Set of uniqueness
Encyclopedia
In mathematics
, a set of uniqueness is a concept relevant to trigonometric expansions which are not necessarily Fourier series
. Their study is a relatively pure
branch of harmonic analysis
.
which converges to zero for
is identically zero; that is, such that
Otherwise E is a set of multiplicity (sometimes called an M-set or a Menshov set). Analogous definitions apply on the real line
, 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".
To understand the importance of the definition it is important to get out of the Fourier
mind-set. In Fourier analysis there is no question of uniqueness, since the coefficients c(n) are derived by integrating the function. Hence in Fourier analysis the order of actions is
In the theory of uniqueness the order is different:
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 c(n) must be zero. As is usual in analysis
, the most interesting questions arise when one discusses pointwise convergence
. Hence the definition above, which arose when it became clear that neither convergence everywhere nor convergence almost everywhere
give a satisfactory answer.
is a set of uniqueness. This is just a fancy way to say that if a trigonometric series converges to zero everywhere then it is trivial. This was proved by Riemann
, using a delicate technique of double formal integration; and showing that the resulting sum has some generalized kind of second derivative using Toeplitz operator
s. Later on, Cantor
generalized Riemann's techniques to show that any countable
, closed set
is a set of uniqueness, a discovery which led him to the development of set theory
. Interestingly, Paul Cohen
, another great innovator in set theory, started his career with a thesis on sets of uniqueness.
As the theory of Lebesgue integration
developed, it was assumed that any set of zero measure
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 D. E. Menshov
who in 1916 constructed an example of a set of multiplicity which has measure zero.
of a set of uniqueness is a set of uniqueness. A union of a countable family of closed 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 Borel
. It is an open problem whether the union of any two Borel sets of uniqueness is a set of uniqueness.
S support
ed on the set (so in particular it must be singular) such that
(
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, Ilya Piatetski-Shapiro 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.
. Salem
and Zygmund
showed that a Cantor-like set with dissection ratio ξ is a set of uniqueness if and only if 1/ξ is a Pisot number
, that is an algebraic integer
with the property that all its conjugates
(if any) are smaller than 1. This was the first demonstration that the property of being a set of uniqueness has to do with arithmetic properties and not just some concept of size (Nina Bary 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).
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 Hausdorff distance
), was located inside the analytical hierarchy
. A crucial part in this research is played by the index of the set, which is an ordinal
between 1 and ω1, first defined by Pyatetskii-Shapiro. Nowadays the research of sets of uniqueness is just as much a branch of descriptive set theory
as it is of harmonic analysis. See the Kechris-Louveau book referenced below.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a set of uniqueness is a concept relevant to trigonometric expansions which are not necessarily Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
. Their study is a relatively pure
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
branch of harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
.
Definition
A subset E of the circle is called a set of uniqueness, or a U-set, if any trigonometric expansionwhich converges to zero for
is identically zero; that is, such thatOtherwise E is a set of multiplicity (sometimes called an M-set or a Menshov set). Analogous definitions apply on the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
, 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".
To understand the importance of the definition it is important to get out of the Fourier
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
mind-set. In Fourier analysis there is no question of uniqueness, since the coefficients c(n) are derived by integrating the function. Hence in Fourier analysis the order of actions is
- Start with a function f.
- Calculate the Fourier coefficients using
- Ask: does the sum converge to f? In which sense?
In the theory of uniqueness the order is different:
- Start with some coefficients c(n) for which the sum converge in some sense
- Ask: does this means that they are the Fourier coefficients of the function?
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 c(n) must be zero. As is usual in analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, the most interesting questions arise when one discusses pointwise convergence
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.-Definition:...
. Hence the definition above, which arose when it became clear that neither convergence everywhere nor convergence almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
give a satisfactory answer.
Early research
The empty setEmpty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
is a set of uniqueness. This is just a fancy way to say that if a trigonometric series converges to zero everywhere then it is trivial. This was proved by Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
, using a delicate technique of double formal integration; and showing that the resulting sum has some generalized kind of second derivative using Toeplitz operator
Toeplitz operator
In operator theory, a Toeplitz operator is the compression of a multiplication operator on the circle to the Hardy space.- Details :Let S1 be the circle, with the standard Lebesgue measure, and L2 be the Hilbert space of square-integrable functions. A bounded measurable function g on S1 defines a...
s. Later on, Cantor
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
generalized Riemann's techniques to show that any countable
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
, closed set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
is a set of uniqueness, a discovery which led him to the development of set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
. Interestingly, Paul Cohen
Paul Cohen (mathematician)
Paul Joseph Cohen was an American mathematician best known for his proof of the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory, the most widely accepted axiomatization of set theory.-Early years:Cohen was born in Long Branch, New Jersey, into a...
, another great innovator in set theory, started his career with a thesis on sets of uniqueness.
As the theory of Lebesgue integration
Lebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...
developed, it was assumed that any set of zero measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
would be a set of uniqueness — in one dimension the locality principle for Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
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 D. E. Menshov
Dmitrii Evgenevich Menshov
Dmitrii Evgenevich Menshov was a Russian mathematician known for his contributions to the theory of trigonometric series.He was a student of Nikolai Luzin. He was an advisor of Sergey Borisovich Stechkin. He received USSR State Prize in 1951...
who in 1916 constructed an example of a set of multiplicity which has measure zero.
Transformations
A translation and dilationDilation
Dilation refers to an enlargement or expansion in bulk or extent, the opposite of contraction. It derives from the Latin dilatare, "to spread wide".In physiology:* Pupillary dilation, dilation of the pupil of the eye...
of a set of uniqueness is a set of uniqueness. A union of a countable family of closed 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 Borel
Borel algebra
In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
. It is an open problem whether the union of any two Borel sets of uniqueness is a set of uniqueness.
Singular distributions
A closed set is a set of uniqueness if and only if there exists a distributionDistribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
S support
Support (mathematics)
In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
ed on the set (so in particular it must be singular) such that
(
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, Ilya Piatetski-Shapiro 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.Complexity of structure
The first evidence that sets of uniqueness have complex structure came from the study of Cantor-like setsCantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
. Salem
Raphaël Salem
Raphaël Salem, was a Greek-Sephardic mathematician after whom are named the Salem numbers and whose widow founded the Salem Prize.-Biography:...
and Zygmund
Antoni Zygmund
Antoni Zygmund was a Polish-born American mathematician.-Life:Born in Warsaw, Zygmund obtained his PhD from Warsaw University and became a professor at Stefan Batory University at Wilno...
showed that a Cantor-like set with dissection ratio ξ is a set of uniqueness if and only if 1/ξ is a Pisot number
Pisot-Vijayaraghavan number
In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1 such that all its Galois conjugates are less than 1 in absolute value. These numbers were discovered by Axel Thue in 1912 and rediscovered by G. H. Hardy in...
, that is an algebraic integer
Algebraic integer
In number theory, an algebraic integer is a complex number that is a root of some monic polynomial with coefficients in . The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A...
with the property that all its conjugates
Conjugate element (field theory)
In mathematics, in particular field theory, the conjugate elements of an algebraic element α, over a field K, are the roots of the minimal polynomialof α over K.-Example:The cube roots of the number one are:...
(if any) are smaller than 1. This was the first demonstration that the property of being a set of uniqueness has to do with arithmetic properties and not just some concept of size (Nina Bary 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).
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 Hausdorff distance
Hausdorff distance
In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right...
), was located inside the analytical hierarchy
Analytical hierarchy
In mathematical logic and descriptive set theory, the analytical hierarchy is a higher type analogue of the arithmetical hierarchy. It thus continues the classification of sets by the formulas that define them.-The analytical hierarchy of formulas:...
. A crucial part in this research is played by the index of the set, which is an ordinal
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
between 1 and ω1, first defined by Pyatetskii-Shapiro. Nowadays the research of sets of uniqueness is just as much a branch of descriptive set theory
Descriptive set theory
In mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces...
as it is of harmonic analysis. See the Kechris-Louveau book referenced below.