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Fejér's Theorem
In mathematics, Fejér's theorem,Leopold FejérUntersuchungen über Fouriersche Reihen ''Mathematische Annalen''vol. 58 1904, 51-69. named after Hungarian mathematician Lipót Fejér, states the following: Explanation of Fejér's Theorem's Explicitly, we can write the Fourier series of ''f'' as f(x)= \sum_^ c_n \, e^where the nth partial sum of the Fourier series of ''f'' may be written as :s_n(f,x)=\sum_^nc_ke^, where the Fourier coefficients c_k are :c_k=\frac\int_^\pi f(t)e^dt. Then, we can define :\sigma_n(f,x)=\frac\sum_^s_k(f,x) = \frac\sum_^n s_k(f,x) with ''F''''n'' being the ''n''th order Fejér kernel. Then, Fejér's theorem asserts that \lim_ \sigma_n (f, x) = f(x) with uniform convergence. With the convergence written out explicitly, the above statement becomes \forall \epsilon > 0 \, \exist\, n_0 \in \mathbb: n \geq n_0 \implies , f(x) - \sigma_n(f,x), 0 \, \exist\, n_0 \in \mathbb: n \geq n_0 \implies , f(x) - \sigma_n(f,x), 0,\exist \delta > 0: , x-y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hungary
Hungary ( hu, Magyarország ) is a landlocked country in Central Europe. Spanning of the Pannonian Basin, 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 people in Hungary, Romani minority. Hungarian language, Hungarian, the Languages of Hungary, official language, is the world's most widely spoken Uralic languages, Uralic language and among the few non-Indo-European languages widely spoken in Europe. Budapest is the country's capital and List of cities and towns of Hungary, 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, Ancient Rome, Romans, Germanic peoples, Germanic trib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hyp ... [...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] |
Partial Sum
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of ... [...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- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Means
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics, because it helps distinguish it from other means, such as the geometric mean and the harmonic mean. In addition to mathematics and statistics, the arithmetic mean is used frequently in many diverse fields such as economics, anthropology and history, and it is used in almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation's population. While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly infl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily small positive number \epsilon, a number N can be found such that each of the functions f_N, f_,f_,\ldots differs from f by no more than \epsilon ''at every point'' x ''in'' E. Described in an informal way, if f_n converges to f uniformly, then the rate at which f_n(x) approaches f(x) is "uniform" throughout its domain in the following sense: in order to guarantee that f_n(x) falls within a certain distance \epsilon of f(x), we do not need to know the value of x\in E in question — there can be found a single value of N=N(\epsilon) ''independent of x'', such that choosing n\geq N will ensure that f_n(x) is within \epsilon of f(x) ''for all x\in E''. In contrast, pointwise convergence of f_n to f merely guarantees that for any x\in E giv ... [...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] |
