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would be a set of uniqueness — in one dimension the locality principle for
Fourier series shows that any set of positive measure is a set of multiplicity (in higher dimensions this is still an open question). This was disproved by
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constructed an example of a set of uniqueness which does not support any measure with
Fourier coefficients tending to zero. In other words, the generalization of distribution is necessary.
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sets of uniqueness is a set of uniqueness. There exists an example of two sets of uniqueness whose union is not a set of uniqueness, but the sets in this example are not
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Since the 50s, much work has gone into formalizing this complexity. The family of sets of uniqueness, considered as a set inside the space of compact sets (see
264:, and in higher dimensions. In the latter case, one needs to specify the order of summation, e.g. "a set of uniqueness with respect to summing over balls".
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In effect, it is usually sufficiently interesting (as in the definition above) to assume that the sum converges to zero and ask if that means that all the
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here are the
Fourier coefficients). In all early examples of sets of uniqueness, the distribution in question was in fact a measure. In 1954, though,
460:, using a delicate technique of double formal integration; and showing that the resulting sum has some generalized kind of second derivative using
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had proved the case of ξ rational -- the Cantor-like set is a set of uniqueness if and only if 1/ξ is an integer -- a few years earlier).
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682:(if any) are smaller than 1. This was the first demonstration that the property of being a set of uniqueness has to do with
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713:, first defined by Pyatetskii-Shapiro. Nowadays the research of sets of uniqueness is just as much a branch of
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426:
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524:. It is an open problem whether the union of any two Borel sets of uniqueness is a set of uniqueness.
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showed that a Cantor-like set with dissection ratio ξ is a set of uniqueness if and only if 1/ξ is a
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279:) are derived by integrating the function. Hence, in Fourier analysis the order of actions is
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761:
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484:, another innovator in set theory, started his career with a thesis on sets of uniqueness.
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is a set of uniqueness. This simply means that if a trigonometric series converges to zero
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The first evidence that sets of uniqueness have complex structure came from the study of
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mind-set. In
Fourier analysis there is no question of uniqueness, since the coefficients
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who in 1916 constructed an example of a set of multiplicity which has measure zero.
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To understand the importance of the definition, it is important to get out of the
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of a set of uniqueness is a set of uniqueness. A union of a countable family of
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746:(London Mathematical Society lecture series 128), Cambridge University Press.
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449:
429:. Hence the definition above, which arose when it became clear that neither
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is a concept relevant to trigonometric expansions which are not necessarily
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Ask: does this mean that they are the
Fourier coefficients of the function?
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is a set of uniqueness, a discovery which led him to the development of
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A closed set is a set of uniqueness if and only if there exists a
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Descriptive set theory and the structure of sets of uniqueness
15:
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on the set (so in particular it must be singular) such that
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Topics in the theory of uniqueness of trigonometrical series
425:, the most interesting questions arise when one discusses
598:{\displaystyle \lim _{n\to \infty }{\widehat {S}}(n)=0}
186:{\displaystyle \sum _{n=-\infty }^{\infty }c(n)e^{int}}
375:{\displaystyle c(n)=\int _{0}^{2\pi }f(t)e^{-int}\,dt}
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In the theory of uniqueness, the order is different:
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701:. A crucial part in this research is played by the
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468:generalized Riemann's techniques to show that any
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491:developed, it was assumed that any set of zero
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406:) for which the sum converges in some sense
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66:Learn how and when to remove this message
290:Calculate the Fourier coefficients using
29:This article includes a list of general
456:then it is trivial. This was proved by
260:). Analogous definitions apply on the
7:
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35:it lacks sufficient corresponding
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717:as it is of harmonic analysis.
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90:. Their study is a relatively
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639:{\displaystyle {\hat {S}}(n)}
398:Start with some coefficients
440:give a satisfactory answer.
196:which converges to zero for
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740:and Alain Louveau (1987),
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110:of the circle is called a
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705:of the set, which is an
654:Complexity of structure
50:more precise citations.
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715:descriptive set theory
648:Ilya Piatetski-Shapiro
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528:Singular distributions
431:convergence everywhere
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283:Start with a function
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427:pointwise convergence
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249:(sometimes called an
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738:Alexander S. Kechris
699:analytical hierarchy
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489:Lebesgue integration
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498:Dimitrii E. Menshov
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247:set of multiplicity
772:, Hermann, Paris.
758:Jean-Pierre Kahane
695:Hausdorff distance
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462:Toeplitz operators
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794:Harmonic analysis
676:algebraic integer
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487:As the theory of
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96:harmonic analysis
84:set of uniqueness
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780:(in French).
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672:Pisot number
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510:translation
258:Menshov set
241:Otherwise,
80:mathematics
48:introducing
788:Categories
721:References
684:arithmetic
680:conjugates
482:Paul Cohen
478:set theory
474:closed set
454:everywhere
102:Definition
94:branch of
56:March 2023
31:references
688:Nina Bari
622:^
575:^
564:∞
561:→
540:supported
470:countable
450:empty set
352:−
330:π
318:∫
262:real line
207:∉
151:∞
146:∞
143:−
133:∑
106:A subset
764:(1994),
729:(1958),
514:dilation
423:analysis
707:ordinal
668:Zygmund
493:measure
458:Riemann
269:Fourier
114:, or a
44:improve
776:
750:
518:closed
33:, but
703:index
522:Borel
256:or a
245:is a
774:ISBN
760:and
748:ISBN
666:and
512:and
448:The
433:nor
254:-set
119:-set
92:pure
82:, a
554:lim
78:In
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