1248:
572:
times continuously differentiable functions. While this result has often been interpreted as saying that a general multiplication of distributions is not possible, in fact it only states that one cannot unrestrictedly combine differentiation, multiplication of continuous functions and the presence of
39:
As a mathematical tool, Colombeau algebras can be said to combine a treatment of singularities, differentiation and nonlinear operations in one framework, lifting the limitations of distribution theory. These algebras have found numerous applications in the fields of partial differential equations,
35:
Such a multiplication of distributions has long been believed to be impossible because of L. Schwartz' impossibility result, which basically states that there cannot be a differential algebra containing the space of distributions and preserving the product of continuous functions. However, if one
1052:
789:
684:
956:
1243:{\displaystyle \sup _{x\in K}\left|{\frac {\partial ^{|\alpha |}}{(\partial x_{1})^{\alpha _{1}}\cdots (\partial x_{n})^{\alpha _{n}}}}f_{\varepsilon }(x)\right|=O(\varepsilon ^{-N})\qquad (\varepsilon \to 0).}
622:
477:
1405:
406:
207:
93:
1304:
845:
164:
1372:
550:
1036:
992:
874:
512:
343:
289:
238:
115:
366:
312:
429:
570:
258:
32:. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras provide a rigorous framework for this.
703:
627:
1532:
1418:
This embedding is non-canonical, because it depends on the choice of the δ-net. However, there are versions of
Colombeau algebras (so called
893:
36:
only wants to preserve the product of smooth functions instead such a construction becomes possible, as demonstrated first by
Colombeau.
483:
However, L. Schwartz' result implies that these requirements cannot hold simultaneously. The same is true even if, in 4., one replaces
1563:
579:
434:
1553:
1377:
1002:
695:
1259:
800:
1343:
371:
172:
58:
29:
1548:
120:
1350:
44:
517:
1330:
1012:
968:
850:
1439:
486:
317:
263:
212:
1513:
1254:
25:
98:
1475:
576:
Colombeau algebras are constructed to satisfy conditions 1.–3. and a condition like 4., but with
1463:
351:
297:
1558:
1528:
414:
962:
686:, i.e., they preserve the product of smooth (infinitely differentiable) functions only.
555:
243:
1542:
784:{\displaystyle C_{M}^{\infty }(\mathbb {R} ^{n})/C_{N}^{\infty }(\mathbb {R} ^{n}).}
1039:
1525:
Geometric Theory of
Generalized Functions with Applications to General Relativity
1458:
L. Schwartz, 1954, "Sur l'impossibilité de la multiplication des distributions",
1310:
is defined in the same way but with the partial derivatives instead bounded by O(
1338:
17:
1422:
algebras) which allow for canonical embeddings of distributions. A well known
1427:
679:{\displaystyle C^{\infty }(\mathbb {R} )\times C^{\infty }(\mathbb {R} )}
951:{\displaystyle {f:}\mathbb {R} _{+}\to C^{\infty }(\mathbb {R} ^{n})}
1527:, Springer Series Mathematics and Its Applications, Vol. 537, 2002;
1474:
Gratus, J. (2013). "Colombeau
Algebra: A pedagogical introduction".
1523:
Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.;
1480:
1500:
New
Generalized Functions and Multiplication of the Distributions
40:
geophysics, microlocal analysis and general relativity so far .
378:
179:
65:
1341:
with any element of the algebra having as representative a
43:
Colombeau algebras are named after French mathematician
1380:
1353:
1262:
1055:
1015:
971:
896:
853:
803:
706:
630:
617:{\displaystyle C(\mathbb {R} )\times C(\mathbb {R} )}
582:
558:
520:
489:
472:{\displaystyle C(\mathbb {R} )\times C(\mathbb {R} )}
437:
417:
374:
354:
320:
300:
266:
246:
215:
175:
123:
101:
61:
1507:
Elementary introduction to new generalized functions
1518:Linear Theory of Colombeau's Generalized Functions
1400:{\displaystyle \varphi _{\varepsilon }\to \delta }
1399:
1366:
1298:
1242:
1030:
1005:" parameter ε), such that for all compact subsets
986:
950:
868:
839:
783:
678:
616:
564:
544:
506:
471:
423:
400:
360:
337:
306:
283:
252:
232:
201:
158:
109:
87:
1299:{\displaystyle C_{N}^{\infty }(\mathbb {R} ^{n})}
840:{\displaystyle C_{M}^{\infty }(\mathbb {R} ^{n})}
166:, the following requirements seem to be natural:
1057:
345:which is linear and satisfies the Leibniz rule,
401:{\displaystyle {\mathcal {D}}'(\mathbb {R} )}
202:{\displaystyle {\mathcal {D}}'(\mathbb {R} )}
88:{\displaystyle {\mathcal {D}}'(\mathbb {R} )}
8:
408:coincides with the usual partial derivative,
159:{\displaystyle (A(\mathbb {R} ),\circ ,+)}
28:of a certain kind containing the space of
1479:
1460:Comptes Rendus de L'Académie des Sciences
1385:
1379:
1358:
1352:
1287:
1283:
1282:
1272:
1267:
1261:
1209:
1176:
1161:
1156:
1146:
1125:
1120:
1110:
1092:
1084:
1083:
1077:
1060:
1054:
1022:
1018:
1017:
1014:
978:
974:
973:
970:
939:
935:
934:
924:
911:
907:
906:
897:
895:
860:
856:
855:
852:
828:
824:
823:
813:
808:
802:
769:
765:
764:
754:
749:
740:
731:
727:
726:
716:
711:
705:
669:
668:
659:
645:
644:
635:
629:
607:
606:
590:
589:
581:
557:
535:
534:
525:
519:
497:
496:
488:
462:
461:
445:
444:
436:
416:
391:
390:
377:
376:
373:
353:
328:
327:
319:
299:
274:
273:
265:
245:
223:
222:
214:
192:
191:
178:
177:
174:
134:
133:
122:
103:
102:
100:
78:
77:
64:
63:
60:
694:The Colombeau Algebra is defined as the
1451:
1367:{\displaystyle \varphi _{\varepsilon }}
573:singular objects like the Dirac delta.
294:There is a partial derivative operator
876:is the algebra of families of smooth
479:coincides with the pointwise product.
7:
1347:, i.e. a family of smooth functions
545:{\displaystyle C^{k}(\mathbb {R} )}
50:
1426:version is obtained by adding the
1273:
1139:
1103:
1080:
925:
814:
755:
717:
660:
636:
355:
301:
14:
1509:. North-Holland, Amsterdam, 1985.
1502:. North Holland, Amsterdam, 1984.
1520:, Addison Wesley, Longman, 1998.
1031:{\displaystyle \mathbb {R} ^{n}}
987:{\displaystyle \mathbb {R} ^{n}}
869:{\displaystyle \mathbb {R} ^{n}}
240:such that the constant function
1221:
507:{\displaystyle C(\mathbb {R} )}
338:{\displaystyle A(\mathbb {R} )}
284:{\displaystyle A(\mathbb {R} )}
233:{\displaystyle A(\mathbb {R} )}
1391:
1293:
1278:
1234:
1228:
1222:
1218:
1202:
1188:
1182:
1153:
1136:
1117:
1100:
1093:
1085:
945:
930:
917:
834:
819:
775:
760:
737:
722:
673:
665:
649:
641:
611:
603:
594:
586:
539:
531:
501:
493:
466:
458:
449:
441:
395:
387:
332:
324:
278:
270:
227:
219:
196:
188:
153:
138:
130:
124:
82:
74:
55:Attempting to embed the space
51:Schwartz' impossibility result
1:
1337:algebra by (component-wise)
117:into an associative algebra
110:{\displaystyle \mathbb {R} }
1001: = (0,∞) is the "
1580:
1325:Embedding of distributions
209:is linearly embedded into
1333:can be embedded into the
361:{\displaystyle \partial }
307:{\displaystyle \partial }
1430:as second indexing set.
45:Jean François Colombeau
1564:Schwartz distributions
1401:
1368:
1331:Schwartz distributions
1300:
1244:
1032:
988:
952:
870:
841:
785:
680:
618:
566:
546:
508:
473:
425:
424:{\displaystyle \circ }
402:
362:
339:
308:
285:
254:
234:
203:
160:
111:
89:
30:Schwartz distributions
1402:
1369:
1301:
1245:
1033:
989:
953:
871:
842:
786:
681:
619:
567:
547:
509:
474:
426:
403:
363:
340:
309:
286:
260:becomes the unity in
255:
235:
204:
161:
112:
90:
1440:Generalized function
1378:
1351:
1308:negligible functions
1260:
1053:
1013:
969:
894:
851:
801:
794:Here the algebra of
704:
628:
580:
556:
518:
487:
435:
415:
372:
352:
318:
298:
264:
244:
213:
173:
121:
99:
95:of distributions on
59:
1554:Functional analysis
1516:, Scarpalezos, D.,
1277:
818:
759:
721:
411:the restriction of
348:the restriction of
1505:Colombeau, J. F.,
1498:Colombeau, J. F.,
1397:
1364:
1296:
1263:
1240:
1071:
1028:
984:
948:
866:
837:
804:
796:moderate functions
781:
745:
707:
676:
614:
562:
542:
504:
469:
421:
398:
358:
335:
304:
281:
250:
230:
199:
156:
107:
85:
1533:978-1-4020-0145-1
1462:239, pp. 847–848
1170:
1056:
1046:> 0 such that
565:{\displaystyle k}
253:{\displaystyle 1}
22:Colombeau algebra
1571:
1549:Smooth functions
1486:
1485:
1483:
1471:
1465:
1456:
1415: → 0.
1406:
1404:
1403:
1398:
1390:
1389:
1373:
1371:
1370:
1365:
1363:
1362:
1329:The space(s) of
1305:
1303:
1302:
1297:
1292:
1291:
1286:
1276:
1271:
1249:
1247:
1246:
1241:
1217:
1216:
1195:
1191:
1181:
1180:
1171:
1169:
1168:
1167:
1166:
1165:
1151:
1150:
1132:
1131:
1130:
1129:
1115:
1114:
1098:
1097:
1096:
1088:
1078:
1070:
1037:
1035:
1034:
1029:
1027:
1026:
1021:
993:
991:
990:
985:
983:
982:
977:
963:smooth functions
957:
955:
954:
949:
944:
943:
938:
929:
928:
916:
915:
910:
904:
875:
873:
872:
867:
865:
864:
859:
846:
844:
843:
838:
833:
832:
827:
817:
812:
790:
788:
787:
782:
774:
773:
768:
758:
753:
744:
736:
735:
730:
720:
715:
696:quotient algebra
685:
683:
682:
677:
672:
664:
663:
648:
640:
639:
623:
621:
620:
615:
610:
593:
571:
569:
568:
563:
551:
549:
548:
543:
538:
530:
529:
513:
511:
510:
505:
500:
478:
476:
475:
470:
465:
448:
430:
428:
427:
422:
407:
405:
404:
399:
394:
386:
382:
381:
367:
365:
364:
359:
344:
342:
341:
336:
331:
313:
311:
310:
305:
290:
288:
287:
282:
277:
259:
257:
256:
251:
239:
237:
236:
231:
226:
208:
206:
205:
200:
195:
187:
183:
182:
165:
163:
162:
157:
137:
116:
114:
113:
108:
106:
94:
92:
91:
86:
81:
73:
69:
68:
1579:
1578:
1574:
1573:
1572:
1570:
1569:
1568:
1539:
1538:
1512:Nedeljkov, M.,
1495:
1490:
1489:
1473:
1472:
1468:
1457:
1453:
1448:
1436:
1381:
1376:
1375:
1354:
1349:
1348:
1327:
1281:
1258:
1257:
1205:
1172:
1157:
1152:
1142:
1121:
1116:
1106:
1099:
1079:
1076:
1072:
1051:
1050:
1042:α, there is an
1016:
1011:
1010:
1000:
972:
967:
966:
933:
920:
905:
892:
891:
885:
878:regularisations
854:
849:
848:
822:
799:
798:
763:
725:
702:
701:
692:
655:
631:
626:
625:
578:
577:
554:
553:
552:, the space of
521:
516:
515:
485:
484:
433:
432:
413:
412:
375:
370:
369:
350:
349:
316:
315:
296:
295:
262:
261:
242:
241:
211:
210:
176:
171:
170:
119:
118:
97:
96:
62:
57:
56:
53:
12:
11:
5:
1577:
1575:
1567:
1566:
1561:
1556:
1551:
1541:
1540:
1537:
1536:
1521:
1510:
1503:
1494:
1491:
1488:
1487:
1466:
1450:
1449:
1447:
1444:
1443:
1442:
1435:
1432:
1396:
1393:
1388:
1384:
1361:
1357:
1326:
1323:
1295:
1290:
1285:
1280:
1275:
1270:
1266:
1251:
1250:
1239:
1236:
1233:
1230:
1227:
1224:
1220:
1215:
1212:
1208:
1204:
1201:
1198:
1194:
1190:
1187:
1184:
1179:
1175:
1164:
1160:
1155:
1149:
1145:
1141:
1138:
1135:
1128:
1124:
1119:
1113:
1109:
1105:
1102:
1095:
1091:
1087:
1082:
1075:
1069:
1066:
1063:
1059:
1025:
1020:
1003:regularization
998:
981:
976:
959:
958:
947:
942:
937:
932:
927:
923:
919:
914:
909:
903:
900:
883:
863:
858:
836:
831:
826:
821:
816:
811:
807:
792:
791:
780:
777:
772:
767:
762:
757:
752:
748:
743:
739:
734:
729:
724:
719:
714:
710:
691:
688:
675:
671:
667:
662:
658:
654:
651:
647:
643:
638:
634:
613:
609:
605:
602:
599:
596:
592:
588:
585:
561:
541:
537:
533:
528:
524:
503:
499:
495:
492:
481:
480:
468:
464:
460:
457:
454:
451:
447:
443:
440:
420:
409:
397:
393:
389:
385:
380:
357:
346:
334:
330:
326:
323:
303:
292:
280:
276:
272:
269:
249:
229:
225:
221:
218:
198:
194:
190:
186:
181:
155:
152:
149:
146:
143:
140:
136:
132:
129:
126:
105:
84:
80:
76:
72:
67:
52:
49:
13:
10:
9:
6:
4:
3:
2:
1576:
1565:
1562:
1560:
1557:
1555:
1552:
1550:
1547:
1546:
1544:
1534:
1530:
1526:
1522:
1519:
1515:
1514:Pilipović, S.
1511:
1508:
1504:
1501:
1497:
1496:
1492:
1482:
1477:
1470:
1467:
1464:
1461:
1455:
1452:
1445:
1441:
1438:
1437:
1433:
1431:
1429:
1425:
1421:
1416:
1414:
1410:
1394:
1386:
1382:
1359:
1355:
1346:
1345:
1340:
1336:
1332:
1324:
1322:
1320:
1317:
1313:
1309:
1288:
1268:
1264:
1256:
1237:
1231:
1225:
1213:
1210:
1206:
1199:
1196:
1192:
1185:
1177:
1173:
1162:
1158:
1147:
1143:
1133:
1126:
1122:
1111:
1107:
1089:
1073:
1067:
1064:
1061:
1049:
1048:
1047:
1045:
1041:
1023:
1008:
1004:
997:
979:
964:
940:
921:
912:
901:
898:
890:
889:
888:
886:
879:
861:
829:
809:
805:
797:
778:
770:
750:
746:
741:
732:
712:
708:
700:
699:
698:
697:
689:
687:
656:
652:
632:
600:
597:
583:
574:
559:
526:
522:
490:
455:
452:
438:
418:
410:
383:
347:
321:
293:
267:
247:
216:
184:
169:
168:
167:
150:
147:
144:
141:
127:
70:
48:
46:
41:
37:
33:
31:
27:
23:
19:
1524:
1517:
1506:
1499:
1469:
1459:
1454:
1423:
1419:
1417:
1412:
1408:
1342:
1334:
1328:
1318:
1315:
1311:
1307:
1252:
1043:
1040:multiindices
1006:
995:
960:
881:
877:
795:
793:
693:
624:replaced by
575:
482:
54:
42:
38:
34:
21:
15:
1339:convolution
18:mathematics
1543:Categories
1493:References
1428:mollifiers
1374:such that
1344:δ-net
1335:simplified
690:Basic idea
1481:1308.0257
1395:δ
1392:→
1387:ε
1383:φ
1360:ε
1356:φ
1274:∞
1229:→
1226:ε
1211:−
1207:ε
1178:ε
1159:α
1140:∂
1134:⋯
1123:α
1104:∂
1090:α
1081:∂
1065:∈
926:∞
918:→
815:∞
756:∞
718:∞
661:∞
653:×
637:∞
598:×
453:×
419:∘
356:∂
302:∂
145:∘
1559:Algebras
1434:See also
1411:as
1321:> 0.
1038:and all
384:′
185:′
71:′
994:(where
26:algebra
1531:
1314:) for
24:is an
1476:arXiv
1446:Notes
1255:ideal
1529:ISBN
1424:full
1420:full
1253:The
20:, a
1409:D'
1407:in
1316:all
1306:of
1058:sup
1009:of
965:on
961:of
847:on
514:by
431:to
368:to
314:on
16:In
1545::
887:)
47:.
1535:.
1484:.
1478::
1413:ε
1319:N
1312:ε
1294:)
1289:n
1284:R
1279:(
1269:N
1265:C
1238:.
1235:)
1232:0
1223:(
1219:)
1214:N
1203:(
1200:O
1197:=
1193:|
1189:)
1186:x
1183:(
1174:f
1163:n
1154:)
1148:n
1144:x
1137:(
1127:1
1118:)
1112:1
1108:x
1101:(
1094:|
1086:|
1074:|
1068:K
1062:x
1044:N
1024:n
1019:R
1007:K
999:+
996:R
980:n
975:R
946:)
941:n
936:R
931:(
922:C
913:+
908:R
902::
899:f
884:ε
882:f
880:(
862:n
857:R
835:)
830:n
825:R
820:(
810:M
806:C
779:.
776:)
771:n
766:R
761:(
751:N
747:C
742:/
738:)
733:n
728:R
723:(
713:M
709:C
674:)
670:R
666:(
657:C
650:)
646:R
642:(
633:C
612:)
608:R
604:(
601:C
595:)
591:R
587:(
584:C
560:k
540:)
536:R
532:(
527:k
523:C
502:)
498:R
494:(
491:C
467:)
463:R
459:(
456:C
450:)
446:R
442:(
439:C
396:)
392:R
388:(
379:D
333:)
329:R
325:(
322:A
291:,
279:)
275:R
271:(
268:A
248:1
228:)
224:R
220:(
217:A
197:)
193:R
189:(
180:D
154:)
151:+
148:,
142:,
139:)
135:R
131:(
128:A
125:(
104:R
83:)
79:R
75:(
66:D
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.