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Fejér Kernel
In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lipót Fejér (1880–1959). Definition The Fejér kernel has many equivalent definitions. We outline three such definitions below: 1) The traditional definition expresses the Fejér kernel F_n(x) in terms of the Dirichlet kernel: where :D_k(x)=\sum_^k ^ is the ''k''th order Dirichlet kernel. 2) The Fejér kernel F_n(x) may also be written in a closed form expression as follows This closed form expression may be derived from the definitions used above. The proof of this result goes as follows. First, we use the fact that the Dirichet kernel may be written as: :D_k(x)=\frac Hence, using the definition of the Fejér kernel above we get: :F_n(x) = \frac \sum_^D_k(x) = \frac \sum_^ \frac = \frac \frac\sum_^ \sin((k +\frac)x) = \f ... [...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] |
Summability Kernel
In mathematics, a summability kernel is a family or sequence of periodic integrable functions satisfying a certain set of properties, listed below. Certain kernels, such as the Fejér kernel, are particularly useful in Fourier analysis. Summability kernels are related to approximation of the identity; definitions of an approximation of identity vary, but sometimes the definition of an approximation of the identity is taken to be the same as for a summability kernel. Definition Let \mathbb:=\mathbb/\mathbb. A summability kernel is a sequence (k_n) in L^1(\mathbb) that satisfies # \int_\mathbbk_n(t)\,dt=1 # \int_\mathbb, k_n(t), \,dt\le M (uniformly bounded) # \int_, k_n(t), \,dt\to0 as n\to\infty, for every \delta>0. Note that if k_n\ge0 for all n, i.e. (k_n) is a positive summability kernel, then the second requirement follows automatically from the first. With the convention \mathbb=\mathbb/2\pi\mathbb, the first equation becomes \frac\int_\mathbbk_n(t)\,dt=1, and the upper l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cesàro Summation
In mathematical analysis, Cesàro summation (also known as the Cesàro mean ) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as ''n'' tends to infinity, of the sequence of arithmetic means of the first ''n'' partial sums of the series. This special case of a matrix summability method is named for the Italian analyst Ernesto Cesàro (1859–1906). The term ''summation'' can be misleading, as some statements and proofs regarding Cesàro summation can be said to implicate the Eilenberg–Mazur swindle. For example, it is commonly applied to Grandi's series with the conclusion that the ''sum'' of that series is 1/2. Definition Let (a_n)_^\infty be a sequence, and let :s_k = a_1 + \cdots + a_k= \sum_^k a_n be its th partial sum. The sequence is called Cesàro summable, with Cesàro sum , if, as tends to infinity, the arithmetic mean of its first ''n'' partial sums tends to : :\lim_ \f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''period''), the number of components, and their amplitudes and phase parameters. With appropriate choices, one cycle (or ''period'') of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). The number of components is theoretically infinite, in which case the other parameters can be chosen to cause the series to converge to almost any ''well behaved'' periodic function (see Pathological and Dirichlet–Jordan test). The components of a particular function are determined by ''analysis'' techniques described in this article. Sometimes the components are known first, and the unknown function is ''synthesized'' by a Fourier series. Such is the case of a discrete-ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximate Identity
In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element. Definition A right approximate identity in a Banach algebra ''A'' is a net \ such that for every element ''a'' of ''A'', \lim_\lVert ae_\lambda - a \rVert = 0. Similarly, a left approximate identity in a Banach algebra ''A'' is a net \ such that for every element ''a'' of ''A'', \lim_\lVert e_\lambda a - a \rVert = 0. An approximate identity is a net which is both a right approximate identity and a left approximate identity. C*-algebras For C*-algebras, a right (or left) approximate identity consisting of self-adjoint elements is the same as an approximate identity. The net of all positive elements in ''A'' of norm ≤ 1 with its natural order is an approximate identity for any C*-algebra. This is called the canonical approximate identity of a C*-algebra. Appr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hungary
Hungary ( hu, Magyarország ) is a landlocked country in Central Europe. Spanning of the Carpathian Basin, it is bordered by Slovakia to the north, Ukraine to the northeast, Romania to the east and southeast, Serbia to the south, Croatia and Slovenia to the southwest, and Austria to the west. Hungary has a population of nearly 9 million, mostly ethnic Hungarians and a significant Romani minority. Hungarian, the official language, is the world's most widely spoken Uralic language and among the few non-Indo-European languages widely spoken in Europe. Budapest is the country's capital and largest city; other major urban areas include Debrecen, Szeged, Miskolc, Pécs, and Győr. The territory of present-day Hungary has for centuries been a crossroads for various peoples, including Celts, Romans, Germanic tribes, Huns, West Slavs and the Avars. The foundation of the Hungarian state was established in the late 9th century AD with the conquest of the Carpathian Basin by Hungar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipót Fejér
Lipót Fejér (or Leopold Fejér, ; 9 February 1880 – 15 October 1959) was a Hungarian mathematician of Jewish heritage. Fejér was born Leopold Weisz, and changed to the Hungarian name Fejér around 1900. Biography Fejér studied mathematics and physics at the University of Budapest and at the University of Berlin, where he was taught by Hermann Schwarz. In 1902 he earned his doctorate from University of Budapest (today Eötvös Loránd University). From 1902 to 1905 Fejér taught there and from 1905 until 1911 he taught at Franz Joseph University in Kolozsvár in Austria-Hungary (now Cluj-Napoca in Romania). In 1911 Fejér was appointed to the chair of mathematics at the University of Budapest and he held that post until his death. He was elected corresponding member (1908), member (1930) of the Hungarian Academy of Sciences. During his period in the chair at Budapest Fejér led a highly successful Hungarian school of analysis. He was the thesis advisor of mathematicians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fejér Kernel
In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lipót Fejér (1880–1959). Definition The Fejér kernel has many equivalent definitions. We outline three such definitions below: 1) The traditional definition expresses the Fejér kernel F_n(x) in terms of the Dirichlet kernel: where :D_k(x)=\sum_^k ^ is the ''k''th order Dirichlet kernel. 2) The Fejér kernel F_n(x) may also be written in a closed form expression as follows This closed form expression may be derived from the definitions used above. The proof of this result goes as follows. First, we use the fact that the Dirichet kernel may be written as: :D_k(x)=\frac Hence, using the definition of the Fejér kernel above we get: :F_n(x) = \frac \sum_^D_k(x) = \frac \sum_^ \frac = \frac \frac\sum_^ \sin((k +\frac)x) = \f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Kernel
In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as D_n(x)= \sum_^n e^ = \left(1+2\sum_^n\cos(kx)\right)=\frac, where is any nonnegative integer. The kernel functions are periodic with period 2\pi. 300px, Plot restricted to one period Dirac delta distributions of the Dirac comb">Dirac comb. The importance of the Dirichlet kernel comes from its relation to Fourier series. The convolution of with any function of period 2 is the ''n''th-degree Fourier series approximation to , i.e., we have (D_n*f)(x)=\int_^\pi f(y)D_n(x-y)\,dy=\sum_^n \hat(k)e^, where \widehat(k)=\frac 1 \int_^\pi f(x)e^\,dx is the th Fourier coefficient of . This implies that in order to study convergence of Fourier series it is enough to study properties of the Dirichlet kernel. ''L''1 norm of the kernel function Of particular importance is the fact that the ''L''1 norm of ''Dn'' on , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see #Properties, commutativity). The integral is evaluated for all values of shift, producing the convolution function. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution (f*g) differs from cross-correlation (f \star g) only in that either or is reflected about the y-axis in convolution; thus it is a cross-c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young's Convolution Inequality
In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions, named after William Henry Young. Statement Euclidean Space In real analysis, the following result is called Young's convolution inequality: Suppose f is in the Lebesgue spaceL^p(\Reals^d) and g is in L^q(\Reals^d) and \frac + \frac = \frac + 1 with 1 \leq p, q, r \leq \infty. Then \, f * g\, _r \leq \, f\, _p \, g\, _q. Here the star denotes convolution, L^p is Lebesgue space, and \, f\, _p = \Bigl(\int_ , f(x), ^p\,dx \Bigr)^ denotes the usual L^p norm. Equivalently, if p, q, r \geq 1 and \frac + \frac + \frac = 2 then \left, \int_ \int_ f(x) g(x - y) h(y) \,\mathrmx \,\mathrmy \ \leq \left(\int_ \vert f\vert^p\right)^\frac \left(\int_ \vert g\vert^q\right)^\frac \left(\int_ \vert h\vert^r\right)^\frac Generalizations Young's convolution inequality has a natural generalization in which we replace \Reals^d by a unimodular group G. If we let \mu be a bi-in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pointwise Convergence
In mathematics, pointwise convergence is one of Modes of convergence (annotated index), various senses in which a sequence of functions can Limit (mathematics), converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Definition Suppose that X is a set and Y is a topological space, such as the Real number, real or complex numbers or a metric space, for example. A Net (mathematics), net or sequence of Function (mathematics), functions \left(f_n\right) all having the same domain X and codomain Y is said to converge pointwise to a given function f : X \to Y often written as \lim_ f_n = f\ \mbox if (and only if) \lim_ f_n(x) = f(x) \text x \text f. The function f is said to be the pointwise limit function of the \left(f_n\right). Sometimes, authors use the term bounded pointwise convergence when there is a constant C such that \forall n,x,\;, f_n(x), Properties This concept is often contra ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
