1066:
1494:
1694:
1042:
438:
1347:
1910:
1575:
600:
317:
667:
366:
1958:
1738:
1391:
528:
475:
1397:
1581:
36:
had not been coined at the time of publishing the discovery of Strömberg wavelet and Strömberg's motivation was to find an orthonormal basis for the
950:
1254:}. Using the defining properties of the Strömbeg wavelet, exact expressions for elements of this set can be computed and they are given below.
32:
was earlier known to be an orthonormal wavelet, Strömberg wavelet was the first smooth orthonormal wavelet to be discovered. The term
371:
2022:
1262:
1824:
1069:
The graph of the Strömberg wavelet of order 0. The graph is scaled such that the value of the wavelet function at 1 is 1.
2049:
2044:
1992:, Conference on Harmonic Analysis in Honor of A. Zygmond, Vol. II, W. Beckner, et al (eds.) Wadsworth, 1983, pp.475-494
1500:
533:
248:
160:
605:
323:
118:
81:
1915:
1699:
1352:
53:
484:
923:
25:
2018:
2014:
1799:
1065:
444:
146:
28:
discovered by Jan-Olov Strömberg and presented in a paper published in 1983. Even though the
1987:
A modified
Franklin system and higher order spline systems on R as unconditional bases for
1073:
In the special case of Strömberg wavelets of order 0, the following facts may be observed:
61:
1489:{\displaystyle S^{0}({\tfrac {1}{2}})=-S^{0}(1)\left({\sqrt {3}}+{\tfrac {1}{2}}\right)}
2038:
2007:
1689:{\displaystyle S^{0}(-{\tfrac {k}{2}})=S^{0}(1)(2{\sqrt {3}}-2)({\sqrt {3}}-2)^{k}}
29:
1988:
1780:
1050:
69:
37:
17:
478:
181:
163:
1037:{\displaystyle \left\{2^{j/2}S^{m}(2^{j}t-k):j,k{\text{ integers }}\right\}}
210:= {. . . , -2, -3/2, -1, -1/2} ∪ {0} ∪ {1, 2, 3, . . .} and
1064:
1765:
The Strömberg wavelet of order 0 has the following properties.
1761:
Some additional information about Strömberg wavelet of order 0
922:
The following result establishes the Strömberg wavelet as an
1053:
system in the space of square integrable functions over
433:{\displaystyle \int _{R}\vert S^{m}(t)\vert ^{2}\,dt=1.}
1602:
1470:
1415:
1342:{\displaystyle S^{0}(k)=S^{0}(1)({\sqrt {3}}-2)^{k-1}}
1918:
1905:{\displaystyle S^{0}(-k/2)=(10-6{\sqrt {3}})S^{0}(k)}
1827:
1702:
1584:
1503:
1400:
1355:
1265:
953:
608:
536:
487:
447:
374:
326:
251:
687:
The following are some of the properties of the set
2006:
1952:
1904:
1732:
1688:
1569:
1488:
1385:
1341:
1036:
661:
594:
522:
469:
432:
360:
311:
1570:{\displaystyle S^{0}(0)=S^{0}(1)(2{\sqrt {3}}-2)}
1223:Computation of the Strömberg wavelet of order 0
1101:) is defined uniquely by the discrete subset {
595:{\displaystyle \int _{R}S^{m}(t)\,f(t)\,dt=0}
8:
408:
385:
349:
327:
1981:
1979:
1977:
1975:
1227:As already observed, the Strömberg wavelet
918:Strömberg wavelet as an orthonormal wavelet
312:{\displaystyle S^{m}(t)\in P^{(m)}(A_{1}).}
1946:
1917:
1887:
1873:
1847:
1832:
1826:
1817:and for negative half-integral values of
1701:
1680:
1663:
1644:
1623:
1601:
1589:
1583:
1551:
1530:
1508:
1502:
1469:
1459:
1439:
1414:
1405:
1399:
1354:
1327:
1310:
1292:
1270:
1264:
1024:
994:
981:
967:
963:
952:
647:
628:
607:
579:
566:
551:
541:
535:
511:
492:
486:
452:
446:
417:
411:
392:
379:
373:
334:
325:
297:
278:
256:
250:
1235:) is completely determined by the set {
2013:. Cambridge University Press. pp.
2009:A Mathematical Introduction to Wavelets
1971:
699:Let the number of distinct elements in
662:{\displaystyle f(t)\in P^{(m)}(A_{0}).}
240:) satisfying the following conditions:
1148:is associated: It is defined by λ
361:{\displaystyle \Vert S^{m}(t)\Vert =1}
2000:
1998:
132:satisfying the following conditions:
7:
1953:{\displaystyle k=1,2,3,\ldots \,.}
1813:) for positive integral values of
938:be the Strömberg wavelet of order
101:denote the interval determined by
14:
1206:. The special simple tent λ
1733:{\displaystyle k=1,2,3,\ldots }
1386:{\displaystyle k=1,2,3,\ldots }
867:) where α is constant and
1899:
1893:
1880:
1861:
1855:
1838:
1677:
1660:
1657:
1638:
1635:
1629:
1613:
1595:
1564:
1545:
1542:
1536:
1520:
1514:
1451:
1445:
1426:
1411:
1324:
1307:
1304:
1298:
1282:
1276:
1009:
987:
653:
640:
635:
629:
618:
612:
576:
570:
563:
557:
523:{\displaystyle P^{(m)}(A_{0})}
517:
504:
499:
493:
464:
458:
404:
398:
346:
340:
303:
290:
285:
279:
268:
262:
1:
1061:Strömberg wavelets of order 0
746:) contains nonzero functions.
734:If the number of elements in
188:+ 1 in each of the intervals
1749:(1) is constant such that ||
1191:. These special elements in
1135:, a special function λ
2066:
109:as the left endpoint. Let
942:. Then the following set
224:∪ { 1/2 } then the
1821:are related as follows:
763:are discrete subsets of
470:{\displaystyle S^{m}(t)}
117:) denote the set of all
2005:P. Wojtaszczyk (1997).
1214:) is denoted by λ(
1985:Janos-Olov Strömberg,
1954:
1906:
1790:The Strömberg wavelet
1771:The Strömberg wavelet
1734:
1690:
1571:
1490:
1387:
1343:
1070:
1038:
738:is three or more than
675:Properties of the set
663:
596:
524:
471:
434:
362:
313:
1955:
1907:
1735:
1691:
1572:
1491:
1388:
1344:
1068:
1039:
664:
597:
525:
472:
435:
363:
314:
80:into non-overlapping
1916:
1825:
1700:
1582:
1501:
1398:
1353:
1263:
1026: integers
951:
606:
534:
485:
445:
372:
324:
249:
166:of all orders up to
54:non-negative integer
2050:Continuous wavelets
2045:Orthogonal wavelets
1757:)|| = 1.
924:orthonormal wavelet
26:orthonormal wavelet
1950:
1902:
1730:
1686:
1611:
1567:
1486:
1479:
1424:
1383:
1339:
1071:
1034:
863:) + α λ(
803:). In particular,
659:
592:
520:
467:
430:
358:
309:
1878:
1800:exponential decay
1668:
1649:
1610:
1556:
1478:
1464:
1423:
1315:
1027:
719:) if and only if
226:Strömberg wavelet
147:square integrable
22:Strömberg wavelet
2057:
2029:
2028:
2012:
2002:
1993:
1983:
1959:
1957:
1956:
1951:
1911:
1909:
1908:
1903:
1892:
1891:
1879:
1874:
1851:
1837:
1836:
1739:
1737:
1736:
1731:
1695:
1693:
1692:
1687:
1685:
1684:
1669:
1664:
1650:
1645:
1628:
1627:
1612:
1603:
1594:
1593:
1576:
1574:
1573:
1568:
1557:
1552:
1535:
1534:
1513:
1512:
1495:
1493:
1492:
1487:
1485:
1481:
1480:
1471:
1465:
1460:
1444:
1443:
1425:
1416:
1410:
1409:
1392:
1390:
1389:
1384:
1348:
1346:
1345:
1340:
1338:
1337:
1316:
1311:
1297:
1296:
1275:
1274:
1043:
1041:
1040:
1035:
1033:
1029:
1028:
1025:
999:
998:
986:
985:
976:
975:
971:
886:) is defined by
668:
666:
665:
660:
652:
651:
639:
638:
601:
599:
598:
593:
556:
555:
546:
545:
529:
527:
526:
521:
516:
515:
503:
502:
476:
474:
473:
468:
457:
456:
439:
437:
436:
431:
416:
415:
397:
396:
384:
383:
367:
365:
364:
359:
339:
338:
318:
316:
315:
310:
302:
301:
289:
288:
261:
260:
2065:
2064:
2060:
2059:
2058:
2056:
2055:
2054:
2035:
2034:
2033:
2032:
2025:
2004:
2003:
1996:
1984:
1973:
1968:
1914:
1913:
1883:
1828:
1823:
1822:
1805:The values of
1763:
1698:
1697:
1676:
1619:
1585:
1580:
1579:
1526:
1504:
1499:
1498:
1458:
1454:
1435:
1401:
1396:
1395:
1351:
1350:
1323:
1288:
1266:
1261:
1260:
1253:
1225:
1209:
1201:
1190:
1171:
1153:
1147:
1140:
1134:
1063:
990:
977:
959:
958:
954:
949:
948:
932:
920:
912:
885:
846:
824:
813:
802:
791:
780:
773:
762:
755:
685:
643:
624:
604:
603:
547:
537:
532:
531:
507:
488:
483:
482:
448:
443:
442:
407:
388:
375:
370:
369:
330:
322:
321:
293:
274:
252:
247:
246:
223:
216:
209:
196:
100:
62:discrete subset
46:
12:
11:
5:
2063:
2061:
2053:
2052:
2047:
2037:
2036:
2031:
2030:
2023:
1994:
1970:
1969:
1967:
1964:
1963:
1962:
1961:
1960:
1949:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1901:
1898:
1895:
1890:
1886:
1882:
1877:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1850:
1846:
1843:
1840:
1835:
1831:
1803:
1788:
1762:
1759:
1743:
1742:
1741:
1740:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1683:
1679:
1675:
1672:
1667:
1662:
1659:
1656:
1653:
1648:
1643:
1640:
1637:
1634:
1631:
1626:
1622:
1618:
1615:
1609:
1606:
1600:
1597:
1592:
1588:
1577:
1566:
1563:
1560:
1555:
1550:
1547:
1544:
1541:
1538:
1533:
1529:
1525:
1522:
1519:
1516:
1511:
1507:
1496:
1484:
1477:
1474:
1468:
1463:
1457:
1453:
1450:
1447:
1442:
1438:
1434:
1431:
1428:
1422:
1419:
1413:
1408:
1404:
1393:
1382:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1358:
1336:
1333:
1330:
1326:
1322:
1319:
1314:
1309:
1306:
1303:
1300:
1295:
1291:
1287:
1284:
1281:
1278:
1273:
1269:
1251:
1224:
1221:
1220:
1219:
1207:
1199:
1188:
1167:
1149:
1145:
1136:
1132:
1122:
1062:
1059:
1049:is a complete
1047:
1046:
1045:
1044:
1032:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
997:
993:
989:
984:
980:
974:
970:
966:
962:
957:
931:
928:
919:
916:
915:
914:
910:
883:
844:
826:
822:
811:
800:
789:
778:
771:
760:
753:
747:
732:
727:) = 0 for all
684:
673:
672:
671:
670:
669:
658:
655:
650:
646:
642:
637:
634:
631:
627:
623:
620:
617:
614:
611:
591:
588:
585:
582:
578:
575:
572:
569:
565:
562:
559:
554:
550:
544:
540:
519:
514:
510:
506:
501:
498:
495:
491:
466:
463:
460:
455:
451:
440:
429:
426:
423:
420:
414:
410:
406:
403:
400:
395:
391:
387:
382:
378:
357:
354:
351:
348:
345:
342:
337:
333:
329:
319:
308:
305:
300:
296:
292:
287:
284:
281:
277:
273:
270:
267:
264:
259:
255:
232:is a function
221:
214:
207:
201:
200:
199:
198:
192:
171:
150:
96:
45:
42:
13:
10:
9:
6:
4:
3:
2:
2062:
2051:
2048:
2046:
2043:
2042:
2040:
2026:
2020:
2016:
2011:
2010:
2001:
1999:
1995:
1991:
1990:
1982:
1980:
1978:
1976:
1972:
1965:
1947:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1896:
1888:
1884:
1875:
1870:
1867:
1864:
1858:
1852:
1848:
1844:
1841:
1833:
1829:
1820:
1816:
1812:
1808:
1804:
1801:
1797:
1793:
1789:
1786:
1782:
1778:
1774:
1770:
1769:
1768:
1767:
1766:
1760:
1758:
1756:
1752:
1748:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1681:
1673:
1670:
1665:
1654:
1651:
1646:
1641:
1632:
1624:
1620:
1616:
1607:
1604:
1598:
1590:
1586:
1578:
1561:
1558:
1553:
1548:
1539:
1531:
1527:
1523:
1517:
1509:
1505:
1497:
1482:
1475:
1472:
1466:
1461:
1455:
1448:
1440:
1436:
1432:
1429:
1420:
1417:
1406:
1402:
1394:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1334:
1331:
1328:
1320:
1317:
1312:
1301:
1293:
1289:
1285:
1279:
1271:
1267:
1259:
1258:
1257:
1256:
1255:
1250:
1246:
1242:
1238:
1234:
1230:
1222:
1217:
1213:
1205:
1202:) are called
1198:
1194:
1187:
1183:
1179:
1175:
1170:
1165:
1161:
1157:
1152:
1144:
1139:
1131:
1127:
1123:
1120:
1116:
1112:
1108:
1104:
1100:
1096:
1092:
1088:
1084:
1080:
1076:
1075:
1074:
1067:
1060:
1058:
1056:
1052:
1030:
1021:
1018:
1015:
1012:
1006:
1003:
1000:
995:
991:
982:
978:
972:
968:
964:
960:
955:
947:
946:
945:
944:
943:
941:
937:
929:
927:
925:
917:
909:
905:
901:
897:
893:
889:
882:
878:
874:
870:
866:
862:
858:
854:
850:
843:
839:
835:
831:
827:
821:
817:
810:
806:
799:
795:
788:
784:
777:
770:
766:
759:
752:
748:
745:
741:
737:
733:
730:
726:
722:
718:
714:
710:
706:
703:be two. Then
702:
698:
697:
696:
694:
690:
682:
678:
674:
656:
648:
644:
632:
625:
621:
615:
609:
589:
586:
583:
580:
573:
567:
560:
552:
548:
542:
538:
512:
508:
496:
489:
480:
461:
453:
449:
441:
427:
424:
421:
418:
412:
401:
393:
389:
380:
376:
355:
352:
343:
335:
331:
320:
306:
298:
294:
282:
275:
271:
265:
257:
253:
245:
244:
243:
242:
241:
239:
235:
231:
227:
220:
213:
206:
195:
191:
187:
183:
179:
175:
172:
169:
165:
162:
158:
154:
151:
148:
144:
140:
137:
136:
135:
134:
133:
131:
127:
123:
120:
116:
112:
108:
104:
99:
95:
91:
87:
83:
79:
75:
71:
67:
63:
59:
55:
51:
43:
41:
39:
35:
31:
27:
24:is a certain
23:
19:
2008:
1989:Hardy spaces
1986:
1818:
1814:
1810:
1806:
1795:
1791:
1784:
1776:
1772:
1764:
1754:
1750:
1746:
1744:
1248:
1244:
1240:
1236:
1232:
1228:
1226:
1215:
1211:
1204:simple tents
1203:
1196:
1192:
1185:
1181:
1177:
1173:
1168:
1163:
1159:
1155:
1150:
1142:
1137:
1129:
1125:
1118:
1114:
1110:
1106:
1102:
1098:
1094:
1090:
1086:
1082:
1078:
1072:
1054:
1048:
939:
935:
933:
921:
907:
903:
899:
895:
891:
887:
880:
876:
872:
868:
864:
860:
856:
852:
848:
841:
837:
833:
829:
819:
815:
808:
804:
797:
793:
786:
782:
775:
768:
764:
757:
750:
743:
739:
735:
728:
724:
720:
716:
712:
708:
704:
700:
692:
688:
686:
680:
676:
530:, that is,
237:
233:
229:
225:
218:
211:
204:
202:
193:
189:
185:
177:
173:
167:
156:
152:
142:
138:
129:
125:
121:
114:
110:
106:
102:
97:
93:
89:
85:
77:
73:
70:real numbers
65:
57:
49:
47:
38:Hardy spaces
33:
30:Haar wavelet
21:
15:
1051:orthonormal
368:, that is,
164:derivatives
64:of the set
18:mathematics
2039:Categories
2024:0521570204
1966:References
1781:oscillates
1176:) = 0 if
1166:and λ
1085:) ∈
875:) ∈
836:) ∈
814:) ⊂
792:) ⊂
767:such that
711:) ∈
479:orthogonal
184:of degree
182:polynomial
161:continuous
84:. For any
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