81:, Abelian theorems give the asymptotic behaviour of the transform based on properties of the original function. Conversely, Tauberian theorems give the asymptotic behaviour of the original function based on properties of the transform but usually require some restrictions on the original function.
73:
There is not yet a clear distinction between
Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean. Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives
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500:, taken to the limit, a Tauberian theorem allows one to discard the weighting, under the correct hypotheses. There are many applications of this kind of result in
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70:)) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods.
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is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/
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279:= 0. Then divide the sequence into an initial segment, and a tail of small terms: given any ε > 0 we can take
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exactly 1: if the radius of convergence is greater than one, the convergence of the power series is
296:/2, while each term in the tail is bounded by ε/2 so that the average is also necessarily bounded.
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conditions for a series summable by some method that allows it to be summable in the usual sense.
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A method for finding the asymptotic behavior of a function from its
Laplace transform
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contains the convergent sequences, and its values there are equal to those of the
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tend to the limit 1 from below along the real axis in the power series with term
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The development of the field of
Tauberian theorems received a fresh turn with
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356:. That theorem has its main interest in the case that the power series has
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385:. When the radius is 1 the power series will have some singularity on |
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785:. Cambridge Studies in Advanced Mathematics. Vol. 97. Cambridge:
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methods, and contains much, though not all, of the previous theory.
450:) and the radial limit exists, then the series obtained by setting
731:. Grundlehren der Mathematischen Wissenschaften. Vol. 329.
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theorem states, under some growth condition, that the domain of
389:| = 1; the assertion is that, nonetheless, if the sum of the
261:{\displaystyle d_{N}={\frac {c_{1}+c_{2}+\cdots +c_{N}}{N}}.}
402:. This therefore fits exactly into the abstract picture.
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large enough to make the initial segment of terms up to
454:= 1 is actually convergent. This was strengthened by
190:
260:
594:"Ein Satz aus der Theorie der unendlichen Reihen"
489:is exactly the convergent sequences and no more.
783:Multiplicative number theory I. Classical theory
42:giving conditions for two methods of summing
8:
729:Tauberian theory. A century of developments
596:[A theorem about infinite series].
523:. The central theorem can now be proved by
570:(Thesis). University of British Columbia.
372:and it follows directly that the limit as
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27:Used in the summation of divergent series
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469:In the abstract setting, therefore, an
376:tends up to 1 is simply the sum of the
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398:exists, it is equal to the limit over
599:Monatshefte fĂĽr Mathematik und Physik
462:). A sweeping generalization is the
368:in so that the sum is automatically
46:to give the same result, named after
7:
566:Froese Fischer, Charlotte (1954).
542:Hardy–Littlewood Tauberian theorem
473:theorem states that the domain of
464:Hardy–Littlewood Tauberian theorem
25:
275:everywhere to reduce to the case
515:'s very general results, namely
426:) stated that if we assume also
414:to Abelian theorems are called
144:is defined as the limit of the
644:(1932). "Tauberian theorems".
36:Abelian and Tauberian theorems
18:Abelian and tauberian theorems
1:
172:, then so does the sequence (
519:and its large collection of
504:, in particular in handling
496:as some generalised type of
54:. The original examples are
715:Encyclopedia of Mathematics
160:tends to infinity. One can
136:An example is given by the
861:
787:Cambridge University Press
537:Wiener's Tauberian theorem
517:Wiener's Tauberian theorem
458:: we need only assume O(1/
741:10.1007/978-3-662-10225-1
418:. The original result of
89:For any summation method
58:showing that if a series
727:Korevaar, Jacob (2004).
547:Haar's Tauberian theorem
456:John Edensor Littlewood
315:(thought of within the
62:to some limit then its
299:The name derives from
271:To see that, subtract
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97:is the result that if
647:Annals of Mathematics
358:radius of convergence
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789:. pp. 147–167.
735:. pp. xvi+483.
710:"Tauberian theorems"
188:
840:Summability methods
835:Mathematical series
775:Montgomery, Hugh L.
292:average to at most
112:convergent sequence
79:integral transforms
845:Summability theory
830:Tauberian theorems
779:Vaughan, Robert C.
612:10.1007/BF01696278
576:10.14288/1.0080631
416:Tauberian theorems
406:Tauberian theorems
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796:978-0-521-84903-6
750:978-3-540-21058-0
492:If one thinks of
448:Little o notation
420:Alfred Tauber
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168:does converge to
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48:Niels Henrik Abel
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521:corollaries
140:, in which
32:mathematics
824:Categories
813:1142.11001
767:1056.40002
692:0004.05905
668:58.0226.02
620:28.0221.02
553:References
370:continuous
720:EMS Press
628:120692627
483:Tauberian
412:converses
320:unit disk
234:⋯
152:terms of
60:converges
781:(2007).
592:(1897).
531:See also
410:Partial
344:and set
181:) where
164:that if
64:Abel sum
40:theorems
805:2378655
759:2073637
722:, 2001
684:1503035
676:1968102
471:Abelian
422: (
362:uniform
317:complex
311:is the
121:, then
114:, with
110:) is a
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690:
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618:
438:= o(1/
93:, its
672:JSTOR
624:S2CID
446:(see
162:prove
156:, as
116:limit
791:ISBN
745:ISBN
424:1897
364:for
129:) =
50:and
38:are
809:Zbl
763:Zbl
737:doi
688:Zbl
664:JFM
656:doi
616:JFM
608:doi
572:doi
479:Lim
303:on
133:.
101:= (
30:In
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807:.
801:MR
799:.
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761:.
755:MR
753:.
743:.
718:,
712:,
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680:MR
678:.
670:.
662:.
652:33
650:.
622:.
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508:.
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348:=
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739::
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658::
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578:.
574::
494:L
487:L
475:L
460:n
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442:)
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400:r
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387:z
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378:a
374:r
366:r
354:e
352:·
350:r
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339:z
335:n
331:a
324:r
309:L
294:ε
289:N
285:c
281:N
277:C
273:C
256:.
251:N
245:N
241:c
237:+
231:+
226:2
222:c
218:+
213:1
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197:N
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174:d
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158:N
154:c
150:N
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131:C
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