2289:
on the edge, given as the boundary values of two holomorphic functions on the two wedges. If a hyperfunction is the boundary value of a holomorphic function on a wedge, then its analytic wave front set lies in the dual of the corresponding cone. So the analytic wave front set of
534:
2247:! different wedges. By applying the edge-of-the-wedge theorem (with the edge given by the set of totally spacelike points) one can deduce that the Wightman functions are all analytic continuations of the same holomorphic function, defined on a connected region containing all
1072:
383:
116:. Indeed, a function is holomorphic provided its integral round any contour vanishes; a contour which crosses the real axis can be broken up into contours in the upper and lower half-planes and the integral round these vanishes by hypothesis.
1328:
394:
973:
1685:
1787:
904:
1854:
1450:
1534:
2151:
1598:
1231:
817:
258:
1394:
764:
714:
2049:
1986:
1953:
2185:
is defined or continuous on the edge: it is sufficient to assume that the functions defined on either of the wedges have the same distributional boundary values on the edge.
623:
250:
208:
1640:
2085:
1493:
1111:
577:
2251:! wedges. (The equality of the boundary values on the edge that we need to apply the edge-of-the-wedge theorem follows from the locality axiom of quantum field theory.)
2020:
1362:
1174:
965:
938:
669:
1742:
be holomorphic functions defined exterior and interior to some arc on the unit circle such that locally they have radial limits in some
Sobolev space, Then, letting
1143:
158:
1924:
1740:
1720:
2305:
In the theory of hyperfunctions there is an extension of the edge-of-the-wedge theorem to the case when there are several wedges instead of two, called
1239:
124:
The more general case is phrased in terms of distributions. This is technically simplest in the case where the common boundary is the unit circle
2181:
is defined on the whole of the wedges: it is enough to assume that it is defined near the edge. It is also not necessary to assume that
529:{\displaystyle f(\theta )=\sum _{-\infty }^{\infty }a_{n}e^{in\theta },\,\,\,\,g(\theta )=\sum _{-\infty }^{\infty }b_{n}e^{in\theta }.}
2801:
2571:
2791:
2220:
in the complexification of
Minkowski spacetime. They are defined and holomorphic in the wedge where the imaginary part of each
2796:
2737:
2681:
2647:
2597:
2549:
2517:
1067:{\displaystyle \langle T_{\pm },\varphi \rangle =\lim _{\varepsilon \downarrow 0}\int f(x\pm i\varepsilon )\varphi (x)\,dx.}
2294:
lies in the duals of two opposite cones. But the intersection of these duals is empty, so the analytic wave front set of
1652:
2786:
1748:
51:
at the
International Conference on Theoretical Physics, Seattle, USA (September, 1956) and also published in the book
388:
absolutely convergent in the same regions and have distributional boundary values given by the formal
Fourier series
2282:
of that point, and can be thought of as describing the directions in which the singularity at that point is moving.
825:
108:
In this example, the two wedges are the upper half-plane and the lower half plane, and their common edge is the
2760:
2770:
2756:
2673:
1798:
1145:-th complex derivative of a holomorphic function which extends to a continuous function on the boundary. If
378:{\displaystyle f(z)=\sum _{-\infty }^{\infty }a_{n}z^{n},\,\,\,\,g(z)=\sum _{-\infty }^{\infty }b_{n}z^{n}}
1501:
1399:
2765:
1647:
1550:
1183:
769:
2729:
2589:
1367:
719:
674:
2025:
1962:
1929:
1872:
2119:
1694:
between the circle and the real line, this argument can be rephrased in a standard way in terms of
586:
213:
171:
1606:
2058:
1458:
1080:
542:
1996:
583:. It is then elementary that the common Laurent series converges absolutely in the whole region
2541:
2177:
The conditions for the theorem to be true can be weakened. It is not necessary to assume that
1340:
1152:
943:
916:
35:
of each other provided they both give the same continuous function on the edge. It is used in
2725:
2193:
In quantum field theory the
Wightman distributions are boundary values of Wightman functions
636:
40:
32:
2691:
2398:
2268:
1643:
1116:
127:
93:
80:
In one dimension, a simple case of the edge-of-the-wedge theorem can be stated as follows.
36:
28:
2747:
2699:
2657:
2607:
2559:
8:
1540:
113:
2402:
2567:
2533:
2414:
2371:
2271:, and can also be thought of as something like a "distribution of infinite order". The
1909:
1725:
1705:
48:
2169:(or more precisely, it can be extended to a holomorphic function on a neighborhood of
2733:
2677:
2643:
2593:
2545:
2513:
2418:
2350:
2346:
2329:
44:
2619:
2743:
2695:
2653:
2603:
2555:
2406:
1691:
101:
97:
2706:
For the application of the edge-of-the-wedge theorem to quantum field theory see:
2719:
2715:
2687:
2672:, CMBS Regional Conference Series in Mathematics, vol. 6, Providence, R.I.:
2639:
2279:
2370:
2274:
1695:
2259:
The edge-of-the-wedge theorem has a natural interpretation in the language of
1323:{\displaystyle \langle F,\varphi \rangle =\iint f(x+iy)\varphi (x,y)\,dx\,dy,}
2780:
2635:
2260:
1699:
1455:
In particular if the hypotheses of the edge-of-the-wedge theorem apply, i.e.
89:
64:
60:
2389:
Jost, R.; Lehmann, H. (1957). "Integral-Darstellung kausaler
Kommutatoren".
2285:
In the edge-of-the-wedge theorem, we have a distribution (or hyperfunction)
2239:
lies in the open positive timelike cone. By permuting the variables we get
1883:
therefore patch to give a holomorphic function near the arc and hence so do
1603:
In this case elliptic regularity can be deduced directly from the fact that
2711:
2665:
2626:, Grundlehren der Mathematischen Wissenschaft, vol. 256 (2 ed.),
2351:
Bogolyubov's “edge of the wedge” theorem, its development and applications
2310:
2512:, Graduate texts in mathematics, vol. 125 (2nd ed.), Springer,
20:
2631:
2410:
47:. The formulation and the first proof of the theorem were presented by
2577:
109:
56:
2724:, Princeton Landmarks in Mathematics and Physics (1978 ed.),
55:. Further proofs and generalizations of the theorem were given by
1867:
tend locally to the same function in a higher
Sobolev space. For
967:
can be defined as distributions on the real axis by the formulas
628:
2627:
2585:
2581:
2576:, Mathematical Physics and Applied Mathematics, vol. 10,
1859:
can be solved locally in such a way that the radial limits of
1077:
Existence can be proved by noting that, under the hypothesis,
119:
2334:
Methods of the Theory of
Functions of Many Complex Variables
2566:
67:(1958), H. Epstein (1960), and by other researchers.
160:
in the complex plane. In that case holomorphic functions
2532:
2624:
The analysis of linear partial differential operators I
2540:, Mathematical Physics Monograph Series, vol. 18,
2188:
1402:
2122:
2061:
2028:
1999:
1965:
1932:
1912:
1801:
1751:
1728:
1708:
1655:
1609:
1553:
1504:
1461:
1370:
1343:
1242:
1186:
1155:
1119:
1083:
976:
946:
919:
828:
772:
722:
677:
639:
589:
545:
397:
261:
216:
174:
130:
2614:
The connection with hyperfunctions is described in:
2570:; Logunov, A.A.; Oksak, A.I.; I.T., Todorov (1990),
2302:
is analytic. This is the edge-of-the-wedge theorem.
2388:
2278:of a hyperfunction at each point is a cone in the
2145:
2079:
2043:
2014:
1980:
1947:
1918:
1848:
1781:
1734:
1714:
1680:{\displaystyle \partial /\partial {\overline {z}}}
1679:
1634:
1592:
1528:
1487:
1444:
1388:
1356:
1322:
1225:
1168:
1137:
1105:
1066:
959:
932:
898:
811:
758:
708:
663:
617:
571:
539:Their distributional boundary values are equal if
528:
377:
244:
202:
152:
2328:
2254:
1871:large enough, this convergence is uniform by the
2778:
2710:
2442:
1782:{\displaystyle D=z{\partial \over \partial z},}
1003:
2378:, Princeton: Institute for Advanced Study Press
2161:. Then the edge-of-the-wedge theorem says that
88:is a continuous complex-valued function on the
2538:Introduction to Axiomatic Quantum Field Theory
2376:Problems in the Theory of Dispersion Relations
53:Problems in the Theory of Dispersion Relations
2507:
2468:
1180:is the distribution defined on the rectangle
899:{\displaystyle |f_{\pm }(x+iy)|<C|y|^{-N}}
629:Distributional boundary values on an interval
75:
31:on two "wedges" with an "edge" in common are
2374:; Medvedev, B. V.; Polivanov, M. K. (1958),
1875:. By the argument for continuous functions,
1255:
1243:
996:
977:
671:on the real axis and holomorphic functions
70:
2755:
2573:General Principles of Quantum Field Theory
2508:Berenstein, Carlos A.; Gay, Roger (1991),
2364:
2243:! different Wightman functions defined in
120:Distributional boundary values on a circle
2670:Lectures on the edge-of-the-wedge theorem
2618:
2492:
2480:
2464:
2462:
2453:
2349:, V. V. Zharinov, A. G. Sergeev (1994). "
2322:
2031:
1968:
1935:
1926:be an open cone in the real vector space
1826:
1825:
1824:
1310:
1303:
1054:
692:
691:
463:
462:
461:
460:
321:
320:
319:
318:
2536:; Logunov, A.A.; Todorov, I.T. (1975),
2267:is roughly a sum of boundary values of
2153:that is holomorphic on both the wedges
1849:{\displaystyle D^{k}F=f,\,\,\,D^{k}G=g}
1364:off the real axis and the distribution
2779:
2721:PCT, Spin and Statistics, and All That
2459:
2116:is a continuous function on the union
1903:is a product of a cone with some set.
2664:
2431:
2307:Martineau's edge-of-the-wedge theorem
1445:{\textstyle {1 \over 2}(T_{+}-T_{-})}
1529:{\displaystyle F_{\overline {z}}=0.}
104:. Then it is holomorphic everywhere.
2189:Application to quantum field theory
1894:
13:
2526:
2510:Complex variables: an introduction
1767:
1763:
1664:
1656:
1593:{\displaystyle (a,b)\times (-R,R)}
1543:it then follows that the function
1226:{\displaystyle (a,b)\times (-R,R)}
1176:above and below the real axis and
812:{\displaystyle (a,b)\times (-R,0)}
633:In general given an open interval
492:
487:
426:
421:
350:
345:
290:
285:
14:
2813:
1955:, with vertex at the origin. Let
1389:{\displaystyle F_{\overline {z}}}
759:{\displaystyle (a,b)\times (0,R)}
709:{\displaystyle f_{+},\,\,\ f_{-}}
112:. This result can be proved from
16:Theorem of analytic continuations
2802:Theorems in mathematical physics
2336:, Cambridge, Mass.: M.I.T. Press
2044:{\displaystyle \mathbb {C} ^{n}}
1981:{\displaystyle \mathbb {R} ^{n}}
1948:{\displaystyle \mathbb {R} ^{n}}
2486:
1396:is induced by the distribution
2792:Axiomatic quantum field theory
2474:
2447:
2436:
2425:
2382:
2340:
2255:Connection with hyperfunctions
2146:{\displaystyle W\cup E\cup W'}
1620:
1610:
1587:
1572:
1566:
1554:
1439:
1413:
1300:
1288:
1282:
1267:
1220:
1205:
1199:
1187:
1132:
1120:
1100:
1094:
1051:
1045:
1039:
1024:
1010:
909:for some non-negative integer
883:
874:
863:
859:
844:
830:
806:
791:
785:
773:
753:
741:
735:
723:
658:
646:
605:
597:
473:
467:
407:
401:
331:
325:
271:
265:
232:
224:
190:
182:
140:
132:
1:
2674:American Mathematical Society
2501:
2298:is empty, which implies that
618:{\displaystyle r<|z|<R}
245:{\displaystyle 1<|z|<R}
203:{\displaystyle r<|z|<1}
2797:Theorems in complex analysis
2443:Streater & Wightman 2000
2022:in the complex vector space
1672:
1635:{\displaystyle (\pi z)^{-1}}
1514:
1380:
7:
2766:Encyclopedia of Mathematics
2080:{\displaystyle E\times -iC}
1702:on the circle. Indeed, let
1488:{\displaystyle T_{+}=T_{-}}
1106:{\displaystyle f_{\pm }(z)}
572:{\displaystyle a_{n}=b_{n}}
10:
2818:
2730:Princeton University Press
2590:Kluwer Academic Publishers
2015:{\displaystyle E\times iC}
76:Continuous boundary values
2787:Several complex variables
2469:Berenstein & Gay 1991
2456:, pp. 63–65, 343–344
2213:) depending on variables
2107:with the tip of the cone.
1989:, called the edge. Write
1873:Sobolev embedding theorem
25:edge-of-the-wedge theorem
2316:
1357:{\displaystyle f_{\pm }}
1169:{\displaystyle f_{\pm }}
960:{\displaystyle f_{\pm }}
933:{\displaystyle T_{\pm }}
252:have Laurent expansions
71:The one-dimensional case
2165:is also holomorphic on
2055:for the opposite wedge
1648:Cauchy–Riemann operator
664:{\displaystyle I=(a,b)}
2761:"Bogolyubov's theorem"
2542:Reading, Massachusetts
2495:, pp. 63, 81, 110
2147:
2087:. Then the two wedges
2081:
2045:
2016:
1982:
1949:
1920:
1850:
1783:
1736:
1716:
1681:
1642:is known to provide a
1636:
1594:
1530:
1489:
1446:
1390:
1358:
1324:
1227:
1170:
1139:
1107:
1068:
961:
934:
913:, the boundary values
900:
813:
760:
710:
665:
619:
573:
530:
496:
430:
379:
354:
294:
246:
204:
154:
33:analytic continuations
2355:Russian Math. Surveys
2269:holomorphic functions
2148:
2082:
2046:
2017:
1983:
1959:be an open subset of
1950:
1921:
1851:
1784:
1737:
1717:
1682:
1637:
1595:
1531:
1490:
1447:
1391:
1359:
1325:
1228:
1171:
1140:
1138:{\displaystyle (N+1)}
1108:
1069:
962:
935:
901:
814:
761:
711:
666:
620:
574:
531:
479:
413:
380:
337:
277:
247:
205:
155:
153:{\displaystyle |z|=1}
41:analytic continuation
29:holomorphic functions
2120:
2103:with the product of
2099:, where we identify
2059:
2026:
1997:
1963:
1930:
1910:
1799:
1749:
1726:
1706:
1653:
1644:fundamental solution
1607:
1551:
1502:
1459:
1400:
1368:
1341:
1240:
1184:
1153:
1117:
1081:
974:
944:
917:
826:
770:
720:
675:
637:
587:
543:
395:
259:
214:
172:
128:
37:quantum field theory
2403:1957NCim....5.1598J
1541:elliptic regularity
2471:, pp. 256–265
2411:10.1007/BF02856049
2309:. See the book by
2143:
2077:
2041:
2012:
1978:
1945:
1916:
1846:
1779:
1732:
1712:
1677:
1632:
1590:
1547:is holomorphic in
1526:
1485:
1452:on the real axis.
1442:
1386:
1354:
1320:
1223:
1166:
1135:
1103:
1064:
1017:
957:
930:
896:
809:
756:
706:
661:
615:
569:
526:
375:
242:
200:
150:
49:Nikolay Bogoliubov
45:Wightman functions
2739:978-0-691-07062-9
2683:978-0-8218-1655-4
2649:978-0-387-52343-9
2599:978-0-7923-0540-8
2551:978-0-8053-0982-9
2544:: W.A. Benjamin,
2519:978-0-387-97349-4
2372:Bogoliubov, N. N.
2330:Vladimirov, V. S.
2204:, ...,
2095:meet at the edge
1919:{\displaystyle C}
1774:
1735:{\displaystyle g}
1715:{\displaystyle f}
1675:
1517:
1411:
1383:
1002:
695:
39:to construct the
2809:
2773:
2757:Vladimirov, V.S.
2750:
2702:
2660:
2610:
2568:Bogoliubov, N.N.
2562:
2534:Bogoliubov, N.N.
2522:
2496:
2490:
2484:
2483:, pp. 63–66
2478:
2472:
2466:
2457:
2451:
2445:
2440:
2434:
2429:
2423:
2422:
2397:(6): 1598–1610.
2386:
2380:
2379:
2368:
2362:
2347:V. S. Vladimirov
2344:
2338:
2337:
2326:
2152:
2150:
2149:
2144:
2142:
2086:
2084:
2083:
2078:
2050:
2048:
2047:
2042:
2040:
2039:
2034:
2021:
2019:
2018:
2013:
1987:
1985:
1984:
1979:
1977:
1976:
1971:
1954:
1952:
1951:
1946:
1944:
1943:
1938:
1925:
1923:
1922:
1917:
1895:The general case
1855:
1853:
1852:
1847:
1836:
1835:
1811:
1810:
1788:
1786:
1785:
1780:
1775:
1773:
1762:
1741:
1739:
1738:
1733:
1721:
1719:
1718:
1713:
1692:Cayley transform
1686:
1684:
1683:
1678:
1676:
1668:
1663:
1641:
1639:
1638:
1633:
1631:
1630:
1599:
1597:
1596:
1591:
1535:
1533:
1532:
1527:
1519:
1518:
1510:
1494:
1492:
1491:
1486:
1484:
1483:
1471:
1470:
1451:
1449:
1448:
1443:
1438:
1437:
1425:
1424:
1412:
1404:
1395:
1393:
1392:
1387:
1385:
1384:
1376:
1363:
1361:
1360:
1355:
1353:
1352:
1329:
1327:
1326:
1321:
1232:
1230:
1229:
1224:
1175:
1173:
1172:
1167:
1165:
1164:
1144:
1142:
1141:
1136:
1112:
1110:
1109:
1104:
1093:
1092:
1073:
1071:
1070:
1065:
1016:
989:
988:
966:
964:
963:
958:
956:
955:
939:
937:
936:
931:
929:
928:
905:
903:
902:
897:
895:
894:
886:
877:
866:
843:
842:
833:
818:
816:
815:
810:
765:
763:
762:
757:
715:
713:
712:
707:
705:
704:
693:
687:
686:
670:
668:
667:
662:
624:
622:
621:
616:
608:
600:
578:
576:
575:
570:
568:
567:
555:
554:
535:
533:
532:
527:
522:
521:
506:
505:
495:
490:
456:
455:
440:
439:
429:
424:
384:
382:
381:
376:
374:
373:
364:
363:
353:
348:
314:
313:
304:
303:
293:
288:
251:
249:
248:
243:
235:
227:
209:
207:
206:
201:
193:
185:
159:
157:
156:
151:
143:
135:
114:Morera's theorem
102:lower half-plane
98:upper half-plane
23:, Bogoliubov's
2817:
2816:
2812:
2811:
2810:
2808:
2807:
2806:
2777:
2776:
2740:
2684:
2650:
2640:Springer-Verlag
2620:Hörmander, Lars
2600:
2552:
2529:
2527:Further reading
2520:
2504:
2499:
2491:
2487:
2479:
2475:
2467:
2460:
2452:
2448:
2441:
2437:
2430:
2426:
2387:
2383:
2369:
2365:
2345:
2341:
2327:
2323:
2319:
2280:cotangent space
2257:
2238:
2228:
2218:
2212:
2203:
2191:
2135:
2121:
2118:
2117:
2060:
2057:
2056:
2035:
2030:
2029:
2027:
2024:
2023:
1998:
1995:
1994:
1972:
1967:
1966:
1964:
1961:
1960:
1939:
1934:
1933:
1931:
1928:
1927:
1911:
1908:
1907:
1897:
1831:
1827:
1806:
1802:
1800:
1797:
1796:
1766:
1761:
1750:
1747:
1746:
1727:
1724:
1723:
1707:
1704:
1703:
1667:
1659:
1654:
1651:
1650:
1623:
1619:
1608:
1605:
1604:
1552:
1549:
1548:
1509:
1505:
1503:
1500:
1499:
1479:
1475:
1466:
1462:
1460:
1457:
1456:
1433:
1429:
1420:
1416:
1403:
1401:
1398:
1397:
1375:
1371:
1369:
1366:
1365:
1348:
1344:
1342:
1339:
1338:
1241:
1238:
1237:
1233:by the formula
1185:
1182:
1181:
1160:
1156:
1154:
1151:
1150:
1118:
1115:
1114:
1088:
1084:
1082:
1079:
1078:
1006:
984:
980:
975:
972:
971:
951:
947:
945:
942:
941:
924:
920:
918:
915:
914:
887:
882:
881:
873:
862:
838:
834:
829:
827:
824:
823:
771:
768:
767:
721:
718:
717:
700:
696:
682:
678:
676:
673:
672:
638:
635:
634:
631:
604:
596:
588:
585:
584:
563:
559:
550:
546:
544:
541:
540:
511:
507:
501:
497:
491:
483:
445:
441:
435:
431:
425:
417:
396:
393:
392:
369:
365:
359:
355:
349:
341:
309:
305:
299:
295:
289:
281:
260:
257:
256:
231:
223:
215:
212:
211:
189:
181:
173:
170:
169:
168:in the regions
139:
131:
129:
126:
125:
122:
78:
73:
17:
12:
11:
5:
2815:
2805:
2804:
2799:
2794:
2789:
2775:
2774:
2752:
2751:
2738:
2716:Wightman, A.S.
2712:Streater, R.F.
2704:
2703:
2682:
2662:
2648:
2612:
2611:
2598:
2564:
2550:
2528:
2525:
2524:
2523:
2518:
2503:
2500:
2498:
2497:
2493:Hörmander 1990
2485:
2481:Hörmander 1990
2473:
2458:
2454:Hörmander 1990
2446:
2435:
2424:
2381:
2363:
2339:
2320:
2318:
2315:
2275:wave front set
2261:hyperfunctions
2256:
2253:
2233:
2224:
2216:
2208:
2201:
2190:
2187:
2175:
2174:
2141:
2138:
2134:
2131:
2128:
2125:
2076:
2073:
2070:
2067:
2064:
2038:
2033:
2011:
2008:
2005:
2002:
1993:for the wedge
1975:
1970:
1942:
1937:
1915:
1896:
1893:
1857:
1856:
1845:
1842:
1839:
1834:
1830:
1823:
1820:
1817:
1814:
1809:
1805:
1792:the equations
1790:
1789:
1778:
1772:
1769:
1765:
1760:
1757:
1754:
1731:
1711:
1700:Sobolev spaces
1696:Fourier series
1674:
1671:
1666:
1662:
1658:
1629:
1626:
1622:
1618:
1615:
1612:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1537:
1536:
1525:
1522:
1516:
1513:
1508:
1482:
1478:
1474:
1469:
1465:
1441:
1436:
1432:
1428:
1423:
1419:
1415:
1410:
1407:
1382:
1379:
1374:
1351:
1347:
1331:
1330:
1319:
1316:
1313:
1309:
1306:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1163:
1159:
1149:is defined as
1134:
1131:
1128:
1125:
1122:
1102:
1099:
1096:
1091:
1087:
1075:
1074:
1063:
1060:
1057:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1015:
1012:
1009:
1005:
1001:
998:
995:
992:
987:
983:
979:
954:
950:
927:
923:
907:
906:
893:
890:
885:
880:
876:
872:
869:
865:
861:
858:
855:
852:
849:
846:
841:
837:
832:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
778:
775:
755:
752:
749:
746:
743:
740:
737:
734:
731:
728:
725:
703:
699:
690:
685:
681:
660:
657:
654:
651:
648:
645:
642:
630:
627:
614:
611:
607:
603:
599:
595:
592:
566:
562:
558:
553:
549:
537:
536:
525:
520:
517:
514:
510:
504:
500:
494:
489:
486:
482:
478:
475:
472:
469:
466:
459:
454:
451:
448:
444:
438:
434:
428:
423:
420:
416:
412:
409:
406:
403:
400:
386:
385:
372:
368:
362:
358:
352:
347:
344:
340:
336:
333:
330:
327:
324:
317:
312:
308:
302:
298:
292:
287:
284:
280:
276:
273:
270:
267:
264:
241:
238:
234:
230:
226:
222:
219:
199:
196:
192:
188:
184:
180:
177:
149:
146:
142:
138:
134:
121:
118:
106:
105:
77:
74:
72:
69:
15:
9:
6:
4:
3:
2:
2814:
2803:
2800:
2798:
2795:
2793:
2790:
2788:
2785:
2784:
2782:
2772:
2768:
2767:
2762:
2758:
2754:
2753:
2749:
2745:
2741:
2735:
2731:
2727:
2726:Princeton, NJ
2723:
2722:
2717:
2713:
2709:
2708:
2707:
2701:
2697:
2693:
2689:
2685:
2679:
2675:
2671:
2667:
2666:Rudin, Walter
2663:
2659:
2655:
2651:
2645:
2641:
2637:
2633:
2629:
2625:
2621:
2617:
2616:
2615:
2609:
2605:
2601:
2595:
2591:
2587:
2583:
2579:
2575:
2574:
2569:
2565:
2561:
2557:
2553:
2547:
2543:
2539:
2535:
2531:
2530:
2521:
2515:
2511:
2506:
2505:
2494:
2489:
2482:
2477:
2470:
2465:
2463:
2455:
2450:
2444:
2439:
2433:
2428:
2420:
2416:
2412:
2408:
2404:
2400:
2396:
2392:
2391:Nuovo Cimento
2385:
2377:
2373:
2367:
2360:
2356:
2352:
2348:
2343:
2335:
2331:
2325:
2321:
2314:
2313:for details.
2312:
2308:
2303:
2301:
2297:
2293:
2288:
2283:
2281:
2277:
2276:
2270:
2266:
2265:hyperfunction
2262:
2252:
2250:
2246:
2242:
2236:
2232:
2227:
2223:
2219:
2211:
2207:
2200:
2196:
2186:
2184:
2180:
2172:
2168:
2164:
2160:
2156:
2139:
2136:
2132:
2129:
2126:
2123:
2115:
2112:Suppose that
2111:
2110:
2109:
2108:
2104:
2100:
2096:
2092:
2088:
2074:
2071:
2068:
2065:
2062:
2052:
2036:
2009:
2006:
2003:
2000:
1990:
1973:
1958:
1940:
1913:
1904:
1902:
1892:
1890:
1886:
1882:
1878:
1874:
1870:
1866:
1862:
1843:
1840:
1837:
1832:
1828:
1821:
1818:
1815:
1812:
1807:
1803:
1795:
1794:
1793:
1776:
1770:
1758:
1755:
1752:
1745:
1744:
1743:
1729:
1709:
1701:
1697:
1693:
1688:
1669:
1660:
1649:
1645:
1627:
1624:
1616:
1613:
1601:
1584:
1581:
1578:
1575:
1569:
1563:
1560:
1557:
1546:
1542:
1523:
1520:
1511:
1506:
1498:
1497:
1496:
1480:
1476:
1472:
1467:
1463:
1453:
1434:
1430:
1426:
1421:
1417:
1408:
1405:
1377:
1372:
1349:
1345:
1336:
1317:
1314:
1311:
1307:
1304:
1297:
1294:
1291:
1285:
1279:
1276:
1273:
1270:
1264:
1261:
1258:
1252:
1249:
1246:
1236:
1235:
1234:
1217:
1214:
1211:
1208:
1202:
1196:
1193:
1190:
1179:
1161:
1157:
1148:
1129:
1126:
1123:
1097:
1089:
1085:
1061:
1058:
1055:
1048:
1042:
1036:
1033:
1030:
1027:
1021:
1018:
1013:
1007:
999:
993:
990:
985:
981:
970:
969:
968:
952:
948:
925:
921:
912:
891:
888:
878:
870:
867:
856:
853:
850:
847:
839:
835:
822:
821:
820:
803:
800:
797:
794:
788:
782:
779:
776:
750:
747:
744:
738:
732:
729:
726:
701:
697:
688:
683:
679:
655:
652:
649:
643:
640:
626:
612:
609:
601:
593:
590:
582:
564:
560:
556:
551:
547:
523:
518:
515:
512:
508:
502:
498:
484:
480:
476:
470:
464:
457:
452:
449:
446:
442:
436:
432:
418:
414:
410:
404:
398:
391:
390:
389:
370:
366:
360:
356:
342:
338:
334:
328:
322:
315:
310:
306:
300:
296:
282:
278:
274:
268:
262:
255:
254:
253:
239:
236:
228:
220:
217:
197:
194:
186:
178:
175:
167:
163:
147:
144:
136:
117:
115:
111:
103:
100:, and on the
99:
95:
91:
90:complex plane
87:
84:Suppose that
83:
82:
81:
68:
66:
65:Freeman Dyson
62:
61:Harry Lehmann
58:
54:
50:
46:
42:
38:
34:
30:
27:implies that
26:
22:
2764:
2720:
2705:
2669:
2623:
2613:
2572:
2537:
2509:
2488:
2476:
2449:
2438:
2427:
2394:
2390:
2384:
2375:
2366:
2358:
2354:
2342:
2333:
2324:
2306:
2304:
2299:
2295:
2291:
2286:
2284:
2272:
2264:
2258:
2248:
2244:
2240:
2234:
2230:
2225:
2221:
2214:
2209:
2205:
2198:
2194:
2192:
2182:
2178:
2176:
2170:
2166:
2162:
2158:
2154:
2113:
2106:
2102:
2098:
2094:
2090:
2054:
2051:, and write
1992:
1988:
1956:
1905:
1900:
1898:
1888:
1884:
1880:
1876:
1868:
1864:
1860:
1858:
1791:
1689:
1602:
1544:
1538:
1454:
1334:
1332:
1177:
1146:
1076:
910:
908:
632:
580:
538:
387:
165:
161:
123:
107:
85:
79:
52:
24:
18:
2361:(5): 51—65.
819:satisfying
716:defined in
94:holomorphic
21:mathematics
2781:Categories
2748:1026.81027
2700:0214.09001
2658:0712.35001
2632:Heidelberg
2608:0732.46040
2560:1114.81300
2502:References
2432:Rudin 1971
1690:Using the
2771:EMS Press
2759:(2001) ,
2578:Dordrecht
2419:123500326
2311:Hörmander
2273:analytic
2133:∪
2127:∪
2069:−
2066:×
2004:×
1768:∂
1764:∂
1673:¯
1665:∂
1657:∂
1625:−
1614:π
1576:−
1570:×
1515:¯
1481:−
1435:−
1427:−
1381:¯
1350:±
1286:φ
1262:∬
1256:⟩
1253:φ
1244:⟨
1209:−
1203:×
1162:±
1090:±
1043:φ
1037:ε
1031:±
1019:∫
1011:↓
1008:ε
997:⟩
994:φ
986:±
978:⟨
953:±
926:±
889:−
840:±
795:−
789:×
739:×
702:−
519:θ
493:∞
488:∞
485:−
481:∑
471:θ
453:θ
427:∞
422:∞
419:−
415:∑
405:θ
351:∞
346:∞
343:−
339:∑
291:∞
286:∞
283:−
279:∑
110:real axis
2718:(2000),
2668:(1971),
2636:New York
2622:(1990),
2332:(1966),
2237:−1
2140:′
1646:for the
579:for all
92:that is
63:(1957),
57:Res Jost
2692:0310288
2399:Bibcode
2229:−
1495:, then
1337:equals
1113:is the
96:on the
2746:
2736:
2698:
2690:
2680:
2656:
2646:
2628:Berlin
2606:
2596:
2586:London
2582:Boston
2558:
2548:
2516:
2417:
694:
2415:S2CID
2317:Notes
1901:wedge
1333:then
2734:ISBN
2678:ISBN
2644:ISBN
2594:ISBN
2546:ISBN
2514:ISBN
2263:. A
2157:and
2091:and
1906:Let
1887:and
1879:and
1863:and
1722:and
1698:and
868:<
766:and
610:<
594:<
237:<
221:<
210:and
195:<
179:<
59:and
2744:Zbl
2696:Zbl
2654:Zbl
2604:Zbl
2556:Zbl
2407:doi
2353:",
2159:W'
2093:W'
2053:W'
1539:By
1004:lim
940:of
43:of
19:In
2783::
2769:,
2763:,
2742:,
2732:,
2728::
2714:;
2694:,
2688:MR
2686:,
2676:,
2652:,
2642:,
2638::
2602:,
2592:,
2588::
2554:,
2461:^
2413:.
2405:.
2393:.
2359:49
2357:,
2173:).
1899:A
1891:.
1687:.
1600:.
1524:0.
625:.
164:,
2661:.
2634:-
2630:-
2584:-
2580:-
2563:.
2421:.
2409::
2401::
2395:5
2300:f
2296:f
2292:f
2287:f
2249:n
2245:n
2241:n
2235:i
2231:z
2226:i
2222:z
2217:i
2215:z
2210:n
2206:z
2202:1
2199:z
2197:(
2195:W
2183:f
2179:f
2171:E
2167:E
2163:f
2155:W
2137:W
2130:E
2124:W
2114:f
2105:E
2101:E
2097:E
2089:W
2075:C
2072:i
2063:E
2037:n
2032:C
2010:C
2007:i
2001:E
1991:W
1974:n
1969:R
1957:E
1941:n
1936:R
1914:C
1889:g
1885:f
1881:G
1877:F
1869:k
1865:F
1861:G
1844:g
1841:=
1838:G
1833:k
1829:D
1822:,
1819:f
1816:=
1813:F
1808:k
1804:D
1777:,
1771:z
1759:z
1756:=
1753:D
1730:g
1710:f
1670:z
1661:/
1628:1
1621:)
1617:z
1611:(
1588:)
1585:R
1582:,
1579:R
1573:(
1567:)
1564:b
1561:,
1558:a
1555:(
1545:F
1521:=
1512:z
1507:F
1477:T
1473:=
1468:+
1464:T
1440:)
1431:T
1422:+
1418:T
1414:(
1409:2
1406:1
1378:z
1373:F
1346:f
1335:F
1318:,
1315:y
1312:d
1308:x
1305:d
1301:)
1298:y
1295:,
1292:x
1289:(
1283:)
1280:y
1277:i
1274:+
1271:x
1268:(
1265:f
1259:=
1250:,
1247:F
1221:)
1218:R
1215:,
1212:R
1206:(
1200:)
1197:b
1194:,
1191:a
1188:(
1178:F
1158:f
1147:f
1133:)
1130:1
1127:+
1124:N
1121:(
1101:)
1098:z
1095:(
1086:f
1062:.
1059:x
1056:d
1052:)
1049:x
1046:(
1040:)
1034:i
1028:x
1025:(
1022:f
1014:0
1000:=
991:,
982:T
949:f
922:T
911:N
892:N
884:|
879:y
875:|
871:C
864:|
860:)
857:y
854:i
851:+
848:x
845:(
836:f
831:|
807:)
804:0
801:,
798:R
792:(
786:)
783:b
780:,
777:a
774:(
754:)
751:R
748:,
745:0
742:(
736:)
733:b
730:,
727:a
724:(
698:f
689:,
684:+
680:f
659:)
656:b
653:,
650:a
647:(
644:=
641:I
613:R
606:|
602:z
598:|
591:r
581:n
565:n
561:b
557:=
552:n
548:a
524:.
516:n
513:i
509:e
503:n
499:b
477:=
474:)
468:(
465:g
458:,
450:n
447:i
443:e
437:n
433:a
411:=
408:)
402:(
399:f
371:n
367:z
361:n
357:b
335:=
332:)
329:z
326:(
323:g
316:,
311:n
307:z
301:n
297:a
275:=
272:)
269:z
266:(
263:f
240:R
233:|
229:z
225:|
218:1
198:1
191:|
187:z
183:|
176:r
166:g
162:f
148:1
145:=
141:|
137:z
133:|
86:f
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