25:
1323:
1537:
1446:
1010:
1486:
1399:
934:
690:
603:
253:
165:
1159:
1081:
643:
430:
357:
1188:
1664:
1570:
1359:
866:
833:
734:
1631:
1050:
963:
1716:
1692:
1598:
1212:
1105:
890:
1220:
1758:
proved that, when G is compact and abelian, a function f defined on a closed convex subset of the plane operates in A(G) if and only if f is real analytic. In 1969
462:
285:
1128:
792:
763:
549:
520:
491:
390:
314:
206:
99:
of these groups. The
Fourier–Stieltjes algebra and the Fourier–Stieltjes transform on the Fourier algebra of a locally compact group were introduced by
1013:
64:
1790:
1731:
1785:
1495:
1404:
968:
1891:
552:
1451:
1364:
899:
648:
561:
211:
123:
42:
35:
1133:
1055:
608:
395:
322:
1854:"Functions which Operate in the Fourier Algebra of a Discrete Group" Leonede de Michele; Paolo M. Soardi,
1020:
1804:
1780:
1164:
464:, the space of bounded continuous complex-valued functions on G with pointwise multiplication. We call
287:, the space of bounded continuous complex-valued functions on G with pointwise multiplication. We call
1636:
1542:
1331:
838:
805:
695:
1361:
of these functions is an algebra under pointwise multiplication is isomorphic to the measure algebra
1607:
1026:
939:
1697:
1673:
1579:
1193:
1086:
871:
100:
92:
81:
1318:{\displaystyle {\widehat {\mu }}(x)=\int _{\widehat {G}}{\overline {X(x)}}\,d\mu (X),\quad x\in G}
1829:
1751:
1747:
1896:
1601:
1489:
85:
1819:
1759:
435:
258:
1573:
1113:
768:
739:
525:
496:
467:
366:
290:
182:
117:
89:
1735:
1727:
172:
96:
46:
1885:
1833:
1762:
proved the result holds when G is compact and contains an infinite abelian subgroup.
1739:
893:
360:
1845:"Functions that Operate in the Fourier Algebra of a Compact Group" Charles F. Dunkl
1755:
1743:
1017:
168:
1726:
Let A(G) be the
Fourier algebra of a compact group G. Building upon the work of
1667:
176:
41:
The references used may be made clearer with a different or consistent style of
167:
is the space of all functions on Äś which are integrable with respect to the
1872:"Centralizers of the Fourier Algebra of an Amenable Group", P. F. Renaud,
1604:, to the Fourier–Stieltjes transform of a non-negative finite measure on
1877:
1868:
1859:
1850:
1824:
692:
function is just the
Fourier transform of that function, we have that
359:
for the measure algebra on Äś, meaning the space of all finite regular
1876:, Vol. 32, No. 2. (Apr., 1972), pp. 539–542. Stable URL:
1867:, Vol. 77, No. 1. (Oct., 1979), pp. 99–102. Stable URL:
1863:"Uniform Closures of Fourier-Stieltjes Algebras", Ching Chou,
18:
1858:, Vol. 45, No. 3. (Sep., 1974), pp. 389–392. Stable URL:
1849:, Vol. 21, No. 3. (Jun., 1969), pp. 540–544. Stable URL:
392:
to be the set of
Fourier-Stieltjes transforms of measures in
1703:
1679:
1648:
1585:
1554:
1516:
1465:
1425:
1378:
1343:
1199:
1141:
1092:
1063:
989:
913:
877:
850:
817:
669:
622:
582:
409:
336:
232:
144:
1803:
H. Helson; J.-P. Kahane; Y. Katznelson; W. Rudin (1959).
868:
be a
Fourier algebra such that the locally compact group
1670:
of the set of continuous positive-definite functions on
1700:
1676:
1639:
1610:
1582:
1545:
1539:
and its image is, by definition, the
Fourier algebra
1498:
1454:
1407:
1367:
1334:
1223:
1196:
1167:
1136:
1116:
1089:
1058:
1029:
971:
942:
902:
874:
841:
808:
771:
742:
698:
651:
611:
564:
528:
499:
470:
438:
398:
369:
325:
293:
261:
214:
185:
126:
1110:
The
Fourier–Stieltjes transform of a finite measure
208:
to be the set of
Fourier transforms of functions in
116:
Let G be a locally compact abelian group, and Äś the
1805:"The functions which operate on Fourier transforms"
1710:
1686:
1658:
1625:
1592:
1564:
1531:
1480:
1440:
1393:
1353:
1317:
1206:
1182:
1153:
1122:
1099:
1075:
1044:
1004:
957:
928:
884:
860:
827:
786:
757:
728:
684:
645:, and since the Fourier-Stieltjes transform of an
637:
597:
543:
514:
493:the Fourier-Stieltjes algebra of G. Equivalently,
485:
456:
424:
384:
351:
308:
279:
247:
200:
159:
1874:Proceedings of the American Mathematical Society
1865:Proceedings of the American Mathematical Society
1856:Proceedings of the American Mathematical Society
1847:Proceedings of the American Mathematical Society
175:structure where the product of two functions is
1532:{\displaystyle L_{1}({\widehat {\mathit {G}}})}
1441:{\displaystyle L_{1}({\widehat {\mathit {G}}})}
1005:{\displaystyle L_{1}({\widehat {\mathit {G}}})}
1633:if and only if it is positive definite. Thus,
936:be the measure algebra of finite measures on
522:can be defined as the linear span of the set
8:
1083:is the character group of the Abelian group
1481:{\displaystyle M({\widehat {\mathit {G}}})}
1394:{\displaystyle M({\widehat {\mathit {G}}})}
929:{\displaystyle M({\widehat {\mathit {G}}})}
685:{\displaystyle L_{1}({\hat {\mathit {G}}})}
598:{\displaystyle L_{1}({\hat {\mathit {G}}})}
248:{\displaystyle L_{1}({\hat {\mathit {G}}})}
160:{\displaystyle L_{1}({\hat {\mathit {G}}})}
1823:
1702:
1701:
1699:
1678:
1677:
1675:
1647:
1646:
1638:
1612:
1611:
1609:
1584:
1583:
1581:
1553:
1552:
1544:
1515:
1513:
1512:
1503:
1497:
1488:, the Fourier–Stieltjes transform is the
1464:
1462:
1461:
1453:
1424:
1422:
1421:
1412:
1406:
1377:
1375:
1374:
1366:
1342:
1341:
1333:
1286:
1265:
1253:
1252:
1225:
1224:
1222:
1198:
1197:
1195:
1169:
1168:
1166:
1140:
1138:
1137:
1135:
1115:
1091:
1090:
1088:
1062:
1060:
1059:
1057:
1031:
1030:
1028:
988:
986:
985:
976:
970:
944:
943:
941:
912:
910:
909:
901:
876:
875:
873:
849:
848:
840:
816:
815:
807:
770:
741:
697:
668:
666:
665:
656:
650:
621:
619:
618:
610:
581:
579:
578:
569:
563:
527:
498:
469:
437:
408:
406:
405:
397:
368:
335:
333:
332:
324:
292:
260:
231:
229:
228:
219:
213:
184:
143:
141:
140:
131:
125:
65:Learn how and when to remove this message
1154:{\displaystyle {\widehat {\mathit {G}}}}
1076:{\displaystyle {\widehat {\mathit {G}}}}
1771:
638:{\displaystyle M({\hat {\mathit {G}}})}
425:{\displaystyle M({\hat {\mathit {G}}})}
352:{\displaystyle M({\hat {\mathit {G}}})}
1722:Helson–Kahane–Katznelson–Rudin theorem
1694:. This definition is still valid when
1576:states that a measurable function on
95:. They play an important role in the
16:Algebras arising in harmonic analysis
7:
255:, and it is a closed sub-algebra of
835:be a Fourier–Stieltjes algebra and
14:
1183:{\displaystyle {\widehat {\mu }}}
1659:{\displaystyle B({\mathit {G}})}
1565:{\displaystyle A({\mathit {G}})}
1354:{\displaystyle B({\mathit {G}})}
861:{\displaystyle A({\mathit {G}})}
828:{\displaystyle B({\mathit {G}})}
729:{\displaystyle A(G)\subset B(G)}
432:. It is a closed sub-algebra of
23:
1305:
1653:
1643:
1626:{\displaystyle {\widehat {G}}}
1559:
1549:
1526:
1509:
1475:
1458:
1435:
1418:
1388:
1371:
1348:
1338:
1299:
1293:
1277:
1271:
1242:
1236:
1045:{\displaystyle {\widehat {G}}}
999:
982:
958:{\displaystyle {\widehat {G}}}
923:
906:
855:
845:
822:
812:
781:
775:
752:
746:
723:
717:
708:
702:
679:
673:
662:
632:
626:
615:
592:
586:
575:
538:
532:
509:
503:
480:
474:
451:
445:
419:
413:
402:
379:
373:
346:
340:
329:
303:
297:
274:
268:
242:
236:
225:
195:
189:
154:
148:
137:
1:
1711:{\displaystyle {\mathit {G}}}
1687:{\displaystyle {\mathit {G}}}
1593:{\displaystyle {\mathit {G}}}
1207:{\displaystyle {\mathit {G}}}
1100:{\displaystyle {\mathit {G}}}
885:{\displaystyle {\mathit {G}}}
1281:
1786:Encyclopedia of Mathematics
553:positive-definite functions
1913:
1448:, viewed as a subspace of
316:the Fourier algebra of G.
605:is naturally included in
1779:Renault, Jean (2001) ,
84:occur naturally in the
1712:
1688:
1666:can be defined as the
1660:
1627:
1594:
1566:
1533:
1482:
1442:
1395:
1355:
1319:
1208:
1184:
1155:
1124:
1101:
1077:
1046:
1006:
959:
930:
886:
862:
829:
788:
759:
730:
686:
639:
599:
545:
516:
487:
458:
426:
386:
353:
310:
281:
249:
202:
161:
1713:
1689:
1661:
1628:
1595:
1567:
1534:
1483:
1443:
1396:
1356:
1320:
1209:
1185:
1156:
1125:
1102:
1078:
1047:
1007:
960:
931:
887:
863:
830:
789:
765:is a closed ideal in
760:
731:
687:
640:
600:
546:
517:
488:
459:
457:{\displaystyle CB(G)}
427:
387:
354:
311:
282:
280:{\displaystyle CB(G)}
250:
203:
162:
1781:"Fourier-algebra(2)"
1698:
1674:
1637:
1608:
1580:
1543:
1496:
1452:
1405:
1365:
1332:
1221:
1194:
1165:
1134:
1123:{\displaystyle \mu }
1114:
1087:
1056:
1027:
969:
940:
900:
872:
839:
806:
787:{\displaystyle B(G)}
769:
758:{\displaystyle A(G)}
740:
696:
649:
609:
562:
544:{\displaystyle P(G)}
526:
515:{\displaystyle B(G)}
497:
486:{\displaystyle B(G)}
468:
436:
396:
385:{\displaystyle B(G)}
367:
323:
319:Similarly, we write
309:{\displaystyle A(G)}
291:
259:
212:
201:{\displaystyle A(G)}
183:
124:
1014:convolution algebra
171:on Äś, and it has a
1825:10.1007/bf02559571
1708:
1684:
1656:
1623:
1590:
1572:. The generalized
1562:
1529:
1478:
1438:
1391:
1351:
1315:
1204:
1180:
1151:
1120:
1097:
1073:
1042:
1002:
955:
926:
882:
858:
825:
784:
755:
726:
682:
635:
595:
541:
512:
483:
454:
422:
382:
349:
306:
277:
245:
198:
157:
1892:Harmonic analysis
1620:
1602:almost everywhere
1523:
1490:Fourier transform
1472:
1432:
1385:
1284:
1261:
1233:
1177:
1148:
1070:
1039:
996:
952:
920:
676:
629:
589:
416:
363:on Äś. We define
343:
239:
151:
86:harmonic analysis
75:
74:
67:
1904:
1838:
1837:
1827:
1818:(1–2): 135–157.
1812:Acta Mathematica
1809:
1800:
1794:
1793:
1776:
1718:is not Abelian.
1717:
1715:
1714:
1709:
1707:
1706:
1693:
1691:
1690:
1685:
1683:
1682:
1665:
1663:
1662:
1657:
1652:
1651:
1632:
1630:
1629:
1624:
1622:
1621:
1613:
1599:
1597:
1596:
1591:
1589:
1588:
1571:
1569:
1568:
1563:
1558:
1557:
1538:
1536:
1535:
1530:
1525:
1524:
1519:
1514:
1508:
1507:
1487:
1485:
1484:
1479:
1474:
1473:
1468:
1463:
1447:
1445:
1444:
1439:
1434:
1433:
1428:
1423:
1417:
1416:
1401:. Restricted to
1400:
1398:
1397:
1392:
1387:
1386:
1381:
1376:
1360:
1358:
1357:
1352:
1347:
1346:
1324:
1322:
1321:
1316:
1285:
1280:
1266:
1264:
1263:
1262:
1254:
1235:
1234:
1226:
1213:
1211:
1210:
1205:
1203:
1202:
1189:
1187:
1186:
1181:
1179:
1178:
1170:
1161:is the function
1160:
1158:
1157:
1152:
1150:
1149:
1144:
1139:
1129:
1127:
1126:
1121:
1106:
1104:
1103:
1098:
1096:
1095:
1082:
1080:
1079:
1074:
1072:
1071:
1066:
1061:
1051:
1049:
1048:
1043:
1041:
1040:
1032:
1011:
1009:
1008:
1003:
998:
997:
992:
987:
981:
980:
964:
962:
961:
956:
954:
953:
945:
935:
933:
932:
927:
922:
921:
916:
911:
891:
889:
888:
883:
881:
880:
867:
865:
864:
859:
854:
853:
834:
832:
831:
826:
821:
820:
793:
791:
790:
785:
764:
762:
761:
756:
735:
733:
732:
727:
691:
689:
688:
683:
678:
677:
672:
667:
661:
660:
644:
642:
641:
636:
631:
630:
625:
620:
604:
602:
601:
596:
591:
590:
585:
580:
574:
573:
550:
548:
547:
542:
521:
519:
518:
513:
492:
490:
489:
484:
463:
461:
460:
455:
431:
429:
428:
423:
418:
417:
412:
407:
391:
389:
388:
383:
358:
356:
355:
350:
345:
344:
339:
334:
315:
313:
312:
307:
286:
284:
283:
278:
254:
252:
251:
246:
241:
240:
235:
230:
224:
223:
207:
205:
204:
199:
166:
164:
163:
158:
153:
152:
147:
142:
136:
135:
97:duality theories
70:
63:
59:
56:
50:
27:
26:
19:
1912:
1911:
1907:
1906:
1905:
1903:
1902:
1901:
1882:
1881:
1842:
1841:
1807:
1802:
1801:
1797:
1778:
1777:
1773:
1768:
1724:
1696:
1695:
1672:
1671:
1635:
1634:
1606:
1605:
1578:
1577:
1574:Bochner theorem
1541:
1540:
1499:
1494:
1493:
1450:
1449:
1408:
1403:
1402:
1363:
1362:
1330:
1329:
1267:
1248:
1219:
1218:
1192:
1191:
1163:
1162:
1132:
1131:
1112:
1111:
1085:
1084:
1054:
1053:
1025:
1024:
972:
967:
966:
938:
937:
898:
897:
870:
869:
837:
836:
804:
803:
800:
767:
766:
738:
737:
694:
693:
652:
647:
646:
607:
606:
565:
560:
559:
524:
523:
495:
494:
466:
465:
434:
433:
394:
393:
365:
364:
321:
320:
289:
288:
257:
256:
215:
210:
209:
181:
180:
127:
122:
121:
114:
109:
90:locally compact
71:
60:
54:
51:
40:
34:has an unclear
28:
24:
17:
12:
11:
5:
1910:
1908:
1900:
1899:
1894:
1884:
1883:
1880:
1879:
1870:
1861:
1852:
1840:
1839:
1795:
1770:
1769:
1767:
1764:
1723:
1720:
1705:
1681:
1655:
1650:
1645:
1642:
1619:
1616:
1587:
1561:
1556:
1551:
1548:
1528:
1522:
1518:
1511:
1506:
1502:
1477:
1471:
1467:
1460:
1457:
1437:
1431:
1427:
1420:
1415:
1411:
1390:
1384:
1380:
1373:
1370:
1350:
1345:
1340:
1337:
1326:
1325:
1314:
1311:
1308:
1304:
1301:
1298:
1295:
1292:
1289:
1283:
1279:
1276:
1273:
1270:
1260:
1257:
1251:
1247:
1244:
1241:
1238:
1232:
1229:
1201:
1176:
1173:
1147:
1143:
1119:
1094:
1069:
1065:
1038:
1035:
1001:
995:
991:
984:
979:
975:
951:
948:
925:
919:
915:
908:
905:
879:
857:
852:
847:
844:
824:
819:
814:
811:
799:
796:
783:
780:
777:
774:
754:
751:
748:
745:
725:
722:
719:
716:
713:
710:
707:
704:
701:
681:
675:
671:
664:
659:
655:
634:
628:
624:
617:
614:
594:
588:
584:
577:
572:
568:
551:of continuous
540:
537:
534:
531:
511:
508:
505:
502:
482:
479:
476:
473:
453:
450:
447:
444:
441:
421:
415:
411:
404:
401:
381:
378:
375:
372:
361:Borel measures
348:
342:
338:
331:
328:
305:
302:
299:
296:
276:
273:
270:
267:
264:
244:
238:
234:
227:
222:
218:
197:
194:
191:
188:
173:Banach algebra
156:
150:
146:
139:
134:
130:
113:
110:
108:
105:
73:
72:
36:citation style
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
1909:
1898:
1895:
1893:
1890:
1889:
1887:
1878:
1875:
1871:
1869:
1866:
1862:
1860:
1857:
1853:
1851:
1848:
1844:
1843:
1835:
1831:
1826:
1821:
1817:
1813:
1806:
1799:
1796:
1792:
1788:
1787:
1782:
1775:
1772:
1765:
1763:
1761:
1757:
1753:
1749:
1745:
1741:
1737:
1733:
1729:
1721:
1719:
1669:
1640:
1617:
1614:
1603:
1575:
1546:
1520:
1504:
1500:
1491:
1469:
1455:
1429:
1413:
1409:
1382:
1368:
1335:
1312:
1309:
1306:
1302:
1296:
1290:
1287:
1274:
1268:
1258:
1255:
1249:
1245:
1239:
1230:
1227:
1217:
1216:
1215:
1174:
1171:
1145:
1117:
1108:
1067:
1036:
1033:
1022:
1019:
1015:
993:
977:
973:
949:
946:
917:
903:
895:
842:
809:
797:
795:
778:
772:
749:
743:
720:
714:
711:
705:
699:
657:
653:
612:
570:
566:
556:
554:
535:
529:
506:
500:
477:
471:
448:
442:
439:
399:
376:
370:
362:
326:
317:
300:
294:
271:
265:
262:
220:
216:
192:
186:
178:
174:
170:
132:
128:
119:
111:
106:
104:
102:
101:Pierre Eymard
98:
94:
91:
87:
83:
79:
69:
66:
58:
55:February 2012
48:
44:
38:
37:
32:This article
30:
21:
20:
1873:
1864:
1855:
1846:
1815:
1811:
1798:
1784:
1774:
1725:
1327:
1109:
801:
557:
318:
179:. We define
169:Haar measure
115:
80:and related
77:
76:
61:
52:
33:
1668:linear span
1214:defined by
736:. In fact,
177:convolution
120:of G. Then
1886:Categories
1766:References
1752:Katznelson
1742:, in 1959
1600:is equal,
1328:The space
1018:integrable
118:dual group
107:Definition
47:footnoting
1834:121739671
1791:EMS Press
1618:^
1521:^
1470:^
1430:^
1383:^
1310:∈
1291:μ
1282:¯
1259:^
1250:∫
1231:^
1228:μ
1175:^
1172:μ
1146:^
1118:μ
1068:^
1037:^
1021:functions
994:^
950:^
918:^
712:⊂
674:^
627:^
587:^
414:^
341:^
237:^
149:^
103:in 1964.
1897:Algebras
1740:Beurling
1052:, where
965:and let
112:Informal
82:algebras
43:citation
1736:Gelfand
1012:be the
894:abelian
78:Fourier
1832:
1754:, and
1748:Kahane
1744:Helson
1738:, and
1728:Wiener
896:. Let
798:Formal
558:Since
555:on G.
93:groups
1830:S2CID
1808:(PDF)
1760:Dunkl
1756:Rudin
1732:LĂ©vy
802:Let
45:and
1820:doi
1816:102
1492:on
1190:on
1130:on
1023:on
1016:of
892:is
88:of
1888::
1828:.
1814:.
1810:.
1789:,
1783:,
1750:,
1746:,
1734:,
1730:,
1107:.
794:.
1836:.
1822::
1704:G
1680:G
1654:)
1649:G
1644:(
1641:B
1615:G
1586:G
1560:)
1555:G
1550:(
1547:A
1527:)
1517:G
1510:(
1505:1
1501:L
1476:)
1466:G
1459:(
1456:M
1436:)
1426:G
1419:(
1414:1
1410:L
1389:)
1379:G
1372:(
1369:M
1349:)
1344:G
1339:(
1336:B
1313:G
1307:x
1303:,
1300:)
1297:X
1294:(
1288:d
1278:)
1275:x
1272:(
1269:X
1256:G
1246:=
1243:)
1240:x
1237:(
1200:G
1142:G
1093:G
1064:G
1034:G
1000:)
990:G
983:(
978:1
974:L
947:G
924:)
914:G
907:(
904:M
878:G
856:)
851:G
846:(
843:A
823:)
818:G
813:(
810:B
782:)
779:G
776:(
773:B
753:)
750:G
747:(
744:A
724:)
721:G
718:(
715:B
709:)
706:G
703:(
700:A
680:)
670:G
663:(
658:1
654:L
633:)
623:G
616:(
613:M
593:)
583:G
576:(
571:1
567:L
539:)
536:G
533:(
530:P
510:)
507:G
504:(
501:B
481:)
478:G
475:(
472:B
452:)
449:G
446:(
443:B
440:C
420:)
410:G
403:(
400:M
380:)
377:G
374:(
371:B
347:)
337:G
330:(
327:M
304:)
301:G
298:(
295:A
275:)
272:G
269:(
266:B
263:C
243:)
233:G
226:(
221:1
217:L
196:)
193:G
190:(
187:A
155:)
145:G
138:(
133:1
129:L
68:)
62:(
57:)
53:(
49:.
39:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.