385:
933:
47:
lie in a given subset of the real line and whose imaginary part is unconstrained; likewise, a tube is the set of complex vectors whose real part is in some given collection of real vectors, and whose imaginary part is unconstrained.
1047:
612:
1211:
769:
511:
165:
64:
1295:
75:
from one variable continues to hold, and characterizes the elements of Hardy spaces as the
Laplace transforms of functions with appropriate integrability properties. Tubes over
816:
626:
948:
92:
566:
1147:
687:
432:
1570:
380:{\displaystyle a=(z_{1},\dots ,z_{n})=(x_{1}+iy_{1},\dots ,x_{n}+iy_{n})=(x_{1},\dots ,x_{n})+i(y_{1},\dots ,y_{n})=x+iy.}
1644:
1540:
1470:
1246:
1465:
17:
1639:
87:
have an especially rich structure, so that precise results are known concerning the boundary values of
1634:
522:
40:
116:
72:
928:{\displaystyle \sup _{x\in A}\int _{\mathbb {R} ^{n}}|f(t)|^{2}e^{-4\pi x\cdot t}\,dt<\infty .}
622:
1460:
140:
1500:
1361:
Some conventions instead define a tube to be a domain such that the imaginary part lies in
666:
618:
532:
80:
52:
8:
1504:
1611:
1559:
1516:
1490:
1597:
1566:
1536:
1512:
120:
104:
56:
1528:
1520:
1601:
1593:
1576:
1508:
1337:
108:
100:
1584:
Carmignani, Robert (1973). "Envelopes of
Holomorphy and Holomorphic Convexity".
1550:
152:
112:
1628:
1342:
36:
1478:
1042:{\displaystyle F(x+iy)=\int _{\mathbb {R} ^{n}}e^{2\pi z\cdot t}f(t)\,dt.}
1554:
650:
545:
68:
60:
24:
1321:), but it requires additional regularity of the cone (specifically, the
43:. A strip can be thought of as the collection of complex numbers whose
1615:
1495:
1126:
124:
84:
76:
32:
1606:
607:{\displaystyle \operatorname {ch} \,T_{A}=T_{\operatorname {ch} \,A}.}
1322:
1206:{\displaystyle x\in A\implies tx\in A\ \ \ {\text{for all}}\ t>0.}
96:
44:
764:{\displaystyle \int _{\mathbb {R} ^{n}}|F(x+iy)|^{p}\,dy<\infty }
1123:
642:
531:
is a connected open set. Then any complex-valued function that is
395:
506:{\displaystyle T_{A}=\{z=x+iy\in \mathbb {C} ^{n}\mid x\in A\}.}
71:
on tubes can be defined in a manner in which a version of the
544:
can be extended uniquely to a holomorphic function on the
95:
is the tube domain associated to the interior of the past
31:
is a generalization of the notion of a vertical strip (or
625:), a convex tube is also a domain of holomorphy. So the
1438:
1436:
1399:
1397:
1533:
Introduction to complex analysis in several variables
1433:
1421:
1249:
1150:
951:
819:
690:
569:
435:
168:
1561:
Introduction to
Fourier Analysis on Euclidean Spaces
1394:
422:consisting of all elements whose real parts lie in
1558:
1382:
1289:
1205:
1041:
927:
763:
606:
505:
379:
1586:Transactions of the American Mathematical Society
1409:
516:
159:can be decomposed into real and imaginary parts:
1626:
1251:
821:
1565:, Princeton, N.J.: Princeton University Press,
1098:) contains a nonzero function if and only if
1085:A corollary of this characterization is that
16:For other uses of "tube" in mathematics, see
1137:, so does the entire ray from the origin to
497:
449:
1583:
1549:
1481:(2000), "Holography and the future tube",
1442:
1427:
1366:
1164:
1160:
1605:
1527:
1494:
1403:
1029:
983:
909:
843:
748:
698:
629:of any tube is equal to its convex hull.
595:
573:
475:
91:functions. In mathematical physics, the
83:. The Hardy spaces on tubes over convex
18:Tube (disambiguation) § Mathematics
1477:
1388:
802:) can be characterized as follows. Let
155:coordinate space. Then any element of
1627:
1458:
1415:
1308:). There is an analogous result for
1290:{\displaystyle \lim _{y\to 0}F(x+iy)}
111:. Certain tubes over cones support a
1328:* needs to have nonempty interior).
560:, which is also a tube, and in fact
1105:
13:
1240:boundary limits in the sense that
919:
758:
65:multidimensional Laplace transform
14:
1656:
1598:10.1090/S0002-9947-1973-0316748-1
1069:). Conversely, every element of
938:The Fourier–Laplace transform of
1220:is a cone, then the elements of
806:be a complex-valued function on
1056:is well-defined and belongs to
632:
617:Since any convex open set is a
1355:
1284:
1269:
1258:
1161:
1026:
1020:
970:
955:
874:
869:
863:
856:
738:
733:
718:
711:
517:Tubes as domains of holomorphy
356:
324:
315:
283:
277:
213:
207:
175:
115:in terms of which they become
1:
1483:Classical and Quantum Gravity
789: = 2, functions in
130:
1376:
7:
1535:, New York: North-Holland,
1466:Encyclopedia of Mathematics
1331:
1102:contains no straight line.
10:
1661:
1513:10.1088/0264-9381/17/5/316
1451:
1114:be an open convex cone in
520:
103:, and has applications in
15:
1645:Several complex variables
117:bounded symmetric domains
59:of a function of several
41:several complex variables
1348:
123:which is fundamental in
785:In the special case of
119:. One of these is the
1459:Chirka, E.M. (2001) ,
1443:Stein & Weiss 1971
1367:Stein & Weiss 1971
1291:
1207:
1043:
929:
765:
623:holomorphically convex
608:
523:Bochner's tube theorem
507:
381:
1292:
1208:
1044:
930:
766:
667:holomorphic functions
609:
508:
382:
141:real coordinate space
81:domains of holomorphy
1247:
1148:
1129:such that, whenever
949:
817:
688:
665:) is the set of all
627:holomorphic envelope
619:domain of holomorphy
567:
433:
166:
73:Paley–Wiener theorem
1505:2000CQGra..17.1071G
1118:. This means that
418:, is the subset of
1287:
1265:
1203:
1039:
925:
835:
761:
604:
503:
377:
1640:Harmonic analysis
1572:978-0-691-08078-9
1250:
1193:
1189:
1185:
1182:
1179:
1141:. Symbolically,
1082:) has this form.
820:
121:Siegel half-space
105:relativity theory
57:Laplace transform
51:Tube domains are
1652:
1635:Fourier analysis
1619:
1609:
1579:
1577:Internet Archive
1564:
1545:
1523:
1498:
1489:(5): 1071–1079,
1473:
1446:
1440:
1431:
1425:
1419:
1413:
1407:
1401:
1392:
1386:
1370:
1359:
1338:Reinhardt domain
1296:
1294:
1293:
1288:
1264:
1212:
1210:
1209:
1204:
1191:
1190:
1187:
1183:
1180:
1177:
1106:Tubes over cones
1048:
1046:
1045:
1040:
1016:
1015:
994:
993:
992:
991:
986:
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932:
931:
926:
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851:
846:
834:
770:
768:
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762:
747:
746:
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714:
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613:
611:
610:
605:
600:
599:
583:
582:
559:
512:
510:
509:
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484:
483:
478:
445:
444:
386:
384:
383:
378:
355:
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336:
335:
314:
313:
295:
294:
276:
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187:
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1659:
1655:
1654:
1653:
1651:
1650:
1649:
1625:
1624:
1623:
1573:
1543:
1529:Hörmander, Lars
1454:
1449:
1441:
1434:
1428:Carmignani 1973
1426:
1422:
1414:
1410:
1402:
1395:
1387:
1383:
1379:
1374:
1373:
1360:
1356:
1351:
1334:
1320:
1254:
1248:
1245:
1244:
1235:
1226:
1186:
1149:
1146:
1145:
1108:
1097:
1081:
1068:
999:
995:
987:
982:
981:
980:
976:
950:
947:
946:
888:
884:
878:
873:
872:
855:
847:
842:
841:
840:
836:
824:
818:
815:
814:
801:
742:
737:
736:
710:
702:
697:
696:
695:
691:
689:
686:
685:
680:
664:
635:
591:
587:
578:
574:
568:
565:
564:
558:
549:
543:
525:
519:
479:
474:
473:
440:
436:
434:
431:
430:
417:
350:
346:
331:
327:
309:
305:
290:
286:
271:
267:
255:
251:
236:
232:
220:
216:
201:
197:
182:
178:
167:
164:
163:
133:
109:quantum gravity
101:Minkowski space
63:variables (see
21:
12:
11:
5:
1658:
1648:
1647:
1642:
1637:
1622:
1621:
1581:
1571:
1547:
1541:
1525:
1496:hep-th/9911027
1475:
1455:
1453:
1450:
1448:
1447:
1432:
1420:
1408:
1404:Hörmander 1990
1393:
1380:
1378:
1375:
1372:
1371:
1353:
1352:
1350:
1347:
1346:
1345:
1340:
1333:
1330:
1316:
1298:
1297:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1263:
1260:
1257:
1253:
1231:
1224:
1214:
1213:
1202:
1199:
1196:
1176:
1173:
1170:
1167:
1163:
1159:
1156:
1153:
1107:
1104:
1093:
1077:
1064:
1050:
1049:
1038:
1035:
1032:
1028:
1025:
1022:
1019:
1014:
1011:
1008:
1005:
1002:
998:
990:
985:
979:
975:
972:
969:
966:
963:
960:
957:
954:
942:is defined by
936:
935:
924:
921:
918:
915:
912:
906:
903:
900:
897:
894:
891:
887:
881:
876:
871:
868:
865:
862:
858:
850:
845:
839:
833:
830:
827:
823:
797:
772:
771:
760:
757:
754:
751:
745:
740:
735:
732:
729:
726:
723:
720:
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713:
705:
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694:
676:
660:
634:
631:
615:
614:
603:
598:
594:
590:
586:
581:
577:
572:
554:
539:
521:Main article:
518:
515:
514:
513:
502:
499:
496:
493:
490:
487:
482:
477:
472:
469:
466:
463:
460:
457:
454:
451:
448:
443:
439:
413:
388:
387:
376:
373:
370:
367:
364:
361:
358:
353:
349:
345:
342:
339:
334:
330:
326:
323:
320:
317:
312:
308:
304:
301:
298:
293:
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279:
274:
270:
266:
263:
258:
254:
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231:
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212:
209:
204:
200:
196:
193:
190:
185:
181:
177:
174:
171:
132:
129:
113:Bergman metric
9:
6:
4:
3:
2:
1657:
1646:
1643:
1641:
1638:
1636:
1633:
1632:
1630:
1617:
1613:
1608:
1603:
1599:
1595:
1591:
1587:
1582:
1578:
1574:
1568:
1563:
1562:
1556:
1552:
1548:
1544:
1542:0-444-88446-7
1538:
1534:
1530:
1526:
1522:
1518:
1514:
1510:
1506:
1502:
1497:
1492:
1488:
1484:
1480:
1479:Gibbons, G.W.
1476:
1472:
1468:
1467:
1462:
1461:"Tube domain"
1457:
1456:
1444:
1439:
1437:
1429:
1424:
1417:
1412:
1405:
1400:
1398:
1390:
1385:
1381:
1368:
1364:
1358:
1354:
1344:
1343:Siegel domain
1341:
1339:
1336:
1335:
1329:
1327:
1324:
1319:
1315:
1311:
1307:
1303:
1281:
1278:
1275:
1272:
1266:
1261:
1255:
1243:
1242:
1241:
1239:
1234:
1230:
1223:
1219:
1200:
1197:
1194:
1174:
1171:
1168:
1165:
1157:
1154:
1151:
1144:
1143:
1142:
1140:
1136:
1132:
1128:
1125:
1121:
1117:
1113:
1103:
1101:
1096:
1092:
1088:
1083:
1080:
1076:
1072:
1067:
1063:
1059:
1055:
1036:
1033:
1030:
1023:
1017:
1012:
1009:
1006:
1003:
1000:
996:
988:
977:
973:
967:
964:
961:
958:
952:
945:
944:
943:
941:
922:
916:
913:
910:
904:
901:
898:
895:
892:
889:
885:
879:
866:
860:
848:
837:
831:
828:
825:
813:
812:
811:
809:
805:
800:
796:
792:
788:
783:
781:
777:
755:
752:
749:
743:
730:
727:
724:
721:
715:
703:
692:
684:
683:
682:
679:
675:
671:
668:
663:
659:
655:
652:
648:
644:
640:
630:
628:
624:
620:
601:
596:
592:
588:
584:
579:
575:
570:
563:
562:
561:
557:
553:
547:
542:
538:
534:
530:
527:Suppose that
524:
500:
494:
491:
488:
485:
480:
470:
467:
464:
461:
458:
455:
452:
446:
441:
437:
429:
428:
427:
425:
421:
416:
412:
408:
407:
401:
397:
393:
374:
371:
368:
365:
362:
359:
351:
347:
343:
340:
337:
332:
328:
321:
318:
310:
306:
302:
299:
296:
291:
287:
280:
272:
268:
264:
261:
256:
252:
248:
245:
242:
237:
233:
229:
226:
221:
217:
210:
202:
198:
194:
191:
188:
183:
179:
172:
169:
162:
161:
160:
158:
154:
150:
146:
143:of dimension
142:
138:
128:
126:
122:
118:
114:
110:
106:
102:
98:
94:
90:
86:
82:
78:
74:
70:
66:
62:
58:
54:
49:
46:
42:
38:
37:complex plane
34:
30:
26:
19:
1589:
1585:
1575:– via
1560:
1555:Weiss, Guido
1551:Stein, Elias
1532:
1486:
1482:
1464:
1423:
1411:
1389:Gibbons 2000
1384:
1362:
1357:
1325:
1317:
1313:
1309:
1305:
1301:
1299:
1237:
1232:
1228:
1221:
1217:
1215:
1138:
1134:
1130:
1119:
1115:
1111:
1109:
1099:
1094:
1090:
1086:
1084:
1078:
1074:
1070:
1065:
1061:
1057:
1053:
1051:
939:
937:
807:
803:
798:
794:
790:
786:
784:
779:
775:
773:
677:
673:
669:
661:
657:
653:
646:
638:
636:
633:Hardy spaces
616:
555:
551:
548:of the tube
540:
536:
528:
526:
423:
419:
414:
410:
405:
403:
399:
391:
389:
156:
148:
144:
136:
134:
88:
69:Hardy spaces
50:
28:
22:
1592:: 415–431.
1416:Chirka 2001
810:satisfying
651:Hardy space
546:convex hull
533:holomorphic
93:future tube
77:convex sets
29:tube domain
25:mathematics
1629:Categories
1607:1911/14576
1300:exists in
1127:convex set
681:such that
535:in a tube
409:, denoted
404:tube over
398:subset of
131:Definition
125:arithmetic
33:half-plane
1471:EMS Press
1377:Citations
1323:dual cone
1259:→
1172:∈
1162:⟹
1155:∈
1010:⋅
1004:π
978:∫
920:∞
902:⋅
896:π
890:−
838:∫
829:∈
759:∞
693:∫
492:∈
486:∣
471:∈
341:…
300:…
246:…
192:…
97:null cone
45:real part
35:) in the
1557:(1971),
1531:(1990),
1521:14045117
1332:See also
1133:lies in
774:for all
643:open set
1616:1996512
1501:Bibcode
1452:Sources
1236:) have
1188:for all
649:. The
402:. The
153:complex
151:denote
139:denote
55:of the
53:domains
1614:
1569:
1539:
1519:
1192:
1184:
1181:
1178:
1122:is an
940:ƒ
804:ƒ
641:be an
394:be an
1612:JSTOR
1517:S2CID
1491:arXiv
1349:Notes
1052:Then
85:cones
1567:ISBN
1537:ISBN
1198:>
1124:open
1110:Let
917:<
756:<
637:Let
396:open
390:Let
147:and
135:Let
107:and
79:are
67:).
61:real
27:, a
1602:hdl
1594:doi
1590:179
1509:doi
1252:lim
1216:If
822:sup
778:in
672:in
645:in
550:ch
99:in
39:to
23:In
1631::
1610:.
1600:.
1588:.
1553:;
1515:,
1507:,
1499:,
1487:17
1485:,
1469:,
1463:,
1435:^
1396:^
1369:).
1201:0.
782:.
593:ch
571:ch
426::
127:.
1620:.
1618:.
1604::
1596::
1580:.
1546:.
1524:.
1511::
1503::
1493::
1474:.
1445:.
1430:.
1418:.
1406:.
1391:.
1365:(
1363:A
1326:A
1318:A
1314:T
1312:(
1310:H
1306:B
1304:(
1302:L
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1282:y
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1267:F
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1116:R
1112:A
1100:A
1095:A
1091:T
1089:(
1087:H
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1058:H
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1024:t
1021:(
1018:f
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1007:z
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997:e
989:n
984:R
974:=
971:)
968:y
965:i
962:+
959:x
956:(
953:F
923:.
914:t
911:d
905:t
899:x
893:4
886:e
880:2
875:|
870:)
867:t
864:(
861:f
857:|
849:n
844:R
832:A
826:x
808:R
799:A
795:T
793:(
791:H
787:p
780:A
776:x
753:y
750:d
744:p
739:|
734:)
731:y
728:i
725:+
722:x
719:(
716:F
712:|
704:n
699:R
678:A
674:T
670:F
662:A
658:T
656:(
654:H
647:R
639:A
621:(
602:.
597:A
589:T
585:=
580:A
576:T
556:A
552:T
541:A
537:T
529:A
501:.
498:}
495:A
489:x
481:n
476:C
468:y
465:i
462:+
459:x
456:=
453:z
450:{
447:=
442:A
438:T
424:A
420:C
415:A
411:T
406:A
400:R
392:A
375:.
372:y
369:i
366:+
363:x
360:=
357:)
352:n
348:y
344:,
338:,
333:1
329:y
325:(
322:i
319:+
316:)
311:n
307:x
303:,
297:,
292:1
288:x
284:(
281:=
278:)
273:n
269:y
265:i
262:+
257:n
253:x
249:,
243:,
238:1
234:y
230:i
227:+
222:1
218:x
214:(
211:=
208:)
203:n
199:z
195:,
189:,
184:1
180:z
176:(
173:=
170:a
157:C
149:C
145:n
137:R
89:H
20:.
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