2116:
2759:
1900:
638:
643:
Since the zeroth cohomology group of any sheaf is simply the global sections of that sheaf, we see that a hyperfunction is a pair of holomorphic functions one each on the upper and lower complex halfplane modulo entire holomorphic functions.
2463:
810:
320:
1456:
1214:
just above the real line - both from left to right. Note that the hyperfunction can be non-trivial, even if the components are analytic continuation of the same function. Also this can be easily checked by differentiating the
Heaviside
1074:
2648:
449:
1391:
Using a partition of unity one can write any continuous function (distribution) as a locally finite sum of functions (distributions) with compact support. This can be exploited to extend the above embedding to an embedding
1358:
1819:
1200:
3186:
3371:
2192:
3584:
1905:
3294:
2596:
2523:
3124:
847:
890:
3074:
3035:
3237:
2836:
3332:
2958:
2338:
1628:
706:
2879:
2264:
1895:
2643:
2795:
956:
457:
3514:
3481:
2991:
2149:
1665:
678:
381:
352:
2111:{\displaystyle {\begin{aligned}h(f_{+},f_{-})&:=(hf_{+},hf_{-})\\{\frac {d}{dz}}(f_{+},f_{-})&:=\left({\frac {df_{+}}{dz}},{\frac {df_{-}}{dz}}\right)\end{aligned}}}
1845:
197:
230:
3428:
3406:
2920:
2333:
919:
1106:
711:
2221:
1587:
1548:
1505:
1257:
246:
3448:
1670:
121:
2307:
2284:
1395:
1126:
3129:
394:
3519:
2754:{\displaystyle {\begin{cases}{\mathcal {B}}_{c}(U)\times {\mathcal {O}}(U)\to \mathbb {C} \\(f,\varphi )\mapsto \int f\cdot \varphi \end{cases}}}
3688:, Princeton Legacy Library (Book 5158), vol. PMS-37, translated by Kato, Goro (Reprint ed.), Princeton University Press,
3337:
2158:
3191:
2528:
3959:
3938:
3809:
3713:
3693:
3670:
3649:
2881:
A special case worth considering is the case of continuous functions or distributions with compact support: If one considers
2468:
3788:
3079:
3242:
74:
A hyperfunction on the real line can be conceived of as the 'difference' between one holomorphic function defined on the
815:
852:
3040:
123:
would be at the real line itself. This difference is not affected by adding the same holomorphic function to both
3924:
3835:
4029:
4003:
2996:
1605:
683:
2841:
2230:
1861:
19:
This article is about hyperfunctions in a mathematical context. For biological hypo- or hyperfunctions, see
3897:
Journal of the
Faculty of Science, University of Tokyo. Sect. 1, Mathematics, Astronomy, Physics, Chemistry
3868:
Journal of the
Faculty of Science, University of Tokyo. Sect. 1, Mathematics, Astronomy, Physics, Chemistry
3616:
2608:
1667:
is a vector space such that addition and multiplication with complex numbers are well-defined. Explicitly:
1203:
633:{\displaystyle H_{\mathbb {R} }^{1}(\mathbb {C} ,{\mathcal {O}})=\left/H^{0}(\mathbb {C} ,{\mathcal {O}}).}
4019:
3998:
2800:
3299:
2925:
2764:
3606:
3486:
3453:
2993:
via the above embedding, then this computes exactly the traditional
Lebesgue-integral. Furthermore: If
2963:
2598:
The integrals are independent of the choice of these curves because the upper and lower half plane are
2121:
1637:
1227:
650:
39:
357:
328:
4024:
1109:
2657:
1826:
178:
3993:
3239:
which are undefined in the usual sense. Moreover: Because the real analytic functions are dense in
210:
3411:
3379:
2884:
2312:
895:
950:
2458:{\displaystyle \int _{a}^{b}f:=-\int _{\gamma _{+}}f_{+}(z)\,dz+\int _{\gamma _{-}}f_{-}(z)\,dz}
921:
giving another reason to think of hyperfunctions as "boundary values" of holomorphic functions.
805:{\displaystyle H^{0}({\tilde {U}}\setminus U,{\mathcal {O}})/H^{0}({\tilde {U}},{\mathcal {O}})}
315:{\displaystyle {\mathcal {B}}(\mathbb {R} )=H_{\mathbb {R} }^{1}(\mathbb {C} ,{\mathcal {O}}).}
63:
4034:
3705:
Hyperfunctions and Pseudo-Differential
Equations, Proceedings of a Conference at Katata, 1971
3635:
1508:
1465:
1079:
2200:
1557:
1518:
1475:
3916:
3887:
3759:
3601:
3433:
1120:
1069:{\displaystyle H(x)=\left(1-{\frac {1}{2\pi i}}\log(z),-{\frac {1}{2\pi i}}\log(z)\right).}
204:
35:
3767:
2761:
one associates to each hyperfunction with compact support a continuous linear function on
8:
3611:
1848:
1589:
looks like a "distribution of infinite order" at 0. (Note that distributions always have
1451:{\displaystyle \textstyle {\mathcal {D}}'(\mathbb {R} )\to {\mathcal {B}}(\mathbb {R} ).}
1223:
200:
100:
2289:
444:{\displaystyle \mathbb {C} ^{+}\cup \mathbb {C} ^{-}=\mathbb {C} \setminus \mathbb {R} }
3777:
3596:
2269:
3954:, Progress in Mathematics (Softcover reprint of the original 1st ed.), Springer,
3745:
78:
and another on the lower half-plane. That is, a hyperfunction is specified by a pair (
3955:
3934:
3855:
3805:
3784:
3783:, Translations of Mathematical Monographs (Book 129), American Mathematical Society,
3709:
3689:
3666:
3645:
20:
3749:
3977:
3904:
3875:
3847:
3763:
3679:
3621:
3188:
Thus this notion of integration gives a precise meaning to formal expressions like
2599:
388:
384:
237:
172:
75:
59:
3912:
3883:
3755:
934:
is any holomorphic function on the whole complex plane, then the restriction of
1353:{\displaystyle f(z)={\frac {1}{2\pi i}}\int _{x\in I}g(x){\frac {1}{z-x}}\,dx.}
3949:
3928:
3799:
3703:
3683:
3660:
3639:
4013:
3859:
3851:
136:
1814:{\displaystyle a(f_{+},f_{-})+b(g_{+},g_{-}):=(af_{+}+bg_{+},af_{-}+bg_{-})}
2645:
be the space of hyperfunctions with compact support. Via the bilinear form
1852:
3801:
Hyperfunctions and
Theoretical Physics, Rencontre de Nice, 21-30 Mai 1973
1385:
1195:{\displaystyle \left({\dfrac {1}{2\pi iz}},{\dfrac {1}{2\pi iz}}\right).}
27:
3181:{\displaystyle \int _{a}^{b}u\cdot \varphi =\langle u,\varphi \rangle .}
3831:
43:
3982:
3908:
3879:
3725:
Relative cohomology of sheaves of solutions of differential equations
2286:
restricts to a real analytic function in some small neighbourhood of
892:. One can show that this definition does not depend on the choice of
233:
2152:
3579:{\displaystyle f\circ \Phi :=(f_{+}\circ \Phi ,f_{-}\circ \Phi )}
1230:) on the real line with support contained in a bounded interval
2335:
are two holomorphic points, then integration is well-defined:
171:
The motivation can be concretely implemented using ideas from
38:
to another at a boundary, and can be thought of informally as
3366:{\displaystyle {\mathcal {E}}'\hookrightarrow {\mathcal {B}}}
2187:{\displaystyle {\mathcal {D}}'\hookrightarrow {\mathcal {B}}}
1464:
is any function that is holomorphic everywhere except for an
938:
to the real axis is a hyperfunction, represented by either (
3334:. This is an alternative description of the same embedding
2747:
3754:, Séminaire Bourbaki, Tome 6 (1960-1961), Exposé no. 214,
3951:
Hyperfunctions and
Harmonic Analysis on Symmetric Spaces
2591:{\displaystyle \gamma _{\pm }(0)=a,\gamma _{\pm }(1)=b.}
3665:, Mathematics and its Applications (Book 3), Springer,
3644:, Mathematics and its Applications (Book 8), Springer,
34:
are generalizations of functions, as a 'jump' from one
3974:
2518:{\displaystyle \gamma _{\pm }:\to \mathbb {C} ^{\pm }}
1399:
1210:
just below the real line, and subtract integration of
90:
is a holomorphic function on the upper half-plane and
3522:
3489:
3456:
3436:
3414:
3382:
3340:
3302:
3245:
3194:
3132:
3119:{\displaystyle \operatorname {supp} (u)\subset (a,b)}
3082:
3043:
2999:
2966:
2928:
2887:
2844:
2803:
2767:
2651:
2611:
2531:
2471:
2341:
2315:
2292:
2272:
2233:
2203:
2161:
2124:
1903:
1864:
1829:
1673:
1640:
1608:
1560:
1521:
1478:
1398:
1260:
1162:
1136:
1129:
1082:
959:
898:
855:
818:
714:
686:
653:
460:
397:
360:
331:
249:
213:
181:
103:
97:
Informally, the hyperfunction is what the difference
42:
of infinite order. Hyperfunctions were introduced by
3895:
3678:
3289:{\displaystyle {\mathcal {E}}(U),{\mathcal {E}}'(U)}
1206:. To verify it one can calculate the integration of
3866:Sato, Mikio (1959), "Theory of Hyperfunctions, I",
94:is a holomorphic function on the lower half-plane.
3776:
3578:
3508:
3475:
3442:
3422:
3400:
3365:
3326:
3288:
3231:
3180:
3118:
3068:
3029:
2985:
2952:
2914:
2873:
2830:
2797:This induces an identification of the dual space,
2789:
2753:
2637:
2590:
2517:
2457:
2327:
2301:
2278:
2258:
2215:
2186:
2143:
2110:
1889:
1839:
1813:
1659:
1622:
1581:
1542:
1499:
1450:
1352:
1194:
1100:
1068:
913:
884:
842:{\displaystyle {\tilde {U}}\subseteq \mathbb {C} }
841:
804:
700:
672:
632:
443:
375:
346:
314:
224:
191:
115:
3818:
3733:Sato, Mikio; Kawai, Takahiro; Kashiwara, Masaki,
4011:
3735:Microfunctions and pseudo-differential equations
1597:
1250:is a holomorphic function on the complement of
885:{\displaystyle {\tilde {U}}\cap \mathbb {R} =U}
3933:, Lecture Notes in Mathematics 126, Springer,
3836:"Cyōkansū no riron (Theory of Hyperfunctions)"
3804:, Lecture Notes in Mathematics 449, Springer,
3744:
3732:
3708:, Lecture Notes in Mathematics 287, Springer,
166:
3947:
3069:{\displaystyle \varphi \in {\mathcal {O}}(U)}
1380:follows from the previous example by writing
3662:Introduction to the Theory of Hyperfunctions
3408:is a real analytic map between open sets of
3172:
3160:
1858:Multiplication with real analytic functions
3682:; Kawai, Takahiro; Kimura, Tatsuo (2017) ,
3232:{\displaystyle \int _{a}^{b}\delta (x)\,dx}
1372:) when crossing the real axis at the point
58:in English), building upon earlier work by
3846:(1), Mathematical Society of Japan: 1–27,
3416:
3222:
2706:
2505:
2448:
2402:
1616:
1437:
1416:
1340:
872:
835:
694:
610:
563:
522:
482:
467:
437:
429:
415:
400:
363:
334:
292:
277:
261:
215:
3923:
3779:An Introduction to Sato's Hyperfunctions
3774:
3037:is a distribution with compact support,
1388:of itself with the Dirac delta function.
1110:principal value of the complex logarithm
3819:Cerezo, A.; Piriou, A.; Chazarain, J.,
3701:
3030:{\displaystyle u\in {\mathcal {E}}'(U)}
1623:{\displaystyle U\subseteq \mathbb {R} }
701:{\displaystyle U\subseteq \mathbb {R} }
135:is a holomorphic function on the whole
4012:
3991:
3658:
2874:{\displaystyle {\mathcal {B}}_{c}(U).}
2259:{\displaystyle f\in {\mathcal {B}}(U)}
1890:{\displaystyle h\in {\mathcal {O}}(U)}
3975:
2638:{\displaystyle {\mathcal {B}}_{c}(U)}
1897:and differentiation are well-defined:
1515:has a pole of finite order at 0 then
3894:
3865:
3830:
3797:
3634:
55:
51:
47:
3722:
2831:{\displaystyle {\mathcal {O}}'(U),}
13:
3570:
3551:
3529:
3492:
3459:
3437:
3383:
3358:
3344:
3327:{\displaystyle {\mathcal {O}}'(U)}
3306:
3268:
3248:
3052:
3009:
2969:
2953:{\displaystyle {\mathcal {E}}'(U)}
2932:
2848:
2807:
2790:{\displaystyle {\mathcal {O}}(U).}
2770:
2688:
2663:
2615:
2242:
2179:
2165:
2127:
1873:
1832:
1823:The obvious restriction maps turn
1643:
1554:has an essential singularity then
1428:
1403:
1238:corresponds to the hyperfunction (
794:
751:
656:
619:
578:
537:
491:
301:
252:
184:
14:
4046:
3968:
3702:Komatsu, Hikosaburo, ed. (1973),
3685:Foundations of Algebraic Analysis
3509:{\displaystyle {\mathcal {B}}(U)}
3476:{\displaystyle {\mathcal {B}}(V)}
3076:is a real analytic function, and
2986:{\displaystyle {\mathcal {B}}(U)}
2144:{\displaystyle {\mathcal {B}}(U)}
1660:{\displaystyle {\mathcal {B}}(U)}
1511:0 that is not a distribution. If
740:
673:{\displaystyle {\mathcal {B}}(U)}
433:
232:Define the hyperfunctions on the
3450:is a well-defined operator from
1202:This is really a restatement of
376:{\displaystyle \mathbb {C} ^{-}}
347:{\displaystyle \mathbb {C} ^{+}}
163:) are defined to be equivalent.
3948:Schlichtkrull, Henrik (2013) ,
3821:Introduction aux hyperfonctions
3573:
3535:
3503:
3497:
3470:
3464:
3392:
3353:
3321:
3315:
3283:
3277:
3259:
3253:
3219:
3213:
3113:
3101:
3095:
3089:
3063:
3057:
3024:
3018:
2980:
2974:
2947:
2941:
2909:
2903:
2865:
2859:
2822:
2816:
2781:
2775:
2729:
2726:
2714:
2702:
2699:
2693:
2680:
2674:
2632:
2626:
2576:
2570:
2548:
2542:
2500:
2497:
2485:
2445:
2439:
2399:
2393:
2253:
2247:
2174:
2138:
2132:
2027:
2001:
1979:
1947:
1937:
1911:
1884:
1878:
1840:{\displaystyle {\mathcal {B}}}
1808:
1744:
1738:
1712:
1703:
1677:
1654:
1648:
1576:
1561:
1537:
1522:
1494:
1479:
1441:
1433:
1423:
1420:
1412:
1319:
1313:
1270:
1264:
1095:
1089:
1055:
1049:
1016:
1010:
969:
963:
905:
862:
825:
799:
783:
774:
756:
734:
725:
667:
661:
647:More generally one can define
624:
606:
583:
558:
542:
517:
496:
478:
306:
288:
265:
257:
192:{\displaystyle {\mathcal {O}}}
69:
1:
3751:Les hyperfonctions de M. Sato
3628:
225:{\displaystyle \mathbb {C} .}
3641:Applied Hyperfunction Theory
3617:Pseudo-differential operator
3423:{\displaystyle \mathbb {R} }
3401:{\displaystyle \Phi :U\to V}
2915:{\displaystyle C_{c}^{0}(U)}
2328:{\displaystyle a\leqslant b}
1598:Operations on hyperfunctions
914:{\displaystyle {\tilde {U}}}
16:Type of generalized function
7:
3999:Encyclopedia of Mathematics
3930:Theories des Hyperfonctions
3589:
2194:is a morphism of D-modules.
1550:is a distribution, so when
924:
167:Definition in one dimension
10:
4051:
3607:Distribution (mathematics)
2525:are arbitrary curves with
18:
3798:Pham, F. L., ed. (1975),
3775:Morimoto, Mitsuo (1993),
1204:Cauchy's integral formula
3852:10.11429/sugaku1947.10.1
3430:, then composition with
1507:is a hyperfunction with
2118:With these definitions
1101:{\displaystyle \log(z)}
951:Heaviside step function
942:, 0) or (0, −
3659:Kaneko, Akira (1988),
3580:
3510:
3477:
3444:
3424:
3402:
3367:
3328:
3290:
3233:
3182:
3120:
3070:
3031:
2987:
2954:
2916:
2875:
2832:
2791:
2755:
2639:
2592:
2519:
2459:
2329:
2303:
2280:
2260:
2217:
2216:{\displaystyle a\in U}
2188:
2145:
2112:
1891:
1841:
1815:
1661:
1624:
1583:
1582:{\displaystyle (f,-f)}
1544:
1543:{\displaystyle (f,-f)}
1501:
1500:{\displaystyle (f,-f)}
1452:
1354:
1196:
1121:Dirac delta "function"
1102:
1070:
953:can be represented as
915:
886:
843:
806:
702:
674:
634:
445:
377:
348:
316:
226:
193:
139:, the hyperfunctions (
117:
4030:Generalized functions
3739:. - It is called SKK.
3723:Komatsu, Hikosaburo,
3581:
3511:
3478:
3445:
3443:{\displaystyle \Phi }
3425:
3403:
3368:
3329:
3291:
3234:
3183:
3121:
3071:
3032:
2988:
2955:
2917:
2876:
2833:
2792:
2756:
2640:
2593:
2520:
2460:
2330:
2304:
2281:
2261:
2218:
2189:
2146:
2113:
1892:
1842:
1816:
1662:
1625:
1584:
1545:
1502:
1466:essential singularity
1453:
1355:
1226:(or more generally a
1197:
1103:
1071:
916:
887:
849:is any open set with
844:
807:
703:
675:
635:
446:
378:
349:
317:
227:
205:holomorphic functions
194:
118:
3992:Kaneko, A. (2001) ,
3602:Generalized function
3520:
3487:
3454:
3434:
3412:
3380:
3338:
3300:
3243:
3192:
3130:
3080:
3041:
2997:
2964:
2926:
2885:
2842:
2801:
2765:
2649:
2609:
2529:
2469:
2339:
2313:
2290:
2270:
2231:
2201:
2159:
2122:
1901:
1862:
1827:
1671:
1638:
1630:be any open subset.
1606:
1593:order at any point.)
1558:
1519:
1476:
1396:
1258:
1127:
1080:
957:
896:
853:
816:
712:
684:
651:
458:
395:
358:
329:
247:
211:
179:
101:
36:holomorphic function
3612:Microlocal analysis
3209:
3147:
2902:
2356:
1468:at 0 (for example,
1224:continuous function
477:
391:respectively. Then
287:
116:{\displaystyle f-g}
4020:Algebraic analysis
3737:, pp. 265–529
3727:, pp. 192–261
3597:Algebraic analysis
3576:
3506:
3473:
3440:
3420:
3398:
3363:
3324:
3286:
3229:
3195:
3178:
3133:
3116:
3066:
3027:
2983:
2950:
2912:
2888:
2871:
2828:
2787:
2751:
2746:
2635:
2588:
2515:
2455:
2342:
2325:
2302:{\displaystyle a.}
2299:
2276:
2256:
2213:
2184:
2155:and the embedding
2141:
2108:
2106:
1887:
1851:(which is in fact
1837:
1811:
1657:
1620:
1579:
1540:
1497:
1448:
1447:
1376:. The formula for
1364:jumps in value by
1350:
1192:
1182:
1156:
1123:is represented by
1098:
1066:
911:
882:
839:
802:
698:
670:
630:
461:
441:
373:
344:
312:
271:
222:
189:
113:
3961:978-1-4612-9775-8
3940:978-3-540-04915-9
3811:978-3-540-37454-1
3715:978-3-540-06218-9
3695:978-0-691-62832-5
3680:Kashiwara, Masaki
3672:978-90-277-2837-1
3651:978-94-010-5125-5
3296:is a subspace of
2960:) as a subset of
2279:{\displaystyle f}
2225:holomorphic point
2097:
2067:
1999:
1338:
1292:
1181:
1155:
1041:
1002:
908:
865:
828:
786:
737:
680:for any open set
21:endocrine disease
4042:
4025:Complex analysis
4006:
3988:
3987:
3964:
3943:
3925:Schapira, Pierre
3919:
3890:
3862:
3824:
3814:
3793:
3790:978-0-82184571-4
3782:
3770:
3746:Martineau, André
3738:
3728:
3718:
3698:
3675:
3654:
3622:Sheaf cohomology
3585:
3583:
3582:
3577:
3566:
3565:
3547:
3546:
3515:
3513:
3512:
3507:
3496:
3495:
3482:
3480:
3479:
3474:
3463:
3462:
3449:
3447:
3446:
3441:
3429:
3427:
3426:
3421:
3419:
3407:
3405:
3404:
3399:
3372:
3370:
3369:
3364:
3362:
3361:
3352:
3348:
3347:
3333:
3331:
3330:
3325:
3314:
3310:
3309:
3295:
3293:
3292:
3287:
3276:
3272:
3271:
3252:
3251:
3238:
3236:
3235:
3230:
3208:
3203:
3187:
3185:
3184:
3179:
3146:
3141:
3125:
3123:
3122:
3117:
3075:
3073:
3072:
3067:
3056:
3055:
3036:
3034:
3033:
3028:
3017:
3013:
3012:
2992:
2990:
2989:
2984:
2973:
2972:
2959:
2957:
2956:
2951:
2940:
2936:
2935:
2921:
2919:
2918:
2913:
2901:
2896:
2880:
2878:
2877:
2872:
2858:
2857:
2852:
2851:
2837:
2835:
2834:
2829:
2815:
2811:
2810:
2796:
2794:
2793:
2788:
2774:
2773:
2760:
2758:
2757:
2752:
2750:
2749:
2709:
2692:
2691:
2673:
2672:
2667:
2666:
2644:
2642:
2641:
2636:
2625:
2624:
2619:
2618:
2600:simply connected
2597:
2595:
2594:
2589:
2569:
2568:
2541:
2540:
2524:
2522:
2521:
2516:
2514:
2513:
2508:
2481:
2480:
2464:
2462:
2461:
2456:
2438:
2437:
2428:
2427:
2426:
2425:
2392:
2391:
2382:
2381:
2380:
2379:
2355:
2350:
2334:
2332:
2331:
2326:
2308:
2306:
2305:
2300:
2285:
2283:
2282:
2277:
2265:
2263:
2262:
2257:
2246:
2245:
2222:
2220:
2219:
2214:
2193:
2191:
2190:
2185:
2183:
2182:
2173:
2169:
2168:
2150:
2148:
2147:
2142:
2131:
2130:
2117:
2115:
2114:
2109:
2107:
2103:
2099:
2098:
2096:
2088:
2087:
2086:
2073:
2068:
2066:
2058:
2057:
2056:
2043:
2026:
2025:
2013:
2012:
2000:
1998:
1987:
1978:
1977:
1962:
1961:
1936:
1935:
1923:
1922:
1896:
1894:
1893:
1888:
1877:
1876:
1846:
1844:
1843:
1838:
1836:
1835:
1820:
1818:
1817:
1812:
1807:
1806:
1791:
1790:
1775:
1774:
1759:
1758:
1737:
1736:
1724:
1723:
1702:
1701:
1689:
1688:
1666:
1664:
1663:
1658:
1647:
1646:
1629:
1627:
1626:
1621:
1619:
1588:
1586:
1585:
1580:
1549:
1547:
1546:
1541:
1506:
1504:
1503:
1498:
1457:
1455:
1454:
1449:
1440:
1432:
1431:
1419:
1411:
1407:
1406:
1359:
1357:
1356:
1351:
1339:
1337:
1323:
1309:
1308:
1293:
1291:
1277:
1201:
1199:
1198:
1193:
1188:
1184:
1183:
1180:
1163:
1157:
1154:
1137:
1115:
1107:
1105:
1104:
1099:
1075:
1073:
1072:
1067:
1062:
1058:
1042:
1040:
1026:
1003:
1001:
987:
920:
918:
917:
912:
910:
909:
901:
891:
889:
888:
883:
875:
867:
866:
858:
848:
846:
845:
840:
838:
830:
829:
821:
811:
809:
808:
803:
798:
797:
788:
787:
779:
773:
772:
763:
755:
754:
739:
738:
730:
724:
723:
708:as the quotient
707:
705:
704:
699:
697:
679:
677:
676:
671:
660:
659:
639:
637:
636:
631:
623:
622:
613:
605:
604:
595:
590:
586:
582:
581:
572:
571:
566:
557:
556:
541:
540:
531:
530:
525:
516:
515:
495:
494:
485:
476:
471:
470:
450:
448:
447:
442:
440:
432:
424:
423:
418:
409:
408:
403:
389:lower half-plane
385:upper half-plane
382:
380:
379:
374:
372:
371:
366:
353:
351:
350:
345:
343:
342:
337:
325:Concretely, let
321:
319:
318:
313:
305:
304:
295:
286:
281:
280:
264:
256:
255:
238:local cohomology
231:
229:
228:
223:
218:
198:
196:
195:
190:
188:
187:
173:sheaf cohomology
122:
120:
119:
114:
76:upper half-plane
60:Laurent Schwartz
4050:
4049:
4045:
4044:
4043:
4041:
4040:
4039:
4010:
4009:
3994:"Hyperfunction"
3978:"Hyperfunction"
3976:Jacobs, Bryan.
3971:
3962:
3941:
3842:(in Japanese),
3823:, pp. 1–53
3812:
3791:
3716:
3696:
3673:
3652:
3631:
3626:
3592:
3561:
3557:
3542:
3538:
3521:
3518:
3517:
3491:
3490:
3488:
3485:
3484:
3458:
3457:
3455:
3452:
3451:
3435:
3432:
3431:
3415:
3413:
3410:
3409:
3381:
3378:
3377:
3357:
3356:
3343:
3342:
3341:
3339:
3336:
3335:
3305:
3304:
3303:
3301:
3298:
3297:
3267:
3266:
3265:
3247:
3246:
3244:
3241:
3240:
3204:
3199:
3193:
3190:
3189:
3142:
3137:
3131:
3128:
3127:
3081:
3078:
3077:
3051:
3050:
3042:
3039:
3038:
3008:
3007:
3006:
2998:
2995:
2994:
2968:
2967:
2965:
2962:
2961:
2931:
2930:
2929:
2927:
2924:
2923:
2897:
2892:
2886:
2883:
2882:
2853:
2847:
2846:
2845:
2843:
2840:
2839:
2806:
2805:
2804:
2802:
2799:
2798:
2769:
2768:
2766:
2763:
2762:
2745:
2744:
2711:
2710:
2705:
2687:
2686:
2668:
2662:
2661:
2660:
2653:
2652:
2650:
2647:
2646:
2620:
2614:
2613:
2612:
2610:
2607:
2606:
2564:
2560:
2536:
2532:
2530:
2527:
2526:
2509:
2504:
2503:
2476:
2472:
2470:
2467:
2466:
2433:
2429:
2421:
2417:
2416:
2412:
2387:
2383:
2375:
2371:
2370:
2366:
2351:
2346:
2340:
2337:
2336:
2314:
2311:
2310:
2291:
2288:
2287:
2271:
2268:
2267:
2241:
2240:
2232:
2229:
2228:
2202:
2199:
2198:
2178:
2177:
2164:
2163:
2162:
2160:
2157:
2156:
2126:
2125:
2123:
2120:
2119:
2105:
2104:
2089:
2082:
2078:
2074:
2072:
2059:
2052:
2048:
2044:
2042:
2041:
2037:
2030:
2021:
2017:
2008:
2004:
1991:
1986:
1983:
1982:
1973:
1969:
1957:
1953:
1940:
1931:
1927:
1918:
1914:
1904:
1902:
1899:
1898:
1872:
1871:
1863:
1860:
1859:
1831:
1830:
1828:
1825:
1824:
1802:
1798:
1786:
1782:
1770:
1766:
1754:
1750:
1732:
1728:
1719:
1715:
1697:
1693:
1684:
1680:
1672:
1669:
1668:
1642:
1641:
1639:
1636:
1635:
1615:
1607:
1604:
1603:
1600:
1559:
1556:
1555:
1520:
1517:
1516:
1477:
1474:
1473:
1436:
1427:
1426:
1415:
1402:
1401:
1400:
1397:
1394:
1393:
1327:
1322:
1298:
1294:
1281:
1276:
1259:
1256:
1255:
1167:
1161:
1141:
1135:
1134:
1130:
1128:
1125:
1124:
1113:
1081:
1078:
1077:
1030:
1025:
991:
986:
979:
975:
958:
955:
954:
927:
900:
899:
897:
894:
893:
871:
857:
856:
854:
851:
850:
834:
820:
819:
817:
814:
813:
793:
792:
778:
777:
768:
764:
759:
750:
749:
729:
728:
719:
715:
713:
710:
709:
693:
685:
682:
681:
655:
654:
652:
649:
648:
618:
617:
609:
600:
596:
591:
577:
576:
567:
562:
561:
552:
548:
536:
535:
526:
521:
520:
511:
507:
506:
502:
490:
489:
481:
472:
466:
465:
459:
456:
455:
436:
428:
419:
414:
413:
404:
399:
398:
396:
393:
392:
367:
362:
361:
359:
356:
355:
338:
333:
332:
330:
327:
326:
300:
299:
291:
282:
276:
275:
260:
251:
250:
248:
245:
244:
214:
212:
209:
208:
183:
182:
180:
177:
176:
169:
102:
99:
98:
72:
24:
17:
12:
11:
5:
4048:
4038:
4037:
4032:
4027:
4022:
4008:
4007:
3989:
3970:
3969:External links
3967:
3966:
3965:
3960:
3945:
3939:
3921:
3903:(2): 387–437,
3892:
3874:(1): 139–193,
3863:
3828:
3827:
3826:
3810:
3795:
3789:
3772:
3742:
3741:
3740:
3730:
3714:
3699:
3694:
3676:
3671:
3656:
3650:
3630:
3627:
3625:
3624:
3619:
3614:
3609:
3604:
3599:
3593:
3591:
3588:
3587:
3586:
3575:
3572:
3569:
3564:
3560:
3556:
3553:
3550:
3545:
3541:
3537:
3534:
3531:
3528:
3525:
3505:
3502:
3499:
3494:
3472:
3469:
3466:
3461:
3439:
3418:
3397:
3394:
3391:
3388:
3385:
3374:
3360:
3355:
3351:
3346:
3323:
3320:
3317:
3313:
3308:
3285:
3282:
3279:
3275:
3270:
3264:
3261:
3258:
3255:
3250:
3228:
3225:
3221:
3218:
3215:
3212:
3207:
3202:
3198:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3145:
3140:
3136:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3091:
3088:
3085:
3065:
3062:
3059:
3054:
3049:
3046:
3026:
3023:
3020:
3016:
3011:
3005:
3002:
2982:
2979:
2976:
2971:
2949:
2946:
2943:
2939:
2934:
2911:
2908:
2905:
2900:
2895:
2891:
2870:
2867:
2864:
2861:
2856:
2850:
2827:
2824:
2821:
2818:
2814:
2809:
2786:
2783:
2780:
2777:
2772:
2748:
2743:
2740:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2712:
2708:
2704:
2701:
2698:
2695:
2690:
2685:
2682:
2679:
2676:
2671:
2665:
2659:
2658:
2656:
2634:
2631:
2628:
2623:
2617:
2603:
2587:
2584:
2581:
2578:
2575:
2572:
2567:
2563:
2559:
2556:
2553:
2550:
2547:
2544:
2539:
2535:
2512:
2507:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2479:
2475:
2454:
2451:
2447:
2444:
2441:
2436:
2432:
2424:
2420:
2415:
2411:
2408:
2405:
2401:
2398:
2395:
2390:
2386:
2378:
2374:
2369:
2365:
2362:
2359:
2354:
2349:
2345:
2324:
2321:
2318:
2298:
2295:
2275:
2255:
2252:
2249:
2244:
2239:
2236:
2212:
2209:
2206:
2195:
2181:
2176:
2172:
2167:
2140:
2137:
2134:
2129:
2102:
2095:
2092:
2085:
2081:
2077:
2071:
2065:
2062:
2055:
2051:
2047:
2040:
2036:
2033:
2031:
2029:
2024:
2020:
2016:
2011:
2007:
2003:
1997:
1994:
1990:
1985:
1984:
1981:
1976:
1972:
1968:
1965:
1960:
1956:
1952:
1949:
1946:
1943:
1941:
1939:
1934:
1930:
1926:
1921:
1917:
1913:
1910:
1907:
1906:
1886:
1883:
1880:
1875:
1870:
1867:
1856:
1834:
1821:
1810:
1805:
1801:
1797:
1794:
1789:
1785:
1781:
1778:
1773:
1769:
1765:
1762:
1757:
1753:
1749:
1746:
1743:
1740:
1735:
1731:
1727:
1722:
1718:
1714:
1711:
1708:
1705:
1700:
1696:
1692:
1687:
1683:
1679:
1676:
1656:
1653:
1650:
1645:
1634:By definition
1618:
1614:
1611:
1599:
1596:
1595:
1594:
1578:
1575:
1572:
1569:
1566:
1563:
1539:
1536:
1533:
1530:
1527:
1524:
1496:
1493:
1490:
1487:
1484:
1481:
1458:
1446:
1443:
1439:
1435:
1430:
1425:
1422:
1418:
1414:
1410:
1405:
1389:
1360:This function
1349:
1346:
1343:
1336:
1333:
1330:
1326:
1321:
1318:
1315:
1312:
1307:
1304:
1301:
1297:
1290:
1287:
1284:
1280:
1275:
1272:
1269:
1266:
1263:
1216:
1191:
1187:
1179:
1176:
1173:
1170:
1166:
1160:
1153:
1150:
1147:
1144:
1140:
1133:
1117:
1097:
1094:
1091:
1088:
1085:
1065:
1061:
1057:
1054:
1051:
1048:
1045:
1039:
1036:
1033:
1029:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1000:
997:
994:
990:
985:
982:
978:
974:
971:
968:
965:
962:
947:
926:
923:
907:
904:
881:
878:
874:
870:
864:
861:
837:
833:
827:
824:
801:
796:
791:
785:
782:
776:
771:
767:
762:
758:
753:
748:
745:
742:
736:
733:
727:
722:
718:
696:
692:
689:
669:
666:
663:
658:
641:
640:
629:
626:
621:
616:
612:
608:
603:
599:
594:
589:
585:
580:
575:
570:
565:
560:
555:
551:
547:
544:
539:
534:
529:
524:
519:
514:
510:
505:
501:
498:
493:
488:
484:
480:
475:
469:
464:
439:
435:
431:
427:
422:
417:
412:
407:
402:
370:
365:
341:
336:
323:
322:
311:
308:
303:
298:
294:
290:
285:
279:
274:
270:
267:
263:
259:
254:
221:
217:
186:
168:
165:
112:
109:
106:
71:
68:
50:in Japanese, (
32:hyperfunctions
15:
9:
6:
4:
3:
2:
4047:
4036:
4033:
4031:
4028:
4026:
4023:
4021:
4018:
4017:
4015:
4005:
4001:
4000:
3995:
3990:
3985:
3984:
3979:
3973:
3972:
3963:
3957:
3953:
3952:
3946:
3942:
3936:
3932:
3931:
3926:
3922:
3918:
3914:
3910:
3906:
3902:
3898:
3893:
3889:
3885:
3881:
3877:
3873:
3869:
3864:
3861:
3857:
3853:
3849:
3845:
3841:
3837:
3833:
3829:
3822:
3817:
3816:
3813:
3807:
3803:
3802:
3796:
3792:
3786:
3781:
3780:
3773:
3769:
3765:
3761:
3757:
3753:
3752:
3748:(1960–1961),
3747:
3743:
3736:
3731:
3726:
3721:
3720:
3717:
3711:
3707:
3706:
3700:
3697:
3691:
3687:
3686:
3681:
3677:
3674:
3668:
3664:
3663:
3657:
3653:
3647:
3643:
3642:
3637:
3633:
3632:
3623:
3620:
3618:
3615:
3613:
3610:
3608:
3605:
3603:
3600:
3598:
3595:
3594:
3567:
3562:
3558:
3554:
3548:
3543:
3539:
3532:
3526:
3523:
3500:
3467:
3395:
3389:
3386:
3375:
3349:
3318:
3311:
3280:
3273:
3262:
3256:
3226:
3223:
3216:
3210:
3205:
3200:
3196:
3175:
3169:
3166:
3163:
3157:
3154:
3151:
3148:
3143:
3138:
3134:
3110:
3107:
3104:
3098:
3092:
3086:
3083:
3060:
3047:
3044:
3021:
3014:
3003:
3000:
2977:
2944:
2937:
2906:
2898:
2893:
2889:
2868:
2862:
2854:
2825:
2819:
2812:
2784:
2778:
2741:
2738:
2735:
2732:
2723:
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2696:
2683:
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2537:
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2357:
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2322:
2319:
2316:
2296:
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2273:
2250:
2237:
2234:
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2210:
2207:
2204:
2196:
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2154:
2135:
2100:
2093:
2090:
2083:
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2069:
2063:
2060:
2053:
2049:
2045:
2038:
2034:
2032:
2022:
2018:
2014:
2009:
2005:
1995:
1992:
1988:
1974:
1970:
1966:
1963:
1958:
1954:
1950:
1944:
1942:
1932:
1928:
1924:
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1915:
1908:
1881:
1868:
1865:
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1822:
1803:
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1787:
1783:
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1741:
1733:
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1725:
1720:
1716:
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1122:
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1111:
1092:
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1034:
1031:
1027:
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1019:
1013:
1007:
1004:
998:
995:
992:
988:
983:
980:
976:
972:
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960:
952:
948:
945:
941:
937:
933:
929:
928:
922:
902:
879:
876:
868:
859:
831:
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789:
780:
769:
765:
760:
746:
743:
731:
720:
716:
690:
687:
664:
645:
627:
614:
601:
597:
592:
587:
573:
568:
553:
549:
545:
532:
527:
512:
508:
503:
499:
486:
473:
462:
454:
453:
452:
425:
420:
410:
405:
390:
386:
368:
339:
309:
296:
283:
272:
268:
243:
242:
241:
239:
236:as the first
235:
219:
206:
202:
174:
164:
162:
159: +
158:
154:
151: +
150:
146:
142:
138:
137:complex plane
134:
130:
126:
110:
107:
104:
95:
93:
89:
85:
81:
77:
67:
65:
61:
57:
53:
49:
45:
41:
40:distributions
37:
33:
29:
22:
4035:Sheaf theory
3997:
3981:
3950:
3929:
3900:
3896:
3871:
3867:
3843:
3839:
3820:
3800:
3778:
3750:
3734:
3724:
3704:
3684:
3661:
3640:
2224:
2223:is called a
1601:
1590:
1551:
1512:
1469:
1461:
1381:
1377:
1373:
1369:
1365:
1361:
1251:
1247:
1243:
1239:
1235:
1231:
1228:distribution
1219:
1211:
1207:
943:
939:
935:
931:
646:
642:
324:
170:
160:
156:
152:
148:
144:
140:
132:
128:
124:
96:
91:
87:
83:
79:
73:
66:and others.
64:Grothendieck
31:
25:
3832:Sato, Mikio
1386:convolution
1254:defined by
70:Formulation
28:mathematics
4014:Categories
3768:0122.34902
3636:Imai, Isao
3629:References
2151:becomes a
44:Mikio Sato
4004:EMS Press
3983:MathWorld
3909:2261/6031
3880:2261/6027
3860:0039-470X
3638:(2012) ,
3571:Φ
3568:∘
3563:−
3552:Φ
3549:∘
3530:Φ
3527:∘
3438:Φ
3393:→
3384:Φ
3354:↪
3211:δ
3197:∫
3173:⟩
3170:φ
3161:⟨
3155:φ
3152:⋅
3135:∫
3099:⊂
3087:
3048:∈
3045:φ
3004:∈
2742:φ
2739:⋅
2733:∫
2730:↦
2724:φ
2703:→
2684:×
2566:±
2562:γ
2538:±
2534:γ
2511:±
2501:→
2478:±
2474:γ
2435:−
2423:−
2419:γ
2414:∫
2373:γ
2368:∫
2364:−
2344:∫
2320:⩽
2238:∈
2208:∈
2175:↪
2084:−
2023:−
1975:−
1933:−
1869:∈
1804:−
1788:−
1734:−
1699:−
1613:⊆
1571:−
1532:−
1489:−
1424:→
1332:−
1303:∈
1296:∫
1286:π
1246:), where
1215:function.
1172:π
1146:π
1087:
1047:
1035:π
1023:−
1008:
996:π
984:−
906:~
869:∩
863:~
832:⊆
826:~
784:~
741:∖
735:~
691:⊆
569:−
546:⊕
434:∖
421:−
411:∪
369:−
234:real line
108:−
86:), where
3927:(1970),
3834:(1958),
3590:See also
3350:′
3312:′
3274:′
3015:′
2938:′
2813:′
2197:A point
2171:′
2153:D-module
1472:), then
1409:′
1242:, −
925:Examples
131:, so if
3917:0132392
3888:0114124
3760:1611794
1847:into a
1509:support
1384:as the
1234:, then
1108:is the
383:be the
240:group:
199:be the
175:. Let
155:,
147:) and (
143:,
82:,
3958:
3937:
3915:
3886:
3858:
3840:Sūgaku
3808:
3787:
3766:
3758:
3712:
3692:
3669:
3648:
2465:where
1853:flabby
1591:finite
1076:where
812:where
3126:then
2838:with
1849:sheaf
1222:is a
201:sheaf
3956:ISBN
3935:ISBN
3856:ISSN
3806:ISBN
3785:ISBN
3710:ISBN
3690:ISBN
3667:ISBN
3646:ISBN
3084:supp
2922:(or
2605:Let
1602:Let
1119:The
949:The
387:and
354:and
127:and
56:1960
52:1959
48:1958
3905:hdl
3876:hdl
3848:doi
3764:Zbl
3483:to
3376:If
2309:If
2266:if
2227:of
1460:If
1218:If
1112:of
1084:log
1044:log
1005:log
930:If
451:so
207:on
203:of
46:in
26:In
4016::
4002:,
3996:,
3980:.
3913:MR
3911:,
3899:,
3884:MR
3882:,
3870:,
3854:,
3844:10
3838:,
3815:.
3762:,
3756:MR
3719:.
3533::=
3516::
2361::=
2035::=
1945::=
1855:).
1742::=
1362:f
946:).
62:,
54:,
30:,
3986:.
3944:.
3920:.
3907::
3901:8
3891:.
3878::
3872:8
3850::
3825:.
3794:.
3771:.
3729:.
3655:.
3574:)
3559:f
3555:,
3544:+
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3536:(
3524:f
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3501:U
3498:(
3493:B
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3359:B
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3201:a
3176:.
3167:,
3164:u
3158:=
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3108:,
3105:a
3102:(
3096:)
3093:u
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3064:)
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2899:0
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2627:(
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2602:.
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2166:D
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2133:(
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2070:,
2064:z
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2054:+
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2039:(
2028:)
2019:f
2015:,
2010:+
2006:f
2002:(
1996:z
1993:d
1989:d
1980:)
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1964:,
1959:+
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1948:(
1938:)
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1800:g
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23:.
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