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Mellin Inversion Theorem
In mathematics, the Mellin inversion formula (named after Hjalmar Mellin) tells us conditions under which the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function. Method If \varphi(s) is analytic in the strip a < \Re(s) < b, and if it tends to zero uniformly as for any real value ''c'' between ''a'' and ''b'', with its integral along such a line converging absolutely, then if : we have that : Conversely, suppose is piecewise continuous on the , taking a value halfway between the limit values at any jump discontinuities, and suppose the integral : |
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] |
Hjalmar Mellin
Robert Hjalmar Mellin (19 June 1854 – 5 April 1933) was a Finnish mathematician and function theorist. Biography Mellin was born on June 19, 1854 to priest and a former teacher Gustaf Robert Mellin (1826-1880) and Sofia Augusta Thérmen (1821-1888) in Liminka, Northern Ostrobothnia, Finland. He was the oldest among his four siblings, He worked as a translator of his father's religious and literary works after his father's expulsion from the priesthood in 1886 because of his heavy drinking, although he was suspended in 1864 because of the same reason. Hjarmar's mother Sofia was the sister of the former Councilor of State, Karl Otto Themén (1818-1893). Mellin studied at the University of Helsinki and later in Berlin under Karl Weierstrass. He is chiefly remembered as the developer of the integral transform known as the ''Mellin transform''. He studied related gamma functions, hypergeometric functions, Dirichlet series and the Riemann ζ function. He was appointed professor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mellin Transform
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions. The Mellin transform of a complex-valued function defined on \mathbf R^_+= (0,\infty) is the function \mathcal M f of complex variable s given (where it exists, see Fundamental strip below) by \mathcal\left\(s) = \varphi(s)=\int_0^\infty x^ f(x) \, dx = \int_f(x) x^s \frac. Notice that dx/x is a Haar measure on the multiplicative group \mathbf R^_+ and x\mapsto x^s is a (in general non-unitary) multiplicative character. The inverse transform is \mathcal^\left\(x) = f(x)=\frac \int_^ x^ \varphi(s)\, ds. The notation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-sided Laplace Transform
In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment-generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin transform, the Z-transform and the ordinary or one-sided Laplace transform. If ''f''(''t'') is a real- or complex-valued function of the real variable ''t'' defined for all real numbers, then the two-sided Laplace transform is defined by the integral :\mathcal\(s) = F(s) = \int_^\infty e^ f(t)\, dt. The integral is most commonly understood as an improper integral, which converges if and only if both integrals :\int_0^\infty e^ f(t) \, dt,\quad \int_^0 e^ f(t)\, dt exist. There seems to be no generally accepted notation for the two-sided transform; the \mathcal used here recalls "bilateral". The two-sided transform used by some authors is :\mathcal\(s) = s\mathcal\(s) = sF(s) = s \int_^\infty e^ f(t)\, dt. In pure mathematics the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Real Numbers
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used for either of these, the notation \R_ or \R^ for \left\ and \R_^ or \R^_ for \left\ has also been widely employed, is aligned with the practice in algebra of denoting the exclusion of the zero element with a star, and should be understandable to most practicing mathematicians. In a complex plane, \R_ is identified with the positive real axis, and is usually drawn as a horizontal ray. This ray is used as reference in the polar form of a complex number. The real positive axis corresponds to complex numbers z = , z, \mathrm^, with argument \varphi = 0. Properties The set \R_ is closed under addition, multiplication, and division. It inherits a topology from the real line and, thus, has the structure of a multiplicative topological group or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Inversion Theorem
In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely. The theorem says that if we have a function f:\R \to \Complex satisfying certain conditions, and we use the convention for the Fourier transform that :(\mathcalf)(\xi):=\int_ e^ \, f(y)\,dy, then :f(x)=\int_ e^ \, (\mathcalf)(\xi)\,d\xi. In other words, the theorem says that :f(x)=\iint_ e^ \, f(y)\,dy\,d\xi. This last equation is called the Fourier integral theorem. Another way to state the theorem is that if R is the flip operator i.e. (Rf)(x) := f(-x), then :\mathcal^=\mathcalR=R\mathcal. The theorem holds if both f and its Fourier transform are absolutely integrable (in the Lebesgue sense) and f is continuous at the point x. However, even under more gener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for treating discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges. They are applied extensively, especially in physics and engineering. Important motivations have been the technical requirements of theories of partial differential equations and group representations. A common feature of some of the approaches is that they build on operator aspects of everyday, numerical functions. The early history is connected with some ideas on operational calculus, and some contemporary developments are closely related to Mikio Sato's algebraic analysis. Some early history In the mathematics of the nineteenth century, aspects of generalized function theory appeared, for example in the def ... [...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] |
Lp Space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Preliminaries The -norm in finite dimensions The Euclidean length of a vector x = (x_1, x_2, \dots, x_n) in the n-dimensional real vector space \Reals^n is given by the Euclidean norm: \, x\, _2 = \left(^2 + ^2 + \dotsb + ^2\right)^. The Euclidean distance between two points x and y is the length \, x - y\, _2 of the straight line b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nachbin's Theorem
In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is a result used to establish bounds on the growth rates for analytic functions. In particular, Nachbin's theorem may be used to give the domain of convergence of the generalized Borel transform, also called Nachbin summation. This article provides a brief review of growth rates, including the idea of a function of exponential type. Classification of growth rates based on type help provide a finer tool than big O or Landau notation, since a number of theorems about the analytic structure of the bounded function and its integral transforms can be stated. Exponential type A function f(z) defined on the complex plane is said to be of exponential type if there exist constants M and \alpha such that :, f(re^), \le Me^ in the limit of r\to\infty. Here, the complex variable z was written as z=re^ to emphasize that the limit must hold in all directions \theta. Letting \alpha stand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science (journal)
''Theoretical Computer Science'' (''TCS'') is a computer science journal published by Elsevier, started in 1975 and covering theoretical computer science. The journal publishes 52 issues a year. It is abstracted and indexed by Scopus and the Science Citation Index. According to the Journal Citation Reports, its 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... is 0.827. References Computer science journals Elsevier academic journals Academic journals established in 1975 {{comp-sci-theory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
