205:. The Stieltjes moment problems and the Hamburger moment problems, if they are solvable, may have infinitely many solutions (indeterminate moment problem) whereas a Hausdorff moment problem always has a unique solution if it is solvable (determinate moment problem). In the indeterminate moment problem case, there are infinite measures corresponding to the same prescribed moments and they consist of a convex set. The set of polynomials may or may not be dense in the associated Hilbert spaces if the moment problem is indeterminate, and it depends on whether measure is extremal or not. But in the determinate moment problem case, the set of polynomials is dense in the associated Hilbert space.
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is such a moment sequence if and only if the sequence is completely monotonic, that is, its difference sequences satisfy the equation
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Feller, W. "An
Introduction to Probability Theory and Its Applications", volume II, John Wiley & Sons, 1971.
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724:{\displaystyle (\Delta ^{4}m)_{6}=m_{6}-4m_{7}+6m_{8}-4m_{9}+m_{10}=\int x^{6}(1-x)^{4}d\mu (x)\geq 0.}
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The essential difference between this and other well-known moment problems is that this is on a
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534:{\displaystyle (-1)^{k}(\Delta ^{k}m)_{n}=\int _{0}^{1}x^{n}(1-x)^{k}d\mu (x),}
32:
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The necessity of this condition is easily seen by the identity
31:, asks for necessary and sufficient conditions that a given
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Absolutely and completely monotonic functions and sequences
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129:{\displaystyle m_{n}=\int _{0}^{1}x^{n}\,d\mu (x)}
128:
784:, American mathematical society, New York, 1943.
302:{\displaystyle (-1)^{k}(\Delta ^{k}m)_{n}\geq 0}
398:{\displaystyle (\Delta m)_{n}=m_{n+1}-m_{n}.}
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163:, this is equivalent to the existence of a
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552:. For example, it is necessary to have
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213:In 1921, Hausdorff showed that
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209:Completely monotonic sequences
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201:one considers the whole line
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193:one considers a half-line
766:Mathematische Zeitschrift
759:Mathematische Zeitschrift
170:supported on , such that
199:Hamburger moment problem
191:Stieltjes moment problem
782:The Problem of Moments
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808:Mathematical problems
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803:Moment (mathematics)
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59:be the sequence of
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548:of a non-negative
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29:Felix Hausdorff
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153:. In the case
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27:, named after
24:moment problem
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775:Shohat, J.A
18:mathematics
792:Categories
752:References
22:Hausdorff
716:≥
704:μ
685:−
666:∫
634:−
602:−
567:Δ
517:μ
498:−
468:∫
442:Δ
422:−
380:−
345:Δ
331:given by
294:≥
272:Δ
252:−
147:supported
115:μ
87:∫
735:See also
550:function
546:integral
323:. Here,
312:for all
139:of some
33:sequence
327:is the
203:(−∞, ∞)
61:moments
325:Δ
236:, ...)
195:[0, ∞)
144:μ
57:, ...)
20:, the
172:E =
777:.;
321:≥ 0
161:= 1
16:In
794::
719:0.
658:10
317:,
229:,
222:,
182:.
50:,
43:,
713:)
710:x
707:(
701:d
696:4
692:)
688:x
682:1
679:(
674:6
670:x
663:=
654:m
650:+
645:9
641:m
637:4
629:8
625:m
621:6
618:+
613:7
609:m
605:4
597:6
593:m
589:=
584:6
580:)
576:m
571:4
563:(
529:,
526:)
523:x
520:(
514:d
509:k
505:)
501:x
495:1
492:(
487:n
483:x
477:1
472:0
464:=
459:n
455:)
451:m
446:k
438:(
433:k
429:)
425:1
419:(
393:.
388:n
384:m
375:1
372:+
369:n
365:m
361:=
356:n
352:)
348:m
342:(
319:k
315:n
297:0
289:n
285:)
281:m
276:k
268:(
263:k
259:)
255:1
249:(
234:2
231:m
227:1
224:m
220:0
217:m
215:(
178:n
174:m
168:X
159:0
156:m
124:)
121:x
118:(
112:d
106:n
102:x
96:1
91:0
83:=
78:n
74:m
55:2
52:m
48:1
45:m
41:0
38:m
36:(
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