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Littlewood's Tauberian Theorem
In mathematics, Littlewood's Tauberian theorem is a strengthening of Tauber's theorem introduced by . Statement Littlewood showed the following: If ''a''''n'' = O(1/''n'' ), and as ''x'' ↑ 1 we have :\sum a_n x^n \to s, then : \sum a_n = s. Hardy and Littlewood later showed that the hypothesis on ''a''''n'' could be weakened to the "one-sided" condition ''a''''n'' ≥ –''C''/''n'' for some constant ''C''. However in some sense the condition is optimal: Littlewood showed that if ''c''''n'' is any unbounded sequence then there is a series with , ''a''''n'', ≤ , ''c''''n'', /''n'' which is divergent but Abel summable. History described his discovery of the proof of his Tauberian theorem. Alfred Tauber's original theorem was similar to Littlewood's, but with the stronger hypothesis that ''a''''n''= o(1/''n''). Hardy had proved a similar theorem for Cesà ro summation with the weaker hypothesis ''a''''n''=O(1/''n''), and suggested to Littlewood that the same weaker hypo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tauber's Theorem
In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series converges to some limit then its Abel sum 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/''n'')) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods. 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 conditions for a series summable by some method that allows it to be summable in the usual sense. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred Tauber
Alfred Tauber (5 November 1866 – 26 July 1942) was a Hungarian-born Austrian mathematician, known for his contribution to mathematical analysis and to the theory of functions of a complex variable: he is the eponym of an important class of theorems with applications ranging from mathematical and harmonic analysis to number theory. He was murdered in the Theresienstadt concentration camp. Life and academic career Born in Pressburg, Kingdom of Hungary, Austrian Empire (now Bratislava, Slovakia), he began studying mathematics at Vienna University in 1884, obtained his Ph.D. in 1889,. and his habilitation in 1891. Starting from 1892, he worked as chief mathematician at the Phönix insurance company until 1908, when he became an a.o. professor at the University of Vienna, though, already from 1901, he had been honorary professor at TU Vienna and director of its insurance mathematics chair.. In 1933, he was awarded the Grand Decoration of Honour in Silver for Services to the Repub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Little O
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jovan Karamata
Jovan Karamata ( sr-cyr, Јован Карамата; February 1, 1902 – August 14, 1967) was a Serbian mathematician. He is remembered for contributions to analysis, in particular, the Tauberian theory and the theory of slowly varying functions. Considered to be among the most influential Serbian mathematicians of the 20th century, Karamata was one of the founders of the Mathematical Institute of the Serbian Academy of Sciences and Arts, established in 1946. Life Jovan Karamata was born in Zagreb on February 1, 1902 into a family descended from merchants based in the city of Zemun, which was then in Austria-Hungary, and now in Serbia. Being of Aromanian origin, the family traced its roots back to Pyrgoi, Eordaia, West Macedonia ''(his father Ioannis Karamatas was the president of the "Greek Community of Zemun")''. Its business affairs on the borders of the Austro-Hungarian and Ottoman empires were very well known. In 1914, he finished most of his primary school in Zemun b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy–Littlewood Tauberian Theorem
In mathematical analysis, the Hardy–Littlewood Tauberian theorem is a Tauberian theorem relating the asymptotics of the partial sums of a series with the asymptotics of its Abel summation. In this form, the theorem asserts that if, as ''y'' ↓ 0, the non-negative sequence ''a''''n'' is such that there is an asymptotic equivalence :\sum_^\infty a_n e^ \sim \frac then there is also an asymptotic equivalence :\sum_^n a_k \sim n as ''n'' → ∞. The integral formulation of the theorem relates in an analogous manner the asymptotics of the cumulative distribution function of a function with the asymptotics of its Laplace transform. The theorem was proved in 1914 by G. H. Hardy and J. E. Littlewood. In 1930, Jovan Karamata gave a new and much simpler proof. Statement of the theorem Series formulation This formulation is from Titchmarsh. Suppose ''a''''n'' ≥ 0 for all ''n'', and as ''x'' ↑ 1 we have :\sum_^\infty a_n x^n \sim \frac. Then as ''n'' goes to ∞ we have : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener's Tauberian Theorem
In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in or can be approximated by linear combinations of translations of a given function.see . Informally, if the Fourier transform of a function vanishes on a certain set , the Fourier transform of any linear combination of translations of also vanishes on . Therefore, the linear combinations of translations of can not approximate a function whose Fourier transform does not vanish on . Wiener's theorems make this precise, stating that linear combinations of translations of are dense if and only if the zero set of the Fourier transform of is empty (in the case of ) or of Lebesgue measure zero (in the case of ). Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the L1 group ring L1(R) of the group R of real numbers is the d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy \, x \, y\, \ \leq \, x\, \, \, y\, \quad \text x, y \in A. This ensures that the multiplication operation is continuous. A Banach algebra is called ''unital'' if it has an identity element for the multiplication whose norm is 1, and ''commutative'' if its multiplication is commutative. Any Banach algebra A (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra A_e so as to form a closed ideal of A_e. Often one assumes ''a priori'' that the algebra under consideration is unital: for one can develop much of the theory by considering A_e and then applying the outcome in the ori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
