385:
639:
149:
450:
380:{\displaystyle f({\mathbf {x} })=\sum _{|\alpha |\leq m}{\frac {D^{\alpha }f({\mathbf {y} })}{\alpha !}}\cdot ({\mathbf {x} }-{\mathbf {y} })^{\alpha }+\sum _{|\alpha |=m}R_{\alpha }({\mathbf {x} },{\mathbf {y} }){\frac {({\mathbf {x} }-{\mathbf {y} })^{\alpha }}{\alpha !}}}
1263:
2092:
1084:
72:
A precise statement of the theorem requires careful consideration of what it means to prescribe the derivative of a function on a closed set. One difficulty, for instance, is that closed subsets of
Euclidean space in general lack a
1886:
634:{\displaystyle f_{\alpha }({\mathbf {x} })=\sum _{|\beta |\leq m-|\alpha |}{\frac {f_{\alpha +\beta }({\mathbf {y} })}{\beta !}}({\mathbf {x} }-{\mathbf {y} })^{\beta }+R_{\alpha }({\mathbf {x} },{\mathbf {y} })}
1562:
1723:
1984:
1628:
1413:
1089:
which is linear, continuous (for the topology of uniform convergence of functions and their derivatives on compacta) and takes functions supported in into functions supported in
1121:
2019:
771:
1007:
1483:
1324:
2115:
1439:
1344:
992:
968:
1919:
1756:
1113:
1768:
2371:
2335:
1631:
2353:
892:
proved a sharpening of the
Whitney extension theorem in the special case of a half space. A smooth function on a half space
2424:
1494:
1647:
1929:
1570:
1351:
2013:
and a local change of variables, the result for a half space implies the existence of an analogous extending map
2137:
74:
2143:
1258:{\displaystyle \displaystyle {E(f)(x)=\sum _{m=1}^{\infty }a_{m}f(-b_{m}x)\varphi (-b_{m}x)\,\,\,(x<0),}}
2087:{\displaystyle \displaystyle {C^{\infty }({\overline {\Omega }})\rightarrow C^{\infty }(\mathbf {R} ^{n})}}
1635:
2385:
37:
2299:, Tata Institute of Fundamental Research Studies in Mathematics, vol. 3, Oxford University Press
1079:{\displaystyle \displaystyle {E:C^{\infty }(\mathbf {R} ^{+})\rightarrow C^{\infty }(\mathbf {R} ),}}
922:
740:
45:
2328:
The analysis of linear partial differential operators. I. Distribution theory and
Fourier analysis
2261:
2131:
2010:
1448:
17:
1296:
2380:
2367:
2349:
2331:
2100:
1418:
1329:
977:
953:
2394:
2313:
2284:
2272:
2251:
2228:
2210:
1894:
1731:
939:
2408:
2224:
52:
is a closed subset of a
Euclidean space, then it is possible to extend a given function of
2404:
2239:
2232:
2220:
1486:
61:
1095:
2318:
2418:
707:
77:. The starting point, then, is an examination of the statement of Taylor's theorem.
2215:
706:
which must be satisfied in order for these functions to be the coefficients of the
2242:(1934), "Analytic extensions of differentiable functions defined in closed sets",
2399:
1881:{\displaystyle M(z)=\sum _{j\geq 1}{(-1)^{j} \over W^{\prime }(2^{j})(z-2^{j})}}
404:
33:
2140: â Continuous maps on a closed subset of a normal space can be extended
699:) may be regarded as purely a compatibility condition between the functions
2304:
Seeley, R. T. (1964), "Extension of Câ functions defined in a half space",
950:. Since Borel's lemma is local in nature, the same argument shows that if
2348:, Studies in Mathematics and Its Applications, vol. 14, Elsevier,
2288:
2265:
2256:
2346:
Introduction to the Theory of Linear
Partial Differential Equations
2201:
McShane, Edward James (1934), "Extension of range of functions",
1445:
A solution to this system of equations can be obtained by taking
974:
with smooth boundary, then any smooth function on the closure of
714:. It is this insight which facilitates the following statement:
56:
in such a way as to have prescribed derivatives at the points of
1001:
Seeley's result for a half line gives a uniform extension map
1832:
2146: â Theorem on extension of bounded linear functionals
1891:
meromorphic with simple poles and prescribed residues at
1762:'(2) are bounded above and below. Similarly the function
1630:
That such a function can be constructed follows from the
2383:(2005), "A sharp form of Whitney's extension theorem",
946:
can be extended to a smooth function on the whole of
2103:
2023:
2022:
1933:
1932:
1897:
1771:
1734:
1650:
1573:
1497:
1451:
1421:
1354:
1332:
1299:
1125:
1124:
1098:
1011:
1010:
980:
956:
743:
453:
152:
1989:
is an entire function with the required properties.
1557:{\displaystyle g(z)=\sum _{m=1}^{\infty }a_{m}z^{m}}
1268:where Ï is a smooth function of compact support on
2174:
2109:
2086:
1978:
1913:
1880:
1750:
1717:
1622:
1556:
1477:
1433:
1407:
1338:
1318:
1257:
1107:
1078:
986:
962:
765:
633:
379:
2244:Transactions of the American Mathematical Society
1718:{\displaystyle W(z)=\prod _{j\geq 1}(1-z/2^{j}),}
729:are a collection of functions on a closed subset
48:. Roughly speaking, the theorem asserts that if
2186:
2277:Bulletin of the Brazilian Mathematical Society
1979:{\displaystyle \displaystyle {g(z)=W(z)M(z)}}
1623:{\displaystyle g\left(2^{j}\right)=(-1)^{j}.}
8:
1408:{\displaystyle \sum a_{m}b_{m}^{j}=(-1)^{j}}
2250:(1), American Mathematical Society: 63â89,
2344:Chazarain, Jacques; Piriou, Alain (1982),
865:Proofs are given in the original paper of
2398:
2317:
2255:
2214:
2162:
2102:
2073:
2068:
2058:
2038:
2029:
2024:
2021:
1934:
1931:
1902:
1896:
1866:
1844:
1831:
1819:
1803:
1791:
1770:
1739:
1733:
1703:
1694:
1670:
1649:
1611:
1585:
1572:
1548:
1538:
1528:
1517:
1496:
1469:
1456:
1450:
1420:
1399:
1377:
1372:
1362:
1353:
1331:
1304:
1298:
1234:
1233:
1232:
1220:
1195:
1176:
1166:
1155:
1126:
1123:
1097:
1063:
1054:
1038:
1033:
1023:
1012:
1009:
979:
955:
878:
874:
870:
752:
744:
742:
622:
621:
612:
611:
602:
589:
579:
578:
569:
568:
545:
544:
529:
522:
515:
507:
493:
485:
484:
468:
467:
458:
452:
360:
350:
349:
340:
339:
333:
324:
323:
314:
313:
304:
287:
279:
278:
265:
255:
254:
245:
244:
218:
217:
205:
198:
185:
177:
176:
160:
159:
151:
96:, Taylor's theorem asserts that for each
2134:gives extensions of Lipschitz functions.
1728:an entire function with simple zeros at
994:can be extended to a smooth function on
773:satisfying the compatibility condition (
2362:Ponnusamy, S.; Silverman, Herb (2006),
2155:
866:
434:. Differentiating (1) with respect to
970:is a (bounded or unbounded) domain in
889:
1272:equal to 1 near 0 and the sequences (
7:
2275:(1980), "Differentiable functions",
444:
143:
29:Partial converse of Taylor's theorem
2364:Complex variables with applications
1992:The definition for a half space in
1641:It can be seen directly by setting
1441:with the sum absolutely convergent.
925:on the half space. On the boundary
853:is real-analytic at every point of
2297:Ideals of differentiable functions
2104:
2059:
2040:
2030:
1529:
1333:
1167:
1055:
1024:
981:
957:
25:
2319:10.1090/s0002-9939-1964-0165392-8
938:restricts to smooth function. By
18:McShaneâWhitney extension theorem
2069:
1064:
1034:
795:. Then there exists a function
623:
613:
580:
570:
546:
469:
351:
341:
325:
315:
256:
246:
219:
161:
2216:10.1090/s0002-9904-1934-05978-0
766:{\displaystyle |\alpha |\leq m}
2175:Ponnusamy & Silverman 2006
2079:
2064:
2051:
2048:
2035:
1971:
1965:
1959:
1953:
1944:
1938:
1872:
1853:
1850:
1837:
1816:
1806:
1781:
1775:
1709:
1682:
1660:
1654:
1608:
1598:
1507:
1501:
1396:
1386:
1247:
1235:
1229:
1210:
1204:
1185:
1145:
1139:
1136:
1130:
1068:
1060:
1047:
1044:
1029:
923:extend to continuous functions
753:
745:
628:
608:
586:
565:
551:
541:
516:
508:
494:
486:
474:
464:
357:
336:
330:
310:
288:
280:
262:
241:
224:
214:
186:
178:
166:
156:
1:
737:for all multi-indices α with
129:) approaching 0 uniformly as
2043:
2009:. Similarly, using a smooth
918:for which the derivatives â
2400:10.4007/annals.2005.161.509
2295:Malgrange, Bernard (1967),
2187:Chazarain & Piriou 1982
1478:{\displaystyle b_{n}=2^{n}}
775:
695:
2441:
1319:{\displaystyle b_{m}>0}
1996:by applying the operator
905:â„ 0 is a smooth function
885:Extension in a half space
438:, and possibly replacing
44:is a partial converse to
42:Whitney extension theorem
2326:Hörmander, Lars (1990),
2138:Tietze extension theorem
75:differentiable structure
2110:{\displaystyle \Omega }
1434:{\displaystyle j\geq 0}
1339:{\displaystyle \infty }
987:{\displaystyle \Omega }
963:{\displaystyle \Omega }
2306:Proc. Amer. Math. Soc.
2203:Bull. Amer. Math. Soc.
2121:with smooth boundary.
2111:
2088:
1980:
1915:
1914:{\displaystyle 2^{j}.}
1882:
1752:
1751:{\displaystyle 2^{j}.}
1719:
1636:Mittag-Leffler theorem
1624:
1558:
1533:
1479:
1435:
1409:
1340:
1320:
1259:
1171:
1109:
1080:
988:
964:
767:
635:
403:where the sum is over
381:
112:, there is a function
2386:Annals of Mathematics
2112:
2089:
2000:to the last variable
1981:
1916:
1883:
1753:
1720:
1625:
1559:
1513:
1480:
1436:
1410:
1341:
1321:
1260:
1151:
1110:
1081:
989:
965:
768:
636:
430:for each multi-index
382:
38:mathematical analysis
2425:Theorems in analysis
2101:
2020:
1930:
1895:
1769:
1732:
1648:
1571:
1495:
1449:
1419:
1352:
1330:
1297:
1122:
1096:
1008:
978:
954:
741:
451:
150:
80:Given a real-valued
60:. It is a result of
2330:, Springer-Verlag,
2144:HahnâBanach theorem
1632:Weierstrass theorem
1382:
857: −
674: −
36:, in particular in
2381:Fefferman, Charles
2289:10.1007/bf02584636
2177:, pp. 442â443
2132:Kirszbraun theorem
2107:
2084:
2083:
2011:partition of unity
1976:
1975:
1911:
1878:
1802:
1748:
1715:
1681:
1620:
1554:
1475:
1431:
1405:
1368:
1336:
1316:
1255:
1254:
1108:{\displaystyle E,}
1105:
1076:
1075:
984:
960:
763:
631:
521:
442:as needed, yields
377:
299:
197:
2273:Bierstone, Edward
2046:
1876:
1787:
1666:
655:
654:
563:
480:
401:
400:
375:
274:
236:
172:
16:(Redirected from
2432:
2411:
2402:
2376:
2358:
2340:
2322:
2321:
2300:
2291:
2268:
2259:
2240:Whitney, Hassler
2235:
2218:
2189:
2184:
2178:
2172:
2166:
2160:
2116:
2114:
2113:
2108:
2093:
2091:
2090:
2085:
2082:
2078:
2077:
2072:
2063:
2062:
2047:
2039:
2034:
2033:
1985:
1983:
1982:
1977:
1974:
1923:By construction
1920:
1918:
1917:
1912:
1907:
1906:
1887:
1885:
1884:
1879:
1877:
1875:
1871:
1870:
1849:
1848:
1836:
1835:
1825:
1824:
1823:
1804:
1801:
1758:The derivatives
1757:
1755:
1754:
1749:
1744:
1743:
1724:
1722:
1721:
1716:
1708:
1707:
1698:
1680:
1629:
1627:
1626:
1621:
1616:
1615:
1594:
1590:
1589:
1563:
1561:
1560:
1555:
1553:
1552:
1543:
1542:
1532:
1527:
1484:
1482:
1481:
1476:
1474:
1473:
1461:
1460:
1440:
1438:
1437:
1432:
1414:
1412:
1411:
1406:
1404:
1403:
1381:
1376:
1367:
1366:
1345:
1343:
1342:
1337:
1325:
1323:
1322:
1317:
1309:
1308:
1264:
1262:
1261:
1256:
1253:
1225:
1224:
1200:
1199:
1181:
1180:
1170:
1165:
1114:
1112:
1111:
1106:
1085:
1083:
1082:
1077:
1074:
1067:
1059:
1058:
1043:
1042:
1037:
1028:
1027:
993:
991:
990:
985:
969:
967:
966:
961:
909:on the interior
896:of points where
879:Hörmander (1990)
875:Bierstone (1980)
871:Malgrange (1967)
779:) at all points
772:
770:
769:
764:
756:
748:
710:of the function
678:|) uniformly as
649:
640:
638:
637:
632:
627:
626:
617:
616:
607:
606:
594:
593:
584:
583:
574:
573:
564:
562:
554:
550:
549:
540:
539:
523:
520:
519:
511:
497:
489:
473:
472:
463:
462:
445:
395:
386:
384:
383:
378:
376:
374:
366:
365:
364:
355:
354:
345:
344:
334:
329:
328:
319:
318:
309:
308:
298:
291:
283:
270:
269:
260:
259:
250:
249:
237:
235:
227:
223:
222:
210:
209:
199:
196:
189:
181:
165:
164:
144:
46:Taylor's theorem
21:
2440:
2439:
2435:
2434:
2433:
2431:
2430:
2429:
2415:
2414:
2379:
2374:
2361:
2356:
2343:
2338:
2325:
2303:
2294:
2271:
2257:10.2307/1989708
2238:
2209:(12): 837â842,
2200:
2197:
2192:
2185:
2181:
2173:
2169:
2161:
2157:
2153:
2127:
2099:
2098:
2097:for any domain
2067:
2054:
2025:
2018:
2017:
2008:
1928:
1927:
1898:
1893:
1892:
1862:
1840:
1827:
1826:
1815:
1805:
1767:
1766:
1735:
1730:
1729:
1699:
1646:
1645:
1607:
1581:
1577:
1569:
1568:
1544:
1534:
1493:
1492:
1487:entire function
1485:and seeking an
1465:
1452:
1447:
1446:
1417:
1416:
1395:
1358:
1350:
1349:
1328:
1327:
1300:
1295:
1294:
1289:
1280:
1216:
1191:
1172:
1120:
1119:
1094:
1093:
1050:
1032:
1019:
1006:
1005:
976:
975:
952:
951:
933:
917:
904:
887:
842:
820:
739:
738:
728:
705:
665:
647:
598:
585:
555:
525:
524:
454:
449:
448:
422:
393:
367:
356:
335:
300:
261:
228:
201:
200:
148:
147:
120:
70:
62:Hassler Whitney
30:
23:
22:
15:
12:
11:
5:
2438:
2436:
2428:
2427:
2417:
2416:
2413:
2412:
2393:(1): 509â577,
2377:
2372:
2366:, BirkhÀuser,
2359:
2354:
2341:
2336:
2323:
2301:
2292:
2283:(2): 139â189,
2269:
2236:
2196:
2193:
2191:
2190:
2179:
2167:
2163:Bierstone 1980
2154:
2152:
2149:
2148:
2147:
2141:
2135:
2126:
2123:
2106:
2095:
2094:
2081:
2076:
2071:
2066:
2061:
2057:
2053:
2050:
2045:
2042:
2037:
2032:
2028:
2004:
1987:
1986:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1910:
1905:
1901:
1889:
1888:
1874:
1869:
1865:
1861:
1858:
1855:
1852:
1847:
1843:
1839:
1834:
1830:
1822:
1818:
1814:
1811:
1808:
1800:
1797:
1794:
1790:
1786:
1783:
1780:
1777:
1774:
1747:
1742:
1738:
1726:
1725:
1714:
1711:
1706:
1702:
1697:
1693:
1690:
1687:
1684:
1679:
1676:
1673:
1669:
1665:
1662:
1659:
1656:
1653:
1619:
1614:
1610:
1606:
1603:
1600:
1597:
1593:
1588:
1584:
1580:
1576:
1565:
1564:
1551:
1547:
1541:
1537:
1531:
1526:
1523:
1520:
1516:
1512:
1509:
1506:
1503:
1500:
1472:
1468:
1464:
1459:
1455:
1443:
1442:
1430:
1427:
1424:
1402:
1398:
1394:
1391:
1388:
1385:
1380:
1375:
1371:
1365:
1361:
1357:
1347:
1335:
1315:
1312:
1307:
1303:
1285:
1276:
1266:
1265:
1252:
1249:
1246:
1243:
1240:
1237:
1231:
1228:
1223:
1219:
1215:
1212:
1209:
1206:
1203:
1198:
1194:
1190:
1187:
1184:
1179:
1175:
1169:
1164:
1161:
1158:
1154:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1104:
1101:
1087:
1086:
1073:
1070:
1066:
1062:
1057:
1053:
1049:
1046:
1041:
1036:
1031:
1026:
1022:
1018:
1015:
983:
959:
929:
913:
900:
886:
883:
867:Whitney (1934)
863:
862:
848:
838:
826:
818:
762:
759:
755:
751:
747:
724:
703:
661:
653:
652:
643:
641:
630:
625:
620:
615:
610:
605:
601:
597:
592:
588:
582:
577:
572:
567:
561:
558:
553:
548:
543:
538:
535:
532:
528:
518:
514:
510:
506:
503:
500:
496:
492:
488:
483:
479:
476:
471:
466:
461:
457:
418:
399:
398:
389:
387:
373:
370:
363:
359:
353:
348:
343:
338:
332:
327:
322:
317:
312:
307:
303:
297:
294:
290:
286:
282:
277:
273:
268:
264:
258:
253:
248:
243:
240:
234:
231:
226:
221:
216:
213:
208:
204:
195:
192:
188:
184:
180:
175:
171:
168:
163:
158:
155:
116:
69:
66:
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2437:
2426:
2423:
2422:
2420:
2410:
2406:
2401:
2396:
2392:
2388:
2387:
2382:
2378:
2375:
2373:0-8176-4457-1
2369:
2365:
2360:
2357:
2351:
2347:
2342:
2339:
2337:3-540-00662-1
2333:
2329:
2324:
2320:
2315:
2311:
2307:
2302:
2298:
2293:
2290:
2286:
2282:
2278:
2274:
2270:
2267:
2263:
2258:
2253:
2249:
2245:
2241:
2237:
2234:
2230:
2226:
2222:
2217:
2212:
2208:
2204:
2199:
2198:
2194:
2188:
2183:
2180:
2176:
2171:
2168:
2165:, p. 143
2164:
2159:
2156:
2150:
2145:
2142:
2139:
2136:
2133:
2129:
2128:
2124:
2122:
2120:
2074:
2055:
2026:
2016:
2015:
2014:
2012:
2007:
2003:
1999:
1995:
1990:
1968:
1962:
1956:
1950:
1947:
1941:
1935:
1926:
1925:
1924:
1921:
1908:
1903:
1899:
1867:
1863:
1859:
1856:
1845:
1841:
1828:
1820:
1812:
1809:
1798:
1795:
1792:
1788:
1784:
1778:
1772:
1765:
1764:
1763:
1761:
1745:
1740:
1736:
1712:
1704:
1700:
1695:
1691:
1688:
1685:
1677:
1674:
1671:
1667:
1663:
1657:
1651:
1644:
1643:
1642:
1639:
1637:
1633:
1617:
1612:
1604:
1601:
1595:
1591:
1586:
1582:
1578:
1574:
1549:
1545:
1539:
1535:
1524:
1521:
1518:
1514:
1510:
1504:
1498:
1491:
1490:
1489:
1488:
1470:
1466:
1462:
1457:
1453:
1428:
1425:
1422:
1400:
1392:
1389:
1383:
1378:
1373:
1369:
1363:
1359:
1355:
1348:
1313:
1310:
1305:
1301:
1293:
1292:
1291:
1288:
1284:
1279:
1275:
1271:
1250:
1244:
1241:
1238:
1226:
1221:
1217:
1213:
1207:
1201:
1196:
1192:
1188:
1182:
1177:
1173:
1162:
1159:
1156:
1152:
1148:
1142:
1133:
1127:
1118:
1117:
1116:
1102:
1099:
1090:
1071:
1051:
1039:
1020:
1016:
1013:
1004:
1003:
1002:
999:
997:
973:
949:
945:
941:
940:Borel's lemma
937:
932:
928:
924:
921:
916:
912:
908:
903:
899:
895:
891:
890:Seeley (1964)
884:
882:
880:
876:
872:
868:
860:
856:
852:
849:
846:
841:
837:
833:
830:
827:
824:
817:
813:
810:
809:
808:
806:
802:
798:
794:
790:
786:
782:
778:
777:
760:
757:
749:
736:
732:
727:
723:
720:Suppose that
719:
715:
713:
709:
708:Taylor series
702:
698:
697:
691:
689:
685:
681:
677:
673:
669:
664:
660:
651:
644:
642:
618:
603:
599:
595:
590:
575:
559:
556:
536:
533:
530:
526:
512:
504:
501:
498:
490:
481:
477:
459:
455:
447:
446:
443:
441:
437:
433:
429:
426:
421:
417:
412:
410:
406:
405:multi-indices
397:
390:
388:
371:
368:
361:
346:
320:
305:
301:
295:
292:
284:
275:
271:
266:
251:
238:
232:
229:
211:
206:
202:
193:
190:
182:
173:
169:
153:
146:
145:
142:
140:
136:
132:
128:
124:
119:
115:
111:
107:
103:
99:
95:
91:
87:
83:
78:
76:
67:
65:
63:
59:
55:
51:
47:
43:
39:
35:
27:
19:
2390:
2384:
2363:
2345:
2327:
2309:
2305:
2296:
2280:
2276:
2247:
2243:
2206:
2202:
2182:
2170:
2158:
2118:
2096:
2005:
2001:
1997:
1993:
1991:
1988:
1922:
1890:
1759:
1727:
1640:
1566:
1444:
1286:
1282:
1277:
1273:
1269:
1267:
1091:
1088:
1000:
995:
971:
947:
943:
935:
930:
926:
919:
914:
910:
906:
901:
897:
893:
888:
864:
858:
854:
850:
844:
839:
835:
831:
828:
822:
815:
811:
804:
800:
796:
792:
788:
784:
780:
774:
734:
730:
725:
721:
717:
716:
711:
700:
694:
692:
687:
683:
679:
675:
671:
667:
662:
658:
656:
645:
439:
435:
431:
427:
424:
419:
415:
413:
408:
402:
391:
138:
134:
130:
126:
122:
117:
113:
109:
105:
101:
97:
93:
89:
85:
81:
79:
71:
57:
53:
49:
41:
31:
26:
2312:: 625â626,
1290:) satisfy:
807:such that:
803:) of class
693:Note that (
34:mathematics
2355:0444864520
2233:0010.34606
2195:References
1567:such that
1092:To define
141:such that
2105:Ω
2060:∞
2052:→
2044:¯
2041:Ω
2031:∞
1860:−
1833:′
1810:−
1796:≥
1789:∑
1689:−
1675:≥
1668:∏
1602:−
1530:∞
1515:∑
1426:≥
1390:−
1356:∑
1334:∞
1326:tends to
1214:−
1208:φ
1189:−
1168:∞
1153:∑
1056:∞
1048:→
1025:∞
982:Ω
958:Ω
869:, and in
758:≤
750:α
604:α
591:β
576:−
557:β
537:β
531:α
513:α
505:−
499:≤
491:β
482:∑
460:α
369:α
362:α
347:−
306:α
285:α
276:∑
267:α
252:−
239:⋅
230:α
207:α
191:≤
183:α
174:∑
84:function
68:Statement
2419:Category
2125:See also
718:Theorem.
2409:2150391
2266:1989708
2225:1562984
2407:
2370:
2352:
2334:
2264:
2231:
2223:
787:, and
657:where
407:
40:, the
2262:JSTOR
2151:Notes
934:= 0,
92:) on
2368:ISBN
2350:ISBN
2332:ISBN
2130:The
1634:and
1415:for
1311:>
1281:), (
1242:<
1115:set
877:and
414:Let
2395:doi
2391:161
2314:doi
2285:doi
2252:doi
2229:Zbl
2211:doi
2117:in
843:on
821:on
791:of
733:of
666:is
32:In
2421::
2405:MR
2403:,
2389:,
2310:15
2308:,
2281:11
2279:,
2260:,
2248:36
2246:,
2227:,
2221:MR
2219:,
2207:40
2205:,
1638:.
998:.
942:,
881:.
873:,
834:=
814:=
783:,
690:.
686:â
670:(|
423:=
411:.
137:â
108:â
104:,
100:,
64:.
2397::
2316::
2287::
2254::
2213::
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2080:)
2075:n
2070:R
2065:(
2056:C
2049:)
2036:(
2027:C
2006:n
2002:x
1998:R
1994:R
1972:)
1969:z
1966:(
1963:M
1960:)
1957:z
1954:(
1951:W
1948:=
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1942:z
1939:(
1936:g
1909:.
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1900:2
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1387:(
1384:=
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1236:(
1230:)
1227:x
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1211:(
1205:)
1202:x
1197:m
1193:b
1186:(
1183:f
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1160:=
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1149:=
1146:)
1143:x
1140:(
1137:)
1134:f
1131:(
1128:E
1103:,
1100:E
1072:,
1069:)
1065:R
1061:(
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1045:)
1040:+
1035:R
1030:(
1021:C
1017::
1014:E
996:R
972:R
948:R
944:f
936:f
931:n
927:x
920:f
915:n
911:x
907:f
902:n
898:x
894:R
861:.
859:A
855:R
851:F
847:.
845:A
840:α
836:f
832:F
829:D
825:.
823:A
819:0
816:f
812:F
805:C
801:x
799:(
797:F
793:A
789:a
785:y
781:x
776:2
761:m
754:|
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735:R
731:A
726:α
722:f
712:f
704:α
701:f
696:2
688:a
684:y
682:,
680:x
676:y
672:x
668:o
663:α
659:R
650:)
648:2
646:(
629:)
624:y
619:,
614:x
609:(
600:R
596:+
587:)
581:y
571:x
566:(
560:!
552:)
547:y
542:(
534:+
527:f
517:|
509:|
502:m
495:|
487:|
478:=
475:)
470:x
465:(
456:f
440:R
436:x
432:α
428:f
425:D
420:α
416:f
409:α
396:)
394:1
392:(
372:!
358:)
352:y
342:x
337:(
331:)
326:y
321:,
316:x
311:(
302:R
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293:=
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281:|
272:+
263:)
257:y
247:x
242:(
233:!
225:)
220:y
215:(
212:f
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194:m
187:|
179:|
170:=
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162:x
157:(
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133:,
131:x
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118:α
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110:R
106:y
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98:a
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88:(
86:f
82:C
58:A
54:A
50:A
20:)
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