1227:
309:
Several constructions of algebras of generalized functions have been proposed, among others those by Yu. M. Shirokov and those by E. Rosinger, Y. Egorov, and R. Robinson. In the first case, the multiplication is determined with some regularization of generalized function. In the second case, the
745:
Such a rule applies to both the space of main functions and the space of operators which act on the space of the main functions. The associativity of multiplication is achieved; and the function signum is defined in such a way, that its square is unity everywhere (including the origin of
976:
727:
794:. Such a formalism includes the conventional theory of generalized functions (without their product) as a special case. However, the resulting algebra is non-commutative: generalized functions signum and delta anticommute. Few applications of the algebra were suggested.
278:
Some solutions to the multiplication problem have been proposed. One is based on a simple definition of by Yu. V. Egorov (see also his article in
Demidov's book in the book list below) that allows arbitrary operations on, and between, generalized functions.
968:
1222:{\displaystyle G_{s}(E,P)={\frac {\{f\in E^{\mathbb {N} }\mid \forall p\in P,\exists m\in \mathbb {Z} :p(f_{n})=o(n^{m})\}}{\{f\in E^{\mathbb {N} }\mid \forall p\in P,\forall m\in \mathbb {Z} :p(f_{n})=o(n^{m})\}}}.}
499:
152:
was introduced, there was for the first time a notion of generalized function central to mathematics. An integrable function, in
Lebesgue's theory, is equivalent to any other which is the same
1608:
449:
792:
399:
2110:
Halperin, I., & Schwartz, L. (1952). Introduction to the Theory of
Distributions. Toronto: University of Toronto Press. (Short lecture by Halperin on Schwartz's theory)
207:
of partial differential equations (i.e. solutions which are generalized functions, but may not be ordinary functions). Others proposing related theories at the time were
355:
862:
258:
This theory was very successful and is still widely used, but suffers from the main drawback that distributions cannot usually be multiplied: unlike most classical
298:
which is invariant under coordinate transformations, this property must be shared by path integrals. This fixes all products of generalized functions as shown by
882:
489:
469:
813:
Various approaches are used today. The simplest one is based on the definition of generalized function given by Yu. V. Egorov. Another approach to construct
1245:
1502:
being (well-)defined for compactly supported generalized functions (component-wise), one can apply the same construction as for distributions, and define
867:
of "moderate" modulo "negligible" nets of functions, where "moderateness" and "negligibility" refers to growth with respect to the index of the family.
1387:, of integral one and have all its derivatives at 0 vanishing. To obtain a canonical injection, the indexing set can be modified to be
90:
In the mathematics of the nineteenth century, aspects of generalized function theory appeared, for example in the definition of the
722:{\displaystyle FG~=~F_{\rm {smooth}}~G_{\rm {smooth}}~+~F_{\rm {smooth}}~G_{\rm {singular}}~+F_{\rm {singular}}~G_{\rm {smooth}}.}
1970:
1949:
1928:
1886:
1857:
1838:
1817:
1790:
2373:
2328:
2283:
200:
137:. They are typical of later application of generalized function methods. An influential book on operational calculus was
1635:. This is on the Schwartz pattern, constructing objects dual to the test objects, smooth sections of a bundle that have
1248:(which can be "infinitely large" and "infinitesimally small" and still allow for rigorous arithmetics, very similar to
2054:
2020:
1994:
1907:
1616:
1470:
For the subsheaf {0}, one gets the usual support (complement of the largest open subset where the function is zero).
2421:
322:
The algebra of generalized functions can be built-up with an appropriate procedure of projection of a function
2326:
O. G. Goryaga; Yu. M. Shirokov (1981). "Energy levels of an oscillator with singular concentrated potential".
1481:, i.e., roughly speaking, the closure of the set where the generalized function is not a smooth function (for
56:
1576:
404:
1689:
1679:
303:
228:
32:
755:
360:
2167:
1799:
H. Komatsu, Introduction to the theory of distributions, Second edition, Iwanami Shoten, Tokyo, 1983.
1674:
1459:
283:
2168:"Rules for integrals over products of distributions from coordinate independence of path integrals"
1520:
240:
1699:
746:
coordinates). Note that the product of singular parts does not appear in the right-hand side of (
31:
on real or complex numbers. There is more than one recognized theory, for example the theory of
1596:
1499:
67:
36:
28:
2278:
2010:
1984:
1897:
1580:
1463:
1249:
188:
115:
1960:
1939:
1918:
1876:
1828:
1807:
1780:
325:
70:
aspects of everyday, numerical functions. The early history is connected with some ideas on
2382:
2337:
2292:
2248:
2192:
2132:
1694:
1669:
1412:
833:
817:
267:
184:
176:
126:
103:
71:
60:
963:{\displaystyle s=\{a_{m}:\mathbb {N} \to \mathbb {R} ,n\mapsto n^{m};~m\in \mathbb {Z} \}}
8:
2227:
1436:
244:
157:
111:
91:
2386:
2341:
2296:
2252:
2196:
2136:
156:. That means its value at each point is (in a sense) not its most important feature. In
2398:
2353:
2308:
2238:
2208:
2182:
2148:
1986:
Path
Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets
1652:
1648:
1540:
474:
454:
79:
2260:
2402:
2357:
2312:
2212:
2152:
2060:
2050:
2016:
1990:
1966:
1945:
1924:
1903:
1882:
1863:
1853:
1834:
1813:
1803:
1786:
1767:
1747:
1732:
1722:
1704:
1644:
1556:
1291:
821:
295:
287:
165:
153:
149:
95:
2144:
2096:
2079:
1503:
2390:
2345:
2300:
2256:
2200:
2140:
2091:
1759:
1664:
1647:. These are homological in nature, in the way that differential forms give rise to
1604:
1516:
1478:
232:
216:
138:
134:
130:
1850:
Multiplication of distributions and applications to partial differential equations
1920:
Geometric theory of generalized functions with applications to general relativity
1640:
1636:
1588:
1548:
1477:(embedded using the canonical (constant) injection), one gets what is called the
252:
212:
208:
169:
40:
203:, defined the first rigorous theory of generalized functions in order to define
2123:
Yu. V. Egorov (1990). "A contribution to the theory of generalized functions".
1980:
1917:
Grosser, M.; Kunzinger, M.; Oberguggenberger, Michael; Steinbauer, R. (2013) .
1536:
1508:
299:
270:. Work of Schwartz from around 1954 showed this to be an intrinsic difficulty.
259:
196:
192:
107:
2002:
291:
2415:
1867:
1771:
1751:
1726:
1684:
1632:
1592:
1572:
1552:
204:
2064:
1600:
55:. Important motivations have been the technical requirements of theories of
2371:
G. K. Tolokonnikov (1982). "Differential rings used in
Shirokov algebras".
1560:
825:
44:
2204:
1916:
1736:
2243:
2187:
2044:
1612:
1544:
1400:
1346:
814:
52:
20:
806:, a limitation of the Schwartz distribution theory, becomes serious for
302:
and A. Chervyakov. The result is equivalent to what can be derived from
2394:
2349:
2304:
807:
236:
180:
75:
1899:
Generalized
Functions in Mathematical Physics: Main Ideas and Concepts
1380:
248:
122:
121:
The intensive use of the
Laplace transform in engineering led to the
175:
During the late 1920s and 1930s further basic steps were taken. The
1584:
970:. Then for any semi normed algebra (E,P), the factor space will be
317:
1282:
is the supremum of all derivatives of order less than or equal to
1941:
A distributional approach to asymptotics. Theory and applications
263:
99:
66:
A common feature of some of the approaches is that they build on
48:
1827:
Vladimirov, V.S.; Drozhzhinov, Yu. N.; Zavâyalov, B.I. (2012) .
1746:(multigraphed lectures). Summer Institute, Stanford University.
1515:
This has an especially important application in the analysis of
164:
feature of an integrable function, namely the way it defines a
2228:"Coordinate Independence of Quantum-Mechanical Path Integrals"
1826:
1458:) will also have this property. This means that the notion of
875:
A simple example is obtained by using the polynomial scale on
2225:
2165:
1809:
New
Generalized Functions and Multiplication of Distributions
282:
Another solution allowing multiplication is suggested by the
2325:
1555:, based (in their initial conception) on boundary values of
1466:
of a generalized function w.r.t. a subsheaf, in particular:
251:
theory, which uses sequences of smooth approximations (the '
133:, these methods were questionable from the point of view of
74:, and some contemporary developments are closely related to
2046:
Elements of the theory of functions and functional analysis
35:. Generalized functions are especially useful for treating
1627:
A further way in which the theory has been extended is as
2005:). See Chapter 11 for products of generalized functions.
2008:
1297:
2272:
2270:
1782:
The
Analysis of Linear Partial Differential Operators
979:
885:
836:
758:
502:
477:
457:
407:
363:
328:
43:, and describing discrete physical phenomena such as
273:
2370:
2279:"Algebra of one-dimensional generalized functions"
2267:
1439:of semi normed algebras on some topological space
1379:in the sense that it depends on the choice of the
1221:
962:
856:
797:
786:
721:
483:
463:
443:
393:
349:
227:The most definitive development was the theory of
820:is based on J.-F. Colombeau's construction: see
290:. Since this is required to be equivalent to the
2413:
2009:Pilipovi, S.; Stankovic, B.; Vindas, J. (2012).
1847:
1758:
318:Non-commutative algebra of generalized functions
168:on other functions. This allows a definition of
2276:
2118:
2116:
2042:
1651:. They can be used to formulate a very general
266:. For example, it is meaningless to square the
1962:Methods of the theory of generalized functions
235:, systematically working out the principle of
47:. They are applied extensively, especially in
2122:
1937:
1744:On quasianalyticity and general distributions
870:
2113:
2012:Asymptotic behavior of generalized functions
1830:Tauberian theorems for generalized functions
1462:will be defined, which allows to define the
1210:
1113:
1108:
1011:
957:
892:
451:parts. The product of generalized functions
129:. Since justifications were given that used
1958:
1302:This algebra "contains" all distributions
2242:
2186:
2095:
2043:Kolmogorov, A. N.; Fomin, S. V. (1999) .
1802:
1778:
1162:
1128:
1060:
1026:
953:
917:
909:
222:
16:Objects extending the notion of functions
2077:
1979:
1878:An introduction to Sato's hyperfunctions
1874:
1741:
1716:
1895:
1639:. The most developed theory is that of
2414:
2319:
2226:H. Kleinert and A. Chervyakov (2000).
2166:H. Kleinert and A. Chervyakov (2001).
1622:
1493:
1607:in this language, characterizing the
1566:
114:. These were disconnected aspects of
85:
2374:Theoretical and Mathematical Physics
2364:
2329:Theoretical and Mathematical Physics
2284:Theoretical and Mathematical Physics
1762:; Vilenkin, Naum JakovleviÄ (1964).
493:
201:partial differential equation theory
160:a clear formulation is given of the
27:are objects extending the notion of
1944:(2nd ed.). BirkhÀuser Boston.
1298:Injection of Schwartz distributions
191:, thought of as densities (such as
13:
1989:(5th ed.). World Scientific.
1766:. Vol. IâVI. Academic Press.
1422:
1152:
1137:
1050:
1035:
710:
707:
704:
701:
698:
695:
680:
677:
674:
671:
668:
665:
662:
659:
641:
638:
635:
632:
629:
626:
623:
620:
605:
602:
599:
596:
593:
590:
569:
566:
563:
560:
557:
554:
539:
536:
533:
530:
527:
524:
444:{\displaystyle F_{\rm {singular}}}
435:
432:
429:
426:
423:
420:
417:
414:
385:
382:
379:
376:
373:
370:
314:. Both cases are discussed below.
14:
2433:
1881:. American Mathematical Society.
1617:explicit formula of an L-function
1615:; and has also applied it to the
1591:. The applications are mostly in
1526:
274:Algebras of generalized functions
215:. Sobolev's work was extended by
106:, which were not necessarily the
1938:Estrada, R.; Kanwal, R. (2012).
1512:also for generalized functions.
787:{\displaystyle \delta (x)^{2}=0}
394:{\displaystyle F_{\rm {smooth}}}
125:use of symbolic methods, called
2145:10.1070/rm1990v045n05abeh002683
2097:10.1090/S0002-9904-1952-09555-0
1721:. Vol. 1. Paris: Hermann.
804:multiplication of distributions
798:Multiplication of distributions
312:multiplication of distributions
2219:
2159:
2104:
2071:
2036:
1292:Colombeau's simplified algebra
1207:
1194:
1185:
1172:
1105:
1092:
1083:
1070:
1002:
990:
927:
913:
769:
762:
344:
338:
57:partial differential equations
1:
2261:10.1016/S0375-9601(00)00475-8
2030:
1848:Oberguggenberger, M. (1992).
1571:Bruhat introduced a class of
1244:,|.|) one gets (Colombeau's)
7:
2080:"Théorie des distributions"
1760:GelÊčfand, IzrailÊč MoiseeviÄ
1658:
1246:generalized complex numbers
748:
735:
10:
2438:
1785:(2nd ed.). Springer.
1690:Laplacian of the indicator
1680:Distribution (mathematics)
1411:) (functions of vanishing
871:Example: Colombeau algebra
310:algebra is constructed as
304:dimensional regularization
195:) like genuine functions.
1959:Vladimirov, V.S. (2002).
1719:Théorie des distributions
1675:Generalized eigenfunction
1577:SchwartzâBruhat functions
284:path integral formulation
241:topological vector spaces
2277:Yu. M. Shirokov (1979).
2049:. Mineola, N.Y.: Dover.
1965:. Taylor & Francis.
1710:
1559:, and now making use of
1779:Hörmander, L. (2015) .
1700:Limit of a distribution
1597:adelic algebraic groups
37:discontinuous functions
1896:Demidov, A.S. (2001).
1581:locally compact groups
1551:; and the theories of
1500:Fourier transformation
1286:on the ball of radius
1223:
964:
858:
788:
723:
485:
465:
445:
395:
351:
350:{\displaystyle F=F(x)}
262:, they do not form an
223:Schwartz distributions
179:was boldly defined by
143:Electromagnetic Theory
2422:Generalized functions
2205:10.1007/s100520100600
2125:Russian Math. Surveys
2084:Bull. Amer. Math. Soc
1875:Morimoto, M. (1993).
1764:Generalized Functions
1742:Beurling, A. (1961).
1717:Schwartz, L. (1950).
1587:that are the typical
1583:that goes beyond the
1399:), with a convenient
1224:
965:
859:
857:{\displaystyle G=M/N}
818:differential algebras
789:
724:
486:
466:
446:
396:
352:
187:); this was to treat
116:mathematical analysis
61:group representations
25:generalized functions
2078:Schwartz, L (1952).
2015:. World Scientific.
1695:Rigged Hilbert space
1670:Dirac delta function
1629:generalized sections
1533:convolution quotient
1232:In particular, for (
977:
883:
834:
756:
500:
475:
455:
405:
361:
326:
268:Dirac delta function
243:. Its main rival in
185:scientific formalism
177:Dirac delta function
127:operational calculus
104:trigonometric series
72:operational calculus
2387:1982TMP....53..952T
2342:1981TMP....46..210G
2297:1979TMP....39..471S
2253:2000PhLA..273....1K
2237:. A 269 (1â2): 63.
2197:2001EPJC...19..743K
2137:1990RuMaS..45....1E
1623:Generalized section
1531:These include: the
1494:Microlocal analysis
1383:Ï, which should be
1250:nonstandard numbers
245:applied mathematics
158:functional analysis
112:integrable function
2395:10.1007/BF01014789
2350:10.1007/BF01032729
2305:10.1007/BF01017992
1649:De Rham cohomology
1645:differential forms
1595:, particularly to
1567:Topological groups
1557:analytic functions
1547:algebras that are
1541:field of fractions
1375:This injection is
1310:via the injection
1219:
960:
854:
784:
752:); in particular,
719:
481:
461:
441:
391:
347:
183:(an aspect of his
86:Some early history
80:algebraic analysis
1972:978-0-415-27356-5
1951:978-0-8176-8130-2
1930:978-94-015-9845-3
1888:978-0-8218-8767-7
1859:978-0-582-08733-0
1840:978-94-009-2831-2
1819:978-0-08-087195-0
1792:978-3-642-61497-2
1705:Generalized space
1609:zeta distribution
1473:For the subsheaf
1214:
945:
822:Colombeau algebra
743:
742:
688:
649:
613:
583:
577:
547:
517:
511:
484:{\displaystyle G}
464:{\displaystyle F}
401:and its singular
296:quantum mechanics
288:quantum mechanics
166:linear functional
154:almost everywhere
150:Lebesgue integral
96:Laplace transform
2429:
2407:
2406:
2368:
2362:
2361:
2323:
2317:
2316:
2274:
2265:
2264:
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2244:quant-ph/0003095
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2217:
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2190:
2188:quant-ph/0002067
2172:
2163:
2157:
2156:
2120:
2111:
2108:
2102:
2101:
2099:
2075:
2069:
2068:
2040:
2026:
2000:
1976:
1955:
1934:
1913:
1902:. Nova Science.
1892:
1871:
1844:
1823:
1804:Colombeau, J.-F.
1796:
1775:
1755:
1730:
1665:Beppo-Levi space
1641:De Rham currents
1589:function domains
1579:, on a class of
1549:integral domains
1479:singular support
1228:
1226:
1225:
1220:
1215:
1213:
1206:
1205:
1184:
1183:
1165:
1133:
1132:
1131:
1111:
1104:
1103:
1082:
1081:
1063:
1031:
1030:
1029:
1009:
989:
988:
969:
967:
966:
961:
956:
943:
939:
938:
920:
912:
904:
903:
863:
861:
860:
855:
850:
793:
791:
790:
785:
777:
776:
737:
728:
726:
725:
720:
715:
714:
713:
686:
685:
684:
683:
647:
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644:
611:
610:
609:
608:
581:
575:
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573:
572:
545:
544:
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542:
515:
509:
494:
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487:
482:
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467:
462:
450:
448:
447:
442:
440:
439:
438:
400:
398:
397:
392:
390:
389:
388:
356:
354:
353:
348:
255:' explanation).
233:Laurent Schwartz
217:Laurent Schwartz
139:Oliver Heaviside
135:pure mathematics
131:divergent series
92:Green's function
41:smooth functions
2437:
2436:
2432:
2431:
2430:
2428:
2427:
2426:
2412:
2411:
2410:
2369:
2365:
2324:
2320:
2275:
2268:
2230:
2224:
2220:
2175:Eur. Phys. J. C
2170:
2164:
2160:
2121:
2114:
2109:
2105:
2076:
2072:
2057:
2041:
2037:
2033:
2023:
1997:
1973:
1952:
1931:
1910:
1889:
1860:
1841:
1820:
1793:
1713:
1661:
1653:Stokes' theorem
1637:compact support
1625:
1569:
1539:, based on the
1529:
1496:
1448:
1425:
1423:Sheaf structure
1358:
1349:operation, and
1345:where â is the
1336:
1326:
1300:
1280:
1273:
1260:) = (
1201:
1197:
1179:
1175:
1161:
1127:
1126:
1122:
1112:
1099:
1095:
1077:
1073:
1059:
1025:
1024:
1020:
1010:
1008:
984:
980:
978:
975:
974:
952:
934:
930:
916:
908:
899:
895:
884:
881:
880:
873:
846:
835:
832:
831:
802:The problem of
800:
772:
768:
757:
754:
753:
694:
693:
689:
658:
657:
653:
619:
618:
614:
589:
588:
584:
553:
552:
548:
523:
522:
518:
501:
498:
497:
476:
473:
472:
456:
453:
452:
413:
412:
408:
406:
403:
402:
369:
368:
364:
362:
359:
358:
357:to its smooth
327:
324:
323:
320:
276:
260:function spaces
253:James Lighthill
225:
213:Kurt Friedrichs
209:Salomon Bochner
170:weak derivative
88:
17:
12:
11:
5:
2435:
2425:
2424:
2409:
2408:
2381:(1): 952â954.
2363:
2336:(3): 321â324.
2318:
2291:(3): 291â301.
2266:
2218:
2181:(4): 743â747.
2158:
2112:
2103:
2070:
2055:
2034:
2032:
2029:
2028:
2027:
2021:
2006:
1995:
1977:
1971:
1956:
1950:
1935:
1929:
1914:
1908:
1893:
1887:
1872:
1858:
1845:
1839:
1824:
1818:
1800:
1797:
1791:
1776:
1756:
1739:
1712:
1709:
1708:
1707:
1702:
1697:
1692:
1687:
1682:
1677:
1672:
1667:
1660:
1657:
1624:
1621:
1573:test functions
1568:
1565:
1553:hyperfunctions
1537:Jan Mikusinski
1528:
1527:Other theories
1525:
1509:wave front set
1504:Lars Hörmander
1495:
1492:
1491:
1490:
1471:
1446:
1424:
1421:
1377:non-canonical
1373:
1372:
1354:
1343:
1342:
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197:Sergei Sobolev
193:charge density
108:Fourier series
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1829:
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1808:
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300:H. Kleinert
292:Schrödinger
53:engineering
21:mathematics
2235:Phys. Lett
2031:References
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1601:André Weil
1535:theory of
810:problems.
808:non-linear
294:theory of
181:Paul Dirac
76:Mikio Sato
39:more like
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249:mollifier
162:essential
148:When the
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123:heuristic
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94:, in the
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