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Dini's Test
In mathematics, the Dini and Dini–Lipschitz tests are highly precise tests that can be used to prove that the Fourier series of a function converges at a given point. These tests are named after Ulisse Dini and Rudolf Lipschitz. Definition Let be a function on ,2 let be some point and let be a positive number. We define the local modulus of continuity at the point by :\left.\right.\omega_f(\delta;t)=\max_ , f(t)-f(t+\varepsilon), Notice that we consider here to be a periodic function, e.g. if and is negative then we define . The global modulus of continuity (or simply the modulus of continuity) is defined by :\omega_f(\delta) = \max_t \omega_f(\delta;t) With these definitions we may state the main results: :Theorem (Dini's test): Assume a function satisfies at a point that ::\int_0^\pi \frac\omega_f(\delta;t)\,\mathrm\delta < \infty. :Then the Fourier series of converges at to . For example, the theorem holds with but does not hold with . :Theorem ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always Convergent series, converge. Well-behaved functions, for example Smoothness, smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called the Domain of a function, domain of the function and the set is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. History of the function concept, Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable function, differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept. A f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ulisse Dini
Ulisse Dini (14 November 1845 – 28 October 1918) was an Italian mathematician and politician, born in Pisa. He is known for his contributions to real analysis, partly collected in his book "''Fondamenti per la teorica delle funzioni di variabili reali''". Life and academic career Dini attended the Scuola Normale Superiore in order to become a teacher. One of his professors was Enrico Betti. In 1865, a scholarship enabled him to visit Paris, where he studied under Charles Hermite as well as Joseph Bertrand, and published several papers. In 1866, he was appointed to the University of Pisa, where he taught algebra and geodesy. In 1871, he succeeded Betti as professor for Mathematical analysis, analysis and geometry. From 1888 until 1890, Dini was ''rettore'' of the Pisa University, and of the ''Scuola Normale Superiore'' from 1908 until his death in 1918. He was also active as a politician: in 1871 he was voted into the Pisa city council and in 1880 became a member of the Parlia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rudolf Lipschitz
Rudolf Otto Sigismund Lipschitz (14 May 1832 – 7 October 1903) was a German mathematician who made contributions to mathematical analysis (where he gave his name to the Lipschitz continuity condition) and differential geometry, as well as number theory, algebras with involution and classical mechanics. Biography Rudolf Lipschitz was born on 14 May 1832 in Königsberg. He was the son of a landowner and was raised at his father's estate at Bönkein which was near Königsberg. He entered the University of Königsberg when he was 15, but later moved to the University of Berlin where he studied with Gustav Dirichlet. Despite having his studies delayed by illness, in 1853 Lipschitz graduated with a PhD in Berlin. After receiving his PhD, Lipschitz started teaching at local Gymnasiums. In 1857 he married Ida Pascha, the daughter of one of the landowners with an estate near to his father's, and earned his habilitation at the University of Bonn, where he remained as a privatdozent. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modulus Of Continuity
In mathematical analysis, a modulus of continuity is a function ω : , ∞→ , ∞used to measure quantitatively the uniform continuity of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if :, f(x)-f(y), \leq\omega(, x-y, ), for all ''x'' and ''y'' in the domain of ''f''. Since moduli of continuity are required to be infinitesimal at 0, a function turns out to be uniformly continuous if and only if it admits a modulus of continuity. Moreover, relevance to the notion is given by the fact that sets of functions sharing the same modulus of continuity are exactly equicontinuous families. For instance, the modulus ω(''t'') := ''kt'' describes the k- Lipschitz functions, the moduli ω(''t'') := ''kt''α describe the Hölder continuity, the modulus ω(''t'') := ''kt''(, log ''t'', +1) describes the almost Lipschitz class, and so on. In general, the role of ω is to fix some explicit functional dependence of ε on δ in the (ε, δ) definition of uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hölder Condition
In mathematics, a real or complex-valued function on -dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants , , such that , f(x) - f(y) , \leq C\, x - y\, ^ for all and in the domain of . More generally, the condition can be formulated for functions between any two metric spaces. The number \alpha is called the ''exponent'' of the Hölder condition. A function on an interval satisfying the condition with is constant (see proof below). If , then the function satisfies a Lipschitz condition. For any , the condition implies the function is uniformly continuous. The condition is named after Otto Hölder. If \alpha = 0, the function is simply bounded (any two values f takes are at most C apart). We have the following chain of inclusions for functions defined on a closed and bounded interval of the real line with : where . Hölder spaces Hölder spaces consisting of functions satisfying a Hölder conditio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a member of a #Related asymptotic notations, family of notations invented by German mathematicians Paul Gustav Heinrich Bachmann, Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for '':wikt:Ordnung#German, Ordnung'', meaning the order of approximation. In computer science, big O notation is used to Computational complexity theory, 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 arithmetic function, arithmetical function and a better understood approximation; one well-known exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence Of Fourier Series
In mathematics, the question of whether the Fourier series of a given periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur. Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, ''L''''p'' spaces, summability methods and the Cesàro mean. Preliminaries Consider ''f'' an integrable function on the interval . For such an ''f'' the Fourier coefficients \widehat(n) are defined by the formula :\widehat(n)=\frac\int_0^f(t)e^\,\mathrmt, \quad n \in \Z. It is common to describe the connection between ''f'' and its Fourier series by :f \sim \sum_n \widehat(n) e^. The notation ~ here means that the sum represents the function in some sense. To investigate this more carefully, the partial sums must be defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dini Continuity
In mathematical analysis, Dini continuity is a refinement of continuity. Every Dini continuous function is continuous. Every Lipschitz continuous function is Dini continuous. Definition Let X be a compact subset of a metric space (such as \mathbb^n), and let f:X\rightarrow X be a function from X into itself. The modulus of continuity of f is :\omega_f(t) = \sup_ d(f(x),f(y)). The function f is called Dini-continuous if :\int_0^1 \frac\,dt < \infty. An equivalent condition is that, for any , : where is the of . See also * Dini test — a condition similar t ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dini Criterion
In mathematics, Dini's criterion is a condition for the pointwise convergence of Fourier series, introduced by . Statement Dini's criterion states that if a periodic function f has the property that (f(t)+f(-t))/t is locally integrable In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions li ... near 0, then the Fourier series of f converges to 0 at t=0. Dini's criterion is in some sense as strong as possible: if g(t) is a positive continuous function such that g(t)/t is not locally integrable near 0, there is a continuous function f with , f(t), \leq g(t) whose Fourier series does not converge at 0. References * *{{SpringerEOM, id=Dini_criterion&oldid=28457, title=Dini criterion, first=B. I., last= Golubov Fourier series ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
