics appear naturally as coefficients of the
208:
is a point on the sphere representing the fixed axis, and
197:{\displaystyle Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )}
1637:
1496:
1327:
1280:
1136:
1012:
743:
706:
538:
378:
327:
283:
218:
159:
64:
1751:
Introduction to
Fourier Analysis on Euclidean Spaces
1748:
1722:
1611:
1407:
1304:
1251:
1101:
969:
725:
692:
485:
350:
309:
255:
196:
126:
1755:, Princeton, N.J.: Princeton University Press,
1312:is the ultraspherical polynomial of degree ℓ.
8:
256:{\displaystyle Z^{(\ell )}(\theta ,\phi ).}
310:{\displaystyle Z_{\mathbf {x} }^{(\ell )}}
1711:
1706:
1690:
1679:
1664:
1649:
1643:
1642:
1636:
1592:
1583:
1576:
1568:
1559:
1549:
1538:
1523:
1508:
1502:
1501:
1495:
1397:
1382:
1376:
1375:
1360:
1342:
1336:
1332:
1326:
1290:
1285:
1279:
1241:
1233:
1218:
1213:
1174:
1163:
1148:
1142:
1141:
1135:
1055:
1041:
1032:
1017:
1011:
959:
954:
949:
944:
939:
924:
918:
913:
908:
903:
898:
897:
872:
867:
861:
856:
848:
843:
837:
832:
829:
817:
807:
796:
774:
769:
763:
755:
750:
744:
742:
711:
705:
679:
664:
658:
657:
647:
637:
626:
610:
605:
599:
588:
583:
575:
562:
548:
539:
537:
467:
459:
445:
430:
424:
423:
405:
400:
385:
377:
340:
326:
295:
289:
288:
282:
223:
217:
186:
171:
165:
164:
158:
103:
69:
63:
351:{\displaystyle P\mapsto P(\mathbf {x} )}
1305:{\displaystyle C_{\ell }^{(\alpha )}}
512:Relationship with harmonic potentials
7:
1415:for every orthogonal transformation
808:
638:
471:
25:
1785:Special hypergeometric functions
1707:
1665:
1644:
1593:
1569:
1524:
1503:
1444:that is a spherical harmonic in
1398:
1377:
1361:
1337:
1242:
1234:
1164:
1143:
955:
940:
914:
899:
862:
838:
764:
756:
680:
659:
600:
589:
460:
446:
425:
386:
341:
290:
187:
166:
1669:
1661:
1656:
1650:
1597:
1589:
1573:
1565:
1528:
1520:
1515:
1509:
1402:
1394:
1389:
1383:
1365:
1354:
1349:
1343:
1297:
1291:
1246:
1230:
1225:
1219:
1168:
1160:
1155:
1149:
1090:
1078:
964:
960:
950:
936:
931:
925:
919:
909:
868:
857:
844:
833:
770:
751:
684:
676:
671:
665:
606:
584:
480:
474:
464:
456:
450:
442:
437:
431:
390:
382:
345:
337:
331:
302:
296:
247:
235:
230:
224:
191:
183:
178:
172:
121:
109:
93:
81:
76:
70:
1:
726:{\displaystyle \omega _{n-1}}
1601:
1419:. Conversely, any function
1274:are the constants above and
1801:
1117:ultraspherical polynomials
358:in the finite-dimensional
321:of the linear functional
45:zonal spherical functions
37:zonal spherical harmonics
1111:The coefficients of the
1724:
1613:
1554:
1409:
1306:
1253:
1103:
971:
812:
727:
694:
642:
487:
352:
311:
257:
198:
128:
1725:
1614:
1534:
1410:
1307:
1254:
1104:
972:
792:
728:
695:
622:
520:for the unit ball in
488:
353:
312:
258:
199:
129:
56:spherical coordinates
1635:
1494:
1325:
1278:
1134:
1010:
741:
704:
536:
376:
370:reproducing property
325:
281:
277:−1)-sphere. Define
216:
157:
62:
1780:Rotational symmetry
1698:
1660:
1519:
1393:
1353:
1301:
1229:
1159:
935:
675:
441:
319:dual representation
306:
273:be a point on the (
182:
147:Legendre polynomial
41:spherical harmonics
33:rotational symmetry
1720:
1675:
1638:
1609:
1497:
1405:
1371:
1328:
1302:
1281:
1249:
1209:
1137:
1099:
991:and the constants
967:
893:
723:
690:
653:
483:
419:
348:
307:
284:
253:
194:
160:
124:
1762:978-0-691-08078-9
1604:
1477:orthonormal basis
1207:
1094:
1053:
891:
787:
617:
560:
16:(Redirected from
1792:
1765:
1754:
1729:
1727:
1726:
1721:
1716:
1715:
1710:
1697:
1689:
1668:
1659:
1648:
1647:
1630:
1618:
1616:
1615:
1610:
1605:
1600:
1596:
1588:
1587:
1577:
1572:
1564:
1563:
1553:
1548:
1527:
1518:
1507:
1506:
1489:
1455:
1443:
1433:
1414:
1412:
1411:
1406:
1401:
1392:
1381:
1380:
1364:
1352:
1341:
1340:
1311:
1309:
1308:
1303:
1300:
1289:
1273:
1258:
1256:
1255:
1250:
1245:
1237:
1228:
1217:
1208:
1206:
1195:
1175:
1167:
1158:
1147:
1146:
1129:
1108:
1106:
1105:
1100:
1095:
1093:
1076:
1056:
1054:
1052:
1051:
1033:
1028:
1027:
1005:
990:
976:
974:
973:
968:
963:
958:
953:
948:
943:
934:
923:
922:
917:
912:
907:
902:
892:
890:
889:
888:
871:
865:
860:
854:
853:
852:
847:
841:
836:
830:
828:
827:
811:
806:
788:
786:
785:
784:
773:
767:
759:
754:
745:
732:
730:
729:
724:
722:
721:
699:
697:
696:
691:
683:
674:
663:
662:
652:
651:
641:
636:
618:
616:
615:
614:
609:
603:
592:
587:
581:
580:
579:
563:
561:
559:
558:
540:
507:
492:
490:
489:
484:
463:
449:
440:
429:
428:
418:
417:
416:
415:
389:
357:
355:
354:
349:
344:
316:
314:
313:
308:
305:
294:
293:
262:
260:
259:
254:
234:
233:
203:
201:
200:
195:
190:
181:
170:
169:
152:
144:
133:
131:
130:
125:
108:
107:
80:
79:
21:
1800:
1799:
1795:
1794:
1793:
1791:
1790:
1789:
1770:
1769:
1763:
1739:
1736:
1705:
1633:
1632:
1622:
1579:
1578:
1555:
1492:
1491:
1488:
1480:
1474:
1465:
1456:zonal harmonic.
1453:
1448:for each fixed
1435:
1420:
1323:
1322:
1318:
1276:
1275:
1272:
1260:
1196:
1176:
1132:
1131:
1120:
1077:
1057:
1037:
1013:
1008:
1007:
1004:
992:
978:
866:
855:
842:
831:
813:
768:
749:
739:
738:
707:
702:
701:
643:
604:
582:
571:
564:
544:
534:
533:
514:
506:
494:
401:
396:
374:
373:
367:
323:
322:
279:
278:
219:
214:
213:
155:
154:
150:
143:
135:
99:
65:
60:
59:
23:
22:
18:Zonal harmonics
15:
12:
11:
5:
1798:
1796:
1788:
1787:
1782:
1772:
1771:
1768:
1767:
1761:
1735:
1732:
1731:
1730:
1719:
1714:
1709:
1704:
1701:
1696:
1693:
1688:
1685:
1682:
1678:
1674:
1671:
1667:
1663:
1658:
1655:
1652:
1646:
1641:
1621:Evaluating at
1619:
1608:
1603:
1599:
1595:
1591:
1586:
1582:
1575:
1571:
1567:
1562:
1558:
1552:
1547:
1544:
1541:
1537:
1533:
1530:
1526:
1522:
1517:
1514:
1511:
1505:
1500:
1484:
1470:
1463:
1457:
1404:
1400:
1396:
1391:
1388:
1385:
1379:
1374:
1370:
1367:
1363:
1359:
1356:
1351:
1348:
1345:
1339:
1335:
1331:
1317:
1314:
1299:
1296:
1293:
1288:
1284:
1264:
1248:
1244:
1240:
1236:
1232:
1227:
1224:
1221:
1216:
1212:
1205:
1202:
1199:
1194:
1191:
1188:
1185:
1182:
1179:
1173:
1170:
1166:
1162:
1157:
1154:
1151:
1145:
1140:
1098:
1092:
1089:
1086:
1083:
1080:
1075:
1072:
1069:
1066:
1063:
1060:
1050:
1047:
1044:
1040:
1036:
1031:
1026:
1023:
1020:
1016:
996:
966:
962:
957:
952:
947:
942:
938:
933:
930:
927:
921:
916:
911:
906:
901:
896:
887:
884:
881:
878:
875:
870:
864:
859:
851:
846:
840:
835:
826:
823:
820:
816:
810:
805:
802:
799:
795:
791:
783:
780:
777:
772:
766:
762:
758:
753:
748:
720:
717:
714:
710:
689:
686:
682:
678:
673:
670:
667:
661:
656:
650:
646:
640:
635:
632:
629:
625:
621:
613:
608:
602:
598:
595:
591:
586:
578:
574:
570:
567:
557:
554:
551:
547:
543:
532:unit vectors,
518:Poisson kernel
513:
510:
502:
482:
479:
476:
473:
470:
466:
462:
458:
455:
452:
448:
444:
439:
436:
433:
427:
422:
414:
411:
408:
404:
399:
395:
392:
388:
384:
381:
365:
347:
343:
339:
336:
333:
330:
304:
301:
298:
292:
287:
252:
249:
246:
243:
240:
237:
232:
229:
226:
222:
193:
189:
185:
180:
177:
174:
168:
163:
139:
123:
120:
117:
114:
111:
106:
102:
98:
95:
92:
89:
86:
83:
78:
75:
72:
68:
49:symmetry group
24:
14:
13:
10:
9:
6:
4:
3:
2:
1797:
1786:
1783:
1781:
1778:
1777:
1775:
1764:
1758:
1753:
1752:
1746:
1742:
1738:
1737:
1733:
1717:
1712:
1702:
1699:
1694:
1691:
1686:
1683:
1680:
1676:
1672:
1653:
1639:
1629:
1625:
1620:
1606:
1584:
1580:
1560:
1556:
1550:
1545:
1542:
1539:
1535:
1531:
1512:
1498:
1487:
1483:
1478:
1473:
1469:
1462:
1458:
1451:
1447:
1442:
1438:
1431:
1427:
1423:
1418:
1386:
1372:
1368:
1357:
1346:
1333:
1329:
1320:
1319:
1315:
1313:
1294:
1286:
1282:
1271:
1267:
1263:
1238:
1222:
1214:
1210:
1203:
1200:
1197:
1192:
1189:
1186:
1183:
1180:
1177:
1171:
1152:
1138:
1127:
1123:
1118:
1114:
1113:Taylor series
1109:
1096:
1087:
1084:
1081:
1073:
1070:
1067:
1064:
1061:
1058:
1048:
1045:
1042:
1038:
1034:
1029:
1024:
1021:
1018:
1014:
1006:are given by
1003:
999:
995:
989:
985:
981:
945:
928:
904:
894:
885:
882:
879:
876:
873:
849:
824:
821:
818:
814:
803:
800:
797:
793:
789:
781:
778:
775:
760:
746:
736:
735:Newton kernel
718:
715:
712:
708:
687:
668:
654:
648:
644:
633:
630:
627:
623:
619:
611:
596:
593:
576:
572:
568:
565:
555:
552:
549:
545:
541:
531:
527:
523:
519:
511:
509:
505:
501:
497:
477:
468:
453:
434:
420:
412:
409:
406:
402:
397:
393:
379:
371:
364:
361:
360:Hilbert space
334:
328:
320:
299:
285:
276:
272:
268:
263:
250:
244:
241:
238:
227:
220:
211:
207:
175:
161:
148:
142:
138:
118:
115:
112:
104:
100:
96:
90:
87:
84:
73:
66:
57:
52:
50:
46:
42:
38:
34:
30:
19:
1750:
1745:Weiss, Guido
1741:Stein, Elias
1627:
1623:
1485:
1481:
1471:
1467:
1460:
1449:
1445:
1440:
1436:
1429:
1425:
1421:
1416:
1269:
1265:
1261:
1125:
1121:
1110:
1001:
997:
993:
987:
983:
979:
529:
525:
521:
515:
503:
499:
495:
362:
274:
270:
266:
264:
209:
205:
140:
136:
53:
39:are special
36:
29:mathematical
26:
1774:Categories
1734:References
1316:Properties
317:to be the
149:of degree
1713:ℓ
1703:
1692:−
1684:−
1677:ω
1654:ℓ
1602:¯
1536:∑
1513:ℓ
1387:ℓ
1347:ℓ
1295:α
1287:ℓ
1239:⋅
1223:α
1215:ℓ
1201:−
1190:−
1187:ℓ
1153:ℓ
1085:−
1071:−
1046:−
1039:ω
883:−
809:∞
794:∑
779:−
761:−
716:−
709:ω
639:∞
624:∑
594:−
569:−
553:−
546:ω
472:Ω
435:ℓ
410:−
398:∫
332:↦
300:ℓ
245:ϕ
239:θ
228:ℓ
176:ℓ
119:θ
116:
105:ℓ
91:ϕ
85:θ
74:ℓ
31:study of
1747:(1971),
986:∈
498:∈
493:for all
204:, where
1490:, then
1466:, ...,
1130:, then
372:holds:
27:In the
1759:
1631:gives
1475:is an
1259:where
977:where
700:where
524:: for
134:where
35:, the
1128:−2)/2
145:is a
1757:ISBN
737:via
528:and
1700:dim
1479:of
1459:If
1434:on
1124:= (
265:In
113:cos
58:by
1776::
1743:;
1626:=
51:.
1766:.
1718:.
1708:H
1695:1
1687:1
1681:n
1673:=
1670:)
1666:x
1662:(
1657:)
1651:(
1645:x
1640:Z
1628:y
1624:x
1607:.
1598:)
1594:y
1590:(
1585:k
1581:Y
1574:)
1570:x
1566:(
1561:k
1557:Y
1551:d
1546:1
1543:=
1540:k
1532:=
1529:)
1525:y
1521:(
1516:)
1510:(
1504:x
1499:Z
1486:ℓ
1482:H
1472:d
1468:Y
1464:1
1461:Y
1454:ℓ
1450:x
1446:y
1441:S
1439:×
1437:S
1432:)
1430:y
1428:,
1426:x
1424:(
1422:f
1417:R
1403:)
1399:y
1395:(
1390:)
1384:(
1378:x
1373:Z
1369:=
1366:)
1362:y
1358:R
1355:(
1350:)
1344:(
1338:x
1334:R
1330:Z
1298:)
1292:(
1283:C
1270:ℓ
1268:,
1266:n
1262:c
1247:)
1243:y
1235:x
1231:(
1226:)
1220:(
1211:C
1204:2
1198:n
1193:2
1184:2
1181:+
1178:n
1172:=
1169:)
1165:y
1161:(
1156:)
1150:(
1144:x
1139:Z
1126:n
1122:α
1097:.
1091:)
1088:2
1082:n
1079:(
1074:2
1068:n
1065:+
1062:k
1059:2
1049:1
1043:n
1035:1
1030:=
1025:k
1022:,
1019:n
1015:c
1002:k
1000:,
998:n
994:c
988:R
984:y
982:,
980:x
965:)
961:|
956:y
951:|
946:/
941:y
937:(
932:)
929:k
926:(
920:|
915:x
910:|
905:/
900:x
895:Z
886:2
880:k
877:+
874:n
869:|
863:y
858:|
850:k
845:|
839:x
834:|
825:k
822:,
819:n
815:c
804:0
801:=
798:k
790:=
782:2
776:n
771:|
765:y
757:x
752:|
747:1
719:1
713:n
688:,
685:)
681:y
677:(
672:)
669:k
666:(
660:x
655:Z
649:k
645:r
634:0
631:=
628:k
620:=
612:n
607:|
601:y
597:r
590:x
585:|
577:2
573:r
566:1
556:1
550:n
542:1
530:y
526:x
522:R
504:ℓ
500:H
496:Y
481:)
478:y
475:(
469:d
465:)
461:y
457:(
454:Y
451:)
447:y
443:(
438:)
432:(
426:x
421:Z
413:1
407:n
403:S
394:=
391:)
387:x
383:(
380:Y
366:ℓ
363:H
346:)
342:x
338:(
335:P
329:P
303:)
297:(
291:x
286:Z
275:n
271:x
267:n
251:.
248:)
242:,
236:(
231:)
225:(
221:Z
210:y
206:x
192:)
188:y
184:(
179:)
173:(
167:x
162:Z
151:ℓ
141:ℓ
137:P
122:)
110:(
101:P
97:=
94:)
88:,
82:(
77:)
71:(
67:Z
20:)
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