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Plancherel Theorem
In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum. That is, if f(x) is a function on the real line, and \widehat(\xi) is its frequency spectrum, then A more precise formulation is that if a function is in both Lp spaces L^1(\mathbb) and L^2(\mathbb), then its Fourier transform is in L^2(\mathbb), and the Fourier transform map is an isometry with respect to the ''L''2 norm. This implies that the Fourier transform map restricted to L^1(\mathbb) \cap L^2(\mathbb) has a unique extension to a linear isometric map L^2(\mathbb) \mapsto L^2(\mathbb), sometimes called the Plancherel transform. This isometry is actually a unitary map. In effect, this makes it possible to speak of Fourier transforms of quadratically integrable functions. ... [...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] |
Non-commutative Harmonic Analysis
In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative. Since locally compact abelian groups have a well-understood theory, Pontryagin duality, which includes the basic structures of Fourier series and Fourier transforms, the major business of non-commutative harmonic analysis is usually taken to be the extension of the theory to all groups ''G'' that are locally compact. The case of compact groups is understood, qualitatively and after the Peter–Weyl theorem from the 1920s, as being generally analogous to that of finite groups and their character theory. The main task is therefore the case of ''G'' that is locally compact, not compact and not commutative. The interesting examples include many Lie groups, and also algebraic groups over p-adic fields. These examples are of interest and frequently applied in mathematical physics, and contemporary number theory, particularly auto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Functional Analysis
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rendiconti Del Circolo Matematico Di Palermo
The Circolo Matematico di Palermo (Mathematical Circle of Palermo) is an Italian mathematical society, founded in Palermo by Sicilian geometer Giovanni B. Guccia in 1884.The Mathematical Circle of Palermo . Retrieved 2011-06-19. It began accepting foreign members in 1888, and by the time of Guccia's death in 1914 it had become the foremost international mathematical society, with approximately one thousand members. However, subsequently to that time it declined in influence. Publications ''Rendiconti del Circolo Matemat ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plancherel Theorem For Spherical Functions
In mathematics, the Plancherel theorem for spherical functions is an important result in the group representation, representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analysis of the Plancherel formula and Fourier inversion formula in the representation theory of the group of real numbers in classical harmonic analysis and has a similarly close interconnection with the theory of differential equations. It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the Spectral theory, eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space ''X''; it also gives the direct integral, direct integral decomposition into irreducible representations of the regular representation on . In the case of hyperbolic space, these expansions were kn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two vectors in the space is a Scalar (mathematics), scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite Dimension (vector space), dimension are widely used in functional analysis. Inner product spaces over the Field (mathematics), field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polarization Identity
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm. The polarization identity shows that a norm can arise from at most one inner product; however, there exist norms that do not arise from any inner product. The norm associated with any inner product space satisfies the parallelogram law: \, x+y\, ^2 + \, x-y\, ^2 = 2\, x\, ^2 + 2\, y\, ^2. In fact, as observed by John von Neumann, the parallelogram law characterizes those norms that arise from inner products. Given a normed space (H, \, \cdot\, ), the parallelogram law holds for \, \cdot\, if and only if there exists an inner product \langle \cdot, \cdot \rangle on H such that \, x\, ^2 = \langle x,\ x\rangle for all x \in H, in which case ... [...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] |
Parseval's Theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after John William Strutt, Lord Rayleigh. Although the term "Parseval's theorem" is often used to describe the unitarity of ''any'' Fourier transform, especially in physics, the most general form of this property is more properly called the Plancherel theorem. Statement of Parseval's theorem Suppose that A(x) and B(x) are two complex-valued functions on \mathbb of period 2 \pi that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series :A(x)=\sum_^\infty a_ne^ and :B(x)=\sum_^\infty b_ne^ respective ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Transformation
In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation. Formal definition More precisely, a unitary transformation is an isomorphism between two inner product spaces (such as Hilbert spaces). In other words, a ''unitary transformation'' is a bijective function U : H \to H_2\, between two inner product spaces, H and H_2, such that \langle Ux, Uy \rangle_ = \langle x, y \rangle_ \quad \text x, y \in H. Properties A unitary transformation is an isometry, as one can see by setting x=y in this formula. Unitary operator In the case when H_1 and H_2 are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator. Antiunitary transformation A closely related notion is that of antiunitary transformation, which is a bijective function :U:H_1\to H_2\, between two co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Compact Abelian Group
In several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped with the discrete topology), or the real numbers or the circle (both with their usual topology) are locally compact abelian groups. Definition and examples A topological group is called ''locally compact'' if the underlying topological space is locally compact and Hausdorff; the topological group is called ''abelian'' if the underlying group is abelian. Examples of locally compact abelian groups include: * \R^n for ''n'' a positive integer, with vector addition as group operation. * The positive real numbers \R^+ with multiplication as operation. This group is isomorphic to (\R, +) by the exponential map. * Any finite abelian group, with the discrete topology. By the structure theorem for finite abelian groups, all such groups are produc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marc-Antoine Parseval
Marc-Antoine Parseval des Chênes (27 April 1755 – 16 August 1836) was a French mathematician, most famous for what is now known as Parseval's theorem, which presaged the unitarity of the Fourier transform. He was born in Rosières-aux-Salines, France, into an aristocratic French family, and married Ursule Guerillot in 1795, but divorced her soon after. A monarchist opposed to the French revolution, imprisoned in 1792, Parseval later fled the country for publishing poetry critical of the government of Napoleon. Later, he was nominated to the French Academy of Sciences five times, from 1796 to 1828, but was never elected. His only mathematical publications were apparently five papers, published in 1806 as ''Mémoires présentés à l'Institut des Sciences, Lettres et Arts, par divers savants, et lus dans ses assemblées. Sciences mathématiques et physiques. (Savants étrangers.)'' This combined the following earlier monographs: # "Mémoire sur la résolution des équation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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