|
Carleman's Theorem
In mathematics, a quasi-analytic class of functions is a generalization of the class of real analytic functions based upon the following fact: If ''f'' is an analytic function on an interval 'a'',''b''nbsp;⊂ R, and at some point ''f'' and all of its derivatives are zero, then ''f'' is identically zero on all of 'a'',''b'' Quasi-analytic classes are broader classes of functions for which this statement still holds true. Definitions Let M=\_^\infty be a sequence of positive real numbers. Then the Denjoy-Carleman class of functions ''C''''M''( 'a'',''b'' is defined to be those ''f'' ∈ ''C''∞( 'a'',''b'' which satisfy :\left , \frac(x) \right , \leq A^ k! M_k for all ''x'' ∈ 'a'',''b'' some constant ''A'', and all non-negative integers ''k''. If ''M''''k'' = 1 this is exactly the class of real analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There ... [...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] |
Analytic Function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about ''x''0 converges to the function in some neighborhood for every ''x''0 in its domain. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write : f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + a_3 (x-x_0)^3 + \cdots in which the coefficients a_0, a_1, \dots are real numbers and the series is convergent to f(x) for x in a neighborhood of x_0. Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point x_0 in its domain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carleman's Inequality
Carleman's inequality is an inequality in mathematics, named after Torsten Carleman, who proved it in 1923 and used it to prove the Denjoy–Carleman theorem on quasi-analytic classes. Statement Let a_1,a_2,a_3,\dots be a sequence of non-negative real numbers, then : \sum_^\infty \left(a_1 a_2 \cdots a_n\right)^ \le \mathrm \sum_^\infty a_n. The constant \mathrm (euler number) in the inequality is optimal, that is, the inequality does not always hold if \mathrm is replaced by a smaller number. The inequality is strict (it holds with "<" instead of "≤") if some element in the sequence is non-zero. Integral version Carleman's inequality has an integral version, which states that : \int_0^\infty \exp\left\ \,\mathrmx \leq \mathrm \int_0^\infty f(x) \,\mathrmx for any ''f'' ≥ 0. Carleson's inequality A generalisation, due to Lennart Carleson, states the following: for any convex function ''g'' with ''g''(0) = 0, and for any -1 1, r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
