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Dirichlet–Jordan Test
In mathematics, the Dirichlet–Jordan test gives sufficient conditions for a real-valued, periodic function ''f'' to be equal to the sum of its Fourier series at a point of continuity. Moreover, the behavior of the Fourier series at points of discontinuity is determined as well (it is the midpoint of the values of the discontinuity). It is one of many conditions for the convergence of Fourier series. The original test was established by Peter Gustav Lejeune Dirichlet in 1829, for piecewise monotone functions. It was extended in the late 19th century by Camille Jordan to functions of bounded variation (any function of bounded variation is the difference of two increasing functions). Dirichlet–Jordan test for Fourier series The Dirichlet–Jordan test states that if a periodic function f(x) is of bounded variation on a period, then the Fourier series S_nf(x) converges, as n\to\infty, at each point of the domain to \lim_\frac. In particular, if f is continuous at x, then the Fouri ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antoni Zygmund
Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician. He worked mostly in the area of mathematical analysis, including especially harmonic analysis, and he is considered one of the greatest analysts of the 20th century. Zygmund was responsible for creating the Chicago school of mathematical analysis together with his doctoral student Alberto Calderón, for which he was awarded the National Medal of Science in 1986. Biography Born in Warsaw, Zygmund obtained his Ph.D. from the University of Warsaw (1923) and was a professor at Stefan Batory University at Wilno from 1930 to 1939, when World War II broke out and Poland was occupied. In 1940 he managed to emigrate to the United States, where he became a professor at Mount Holyoke College in South Hadley, Massachusetts. In 1945–1947 he was a professor at the University of Pennsylvania, and from 1947, until his retirement, at the University of Chicago. He was a member of several scientific societies. From ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolutely Integrable
In mathematics, an absolutely integrable function is a function whose absolute value is integrable, meaning that the integral of the absolute value over the whole domain is finite. For a real-valued function, since \int , f(x), \, dx = \int f^+(x) \, dx + \int f^-(x) \, dx where f^+(x) = \max (f(x),0), \ \ \ f^-(x) = \max(-f(x),0), both \int f^+(x) \, dx and \int f^-(x) \, dx must be finite. In Lebesgue integration, this is exactly the requirement for any measurable function ''f'' to be considered integrable, with the integral then equaling \int f^+(x) \, dx - \int f^-(x) \, dx, so that in fact "absolutely integrable" means the same thing as "Lebesgue integrable" for measurable functions. The same thing goes for a complex-valued function. Let us define f^+(x) = \max(\Re f(x),0) f^-(x) = \max(-\Re f(x),0) f^(x) = \max(\Im f(x),0) f^(x) = \max(-\Im f(x),0) where \Re f(x) and \Im f(x) are the real and imaginary parts In mathematics, a complex number is an element of a numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cornelius Lanczos
__NOTOC__ Cornelius (Cornel) Lanczos ( hu, Lánczos Kornél, ; born as Kornél Lőwy, until 1906: ''Löwy (Lőwy) Kornél''; February 2, 1893 – June 25, 1974) was a Hungarian-American and later Hungarian-Irish mathematician and physicist. According to György Marx he was one of The Martians. Biography He was born in Fehérvár (Alba Regia), Fejér County, Kingdom of Hungary to Károly Lőwy and Adél Hahn. Lanczos' Ph.D. thesis (1921) was on relativity theory. He sent his thesis copy to Albert Einstein, and Einstein wrote back, saying: "I studied your paper as far as my present overload allowed. I believe I may say this much: this does involve competent and original brainwork, on the basis of which a doctorate should be obtainable ... I gladly accept the honorable dedication."Barbara Gellai (2010) ''The Intrinsic Nature of Things: the life and science of Cornelius Lanczos'', American Mathematical Society In 1924 he discovered an exact solution of the Einstein field equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signal Processing and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. They further state that the digital re ...
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, digital storage efficiency, correcting distorted signals, subjective video quality and to also detect or pinpoint components of interest in a measured signal. History According to Alan V. Oppenheim Alan Victor Oppenheim''Alan Victor Oppenheim'' was elected in 1987 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jaak Peetre
Jaak Peetre (29 July 1935, Tallinn – 1 April 2019, Lund) was an Estonian-born Swedish mathematician.Eesti teaduse biograafiline leksikon (Biographical Lexicon of Estonian Science), 3. köide (volume 3) He is known for the Peetre theorem and Peetre's inequality. Biography Jaak Peetre's father was Arthur Peetre (1907–1989), an Estonian jurist, historian, archivist, and from 1941 to 1942 mayor of Pärnu. Jaak Peetre went to Sweden with his parents and brother in 1944. At Lund University he graduated with BSc in 1956 and PhD in 1959. His thesis advisor was Åke Pleijel. At Lund University, Peetre was an assistant professor from 1956 to 1959, an associate professor from 1960 to 1963, and full professor from 1963 to 1988. He was briefly in 1988 a visiting professor at the University of Madrid and was from 1988 to 1992 a visiting professor at Stockholm University. At Lund University he was a lecturer from 1993 to 1997, an assistant professor from 1997 to 2000, and professor emeritu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Fourier
Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and harmonic analysis, and their applications to problems of heat transfer and vibrations. The Fourier transform and Thermal conduction#Fourier.27s law, Fourier's law of conduction are also named in his honour. Fourier is also generally credited with the discovery of the greenhouse effect. Biography Fourier was born at Auxerre (now in the Yonne département of France), the son of a tailor. He was orphaned at the age of nine. Fourier was recommended to the Bishop of Auxerre and, through this introduction, he was educated by the Benedictine Order of the Convent of St. Mark. The commissions in the scientific corps of the army were reserved for those of good birth, and being thus ineligible, he accepted a military lectureship on mathematics. He took ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called ''analysis''. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. The term ''Fourier transform'' refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude ( absolute value) of the complex value represents the amplitude of a constituent complex sinusoid wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Variation
In mathematical analysis, a function of bounded variation, also known as ' function, is a real number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single Variable (mathematics), variable, being of bounded variation means that the distance along the Direction (geometry, geography), direction of the y-axis, -axis, neglecting the contribution of motion along x-axis, -axis, traveled by a point (mathematics), point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a Glossary of differential geometry and topology#H, hypersurface in this case), but can be every Intersection (set theory), intersection of the graph itself with a hyperplan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sufficient Condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of is guaranteed by the truth of (equivalently, it is impossible to have without ). Similarly, is sufficient for , because being true always implies that is true, but not being true does not always imply that is not true. In general, a necessary condition is one that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition. The assertion that a statement is a "necessary ''and'' sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false. In ordinary English (also natural language) "necessary" and "sufficient" indicate relations be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''. Biography Jordan was born in Lyon and educated at the École polytechnique. He was an engineer by profession; later in life he taught at the École polytechnique and the Collège de France, where he had a reputation for eccentric choices of notation. He is remembered now by name in a number of results: * The Jordan curve theorem, a topological result required in complex analysis * The Jordan normal form and the Jordan matrix in linear algebra * In mathematical analysis, Jordan measure (or ''Jordan content'') is an area measure that predates measure theory * In group theory, the Jordan–Hölder theorem on composition series is a basic result. * Jordan's theorem on finite linear groups Jordan's work did much to bring Galois theory into the mainstream. He also investigated t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotone Function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus and analysis In calculus, a function f defined on a subset of the real numbers with real values is called ''monotonic'' if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is called ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'') if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'') if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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