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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] |
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] |
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] |
Carleson's Theorem
Carleson's theorem is a fundamental result in mathematical analysis establishing the ( Lebesgue) pointwise almost everywhere convergence of Fourier series of functions, proved by . The name is also often used to refer to the extension of the result by to functions for (also known as the ''Carleson–Hunt theorem'') and the analogous results for pointwise almost everywhere convergence of Fourier integrals, which can be shown to be equivalent by transference methods. Statement of the theorem The result, as extended by Hunt, can be formally stated as follows: The analogous result for Fourier integrals is: History A fundamental question about Fourier series, asked by Fourier himself at the beginning of the 19th century, is whether the Fourier series of a continuous function converges pointwise to the function. By strengthening the continuity assumption slightly one can easily show that the Fourier series converges everywhere. For example, if a function has bounde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space". Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baire Space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of Baire spaces. The Baire category theorem combined with the properties of Baire spaces has numerous applications in topology, geometry, and analysis, in particular functional analysis. For more motivation and applications, see the article Baire category theorem. The current article focuses more on characterizations and basic properties of Baire spaces per se. Bourbaki introduced the term "Baire space" in honor of René Baire, who investigated the Baire category theorem in the context of Euclidean space \R^n in his 1899 thesis. Definition The definition that follows is based on the notions of meagre (or first category) set (namely, a set that is a countable union of sets whose closure has empty interior) and nonmeagre ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baire Category Theorem
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the intersection of countably many dense open sets is still dense). It is used in the proof of results in many areas of analysis and geometry, including some of the fundamental theorems of functional analysis. Versions of the Baire category theorem were first proved independently in 1897 by Osgood for the real line \R and in 1899 by Baire for Euclidean space \R^n. The more general statement for completely metrizable spaces was first shown by Hausdorff in 1914. Statement A Baire space is a topological space X in which every countable intersection of open dense sets is dense in X. See the corresponding article for a list of equivalent characterizations, as some are more useful than others depending on the application. * (BCT1) Every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Boundedness Principle
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm. The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus, but it was also proven independently by Hans Hahn. Theorem The first inequality (that is, \sup_ \, T(x)\, rather than \infty, \infty/math>) then closed unit ball can be replaced with the unit sphere \sup_ \, T\, _ = \sup_ \, T(x)\, _Y. The completeness of the Banach space X enables the following short proof, using the Baire category theorem. There are also simple proofs not using the Baire theorem . Corollaries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Series
In mathematics, trigonometric series are a special class of orthogonal series of the form : A_0 + \sum_^\infty A_n \cos + B_n \sin, where x is the variable and \ and \ are coefficients. It is an infinite version of a trigonometric polynomial. A trigonometric series is called the Fourier series of the integrable function f if the coefficients have the form: :A_n=\frac1\pi \int^_0\! f(x) \cos \,dx :B_n=\frac\displaystyle\int^_0\! f(x) \sin \, dx Examples Every Fourier series gives an example of a trigonometric series. Let the function f(x) = x on \pi,\pi/math> be extended periodically (see sawtooth wave). Then its Fourier coefficients are: :\begin A_n &= \frac1\pi\int_^ x \cos \,dx = 0, \quad n \ge 0. \\ ptB_n &= \frac1\pi\int_^ x \sin \, dx \\ pt&= -\frac \cos + \frac1\sin \Bigg\vert_^\pi \\ mu&= \frac, \quad n \ge 1.\end Which gives an example of a trigonometric series: :2\sum_^\infty \frac \sin = 2\sin - \frac22\sin + \frac23\sin - \frac24\sin + \cdots However ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass M-test
In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers. It is named after the German mathematician Karl Weierstrass (1815–1897). Statement Weierstrass M-test. Suppose that (''f''''n'') is a sequence of real- or complex-valued functions defined on a set ''A'', and that there is a sequence of non-negative numbers (''M''''n'') satisfying the conditions * , f_n(x), \leq M_n for all n \geq 1 and all x \in A, and * \sum_^ M_n converges. Then the series :\sum_^ f_n (x) converges absolutely and uniformly on ''A''. A series satisfying the hypothesis is called '' normally convergent''. The result is often used in combination with the uniform limit theorem. Together they say that if, in addition to the ... [...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] |
Gibbs Phenomenon
In mathematics, the Gibbs phenomenon is the oscillatory behavior of the Fourier series of a piecewise continuously differentiable periodic function around a jump discontinuity. The Nth partial Fourier series of the function (formed by summing the N lowest constituent sinusoids of the Fourier series of the function) produces large peaks around the jump which overshoot and undershoot the function values. As more sinusoids are used, this approximation error approaches a limit of about 9% of the jump, though the infinite Fourier series sum does eventually converge almost everywhere. The Gibbs phenomenon was observed by experimental physicists and was believed to be due to imperfections in the measuring apparatus, but it is in fact a mathematical result. It is one cause of ringing artifacts in signal processing. It is named after Josiah Willard Gibbs. Description The Gibbs phenomenon is a behavior of the Fourier series of a function with a jump discontinuity and is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
