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Harmonic Analysis
HARMONIC ANALYSIS is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves , and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis ). In the past two centuries, it has become a vast subject with applications in areas as diverse as number theory , representation theory , signal processing , quantum mechanics , tidal analysis and neuroscience . The term "harmonics " originated as the ancient Greek word, "harmonikos," meaning "skilled in music." In physical eigenvalue problems it began to mean waves whose frequencies are integer multiples of one another, as are the frequencies of the harmonics of music notes , but the term has been generalized beyond its original meaning
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Harmony
In music , HARMONY considers the process by which the composition of individual sounds, or superpositions of sounds, is analysed by hearing. Usually, this means simultaneously occurring frequencies , pitches (tones , notes ), or chords . The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them. Harmony
Harmony
is often said to refer to the "vertical" aspect of music, as distinguished from melodic line , or the "horizontal" aspect. Counterpoint , which refers to the relationship between melodic lines, and polyphony , which refers to the simultaneous sounding of separate independent voices, are thus sometimes distinguished from harmony. In popular and jazz harmony , chords are named by their root plus various terms and characters indicating their qualities. In many types of music, notably baroque, romantic, modern, and jazz, chords are often augmented with "tensions"
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Functional Analysis
FUNCTIONAL ANALYSIS is a branch of mathematical analysis , the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product , norm , topology , etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous , unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations . The usage of the word functional goes back to the calculus of variations , implying a function whose argument is a function and the name was first used in Hadamard 's 1910 book on that subject
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Bessel Function
BESSEL FUNCTIONS, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel , are the canonical solutions y ( x ) {displaystyle y(x)} of Bessel's differential equation x 2 d 2 y d x 2 + x d y d x + ( x 2 2 ) y = 0 {displaystyle x^{2}{frac {d^{2}y}{dx^{2}}}+x{frac {dy}{dx}}+(x^{2}-alpha ^{2})y=0} for an arbitrary complex number α, the order of the Bessel function. Although α and −α produce the same differential equation for real α, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of α. The most important cases are for α an integer or half-integer . Bessel functions for integer α are also known as CYLINDER FUNCTIONS or the CYLINDRICAL HARMONICS because they appear in the solution to Laplace\'s equation in cylindrical coordinates
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Convergence Of Fourier Series
In mathematics , the question of whether the FOURIER SERIES OF A PERIODIC FUNCTION CONVERGES to the given function is researched by a field known as CLASSICAL HARMONIC ANALYSIS, a branch of pure mathematics . Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur. Determination of convergence requires the comprehension of pointwise convergence , uniform convergence , absolute convergence , Lp spaces , summability methods and the Cesàro mean . CONTENTS * 1 Preliminaries * 2 Magnitude of Fourier coefficients * 3 Pointwise convergence * 4 Uniform convergence * 5 Absolute convergence * 6 Norm convergence * 7 Convergence almost everywhere * 8 Summability * 9 Order of growth * 10 Multiple dimensions * 11 Notes * 12 References * 12.1 Textbooks * 12.2 Articles referred to in the text PRELIMINARIESConsider ƒ an integrable function on the interval
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Compact Support
In mathematics , the SUPPORT of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero. If the domain of f is a topological space, the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis . CONTENTS * 1 Formulation * 2 Closed support * 3 Compact support * 4 Essential support * 5 Generalization * 6 In probability and measure theory * 7 Support of a distribution * 8 Singular support * 9 Family of supports * 10 See also * 11 References FORMULATIONSuppose that f : X → R is a real-valued function whose domain is an arbitrary set X. The SET-THEORETIC SUPPORT of f, written SUPP(F), is the set of points in X where f is non-zero supp ( f ) = { x X f ( x ) 0 }
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Harmonic Series (music)
A HARMONIC SERIES is the sequence of sounds —pure tones, represented by sinusoidal waves—in which the frequency of each sound is an integer multiple of the fundamental, the lowest frequency. Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous modes simultaneously. At the frequencies of each vibrating mode, waves travel in both directions along the string or air column, reinforcing and canceling each other to form standing waves . Interaction with the surrounding air causes audible sound waves , which travel away from the instrument. Because of the typical spacing of the resonances , these frequencies are mostly limited to integer multiples, or harmonics , of the lowest frequency, and such multiples form the harmonic series (see harmonic series (mathematics) )
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Paley–Wiener Theorem
In mathematics , a PALEY–WIENER THEOREM is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform . The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (1894–1964). The original theorems did not use the language of distributions , and instead applied to square-integrable functions . The first such theorem using distributions was due to Laurent Schwartz . CONTENTS * 1 Holomorphic Fourier transforms * 2 Schwartz\'s Paley–Wiener theorem * 3 Notes * 4 References HOLOMORPHIC FOURIER TRANSFORMSThe classical Paley–Wiener theorems make use of the holomorphic Fourier transform on classes of square-integrable functions supported on the real line
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Abelian Group
In abstract algebra , an ABELIAN GROUP, also called a COMMUTATIVE GROUP, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of commutativity . Abelian groups generalize the arithmetic of addition of integers . They are named after Niels Henrik Abel . The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, from which many other basic concepts, such as modules and vector spaces are developed. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood. On the other hand, the theory of infinite abelian groups is an area of current research
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Hearing The Shape Of A Drum
To HEAR THE SHAPE OF A DRUM is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of overtones , via the use of mathematical theory. "Can One Hear the Shape of a Drum?" was the title of an article by Mark Kac in the American Mathematical Monthly in 1966, but the phrasing of the title is due to Lipman Bers . These questions can be traced back all the way to Hermann Weyl . For the 1966 paper that made the question famous, Kac was given the Lester R. Ford Award in 1967 and the Chauvenet Prize in 1968. The frequencies at which a drumhead can vibrate depends on its shape. The Helmholtz equation calculates the frequencies if the shape is known. These frequencies are the eigenvalues of the Laplacian in the space. A central question is whether the shape can be predicted if the frequencies are known. No other shape than a square vibrates at the same frequencies as a square
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Eigenvector
In linear algebra , an EIGENVECTOR or CHARACTERISTIC VECTOR of a linear transformation is a non-zero vector whose direction does not change when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and V is a vector in V that is not the zero vector , then V is an eigenvector of T if T(V) is a scalar multiple of V. This condition can be written as the equation T ( v ) = v , {displaystyle T(mathbf {v} )=lambda mathbf {v} ,} where λ is a scalar in the field F, known as the EIGENVALUE, CHARACTERISTIC VALUE, or CHARACTERISTIC ROOT associated with the eigenvector V
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Laplacian
* v * t * e In mathematics , the LAPLACE OPERATOR or LAPLACIAN is a differential operator given by the divergence of the gradient of a function on Euclidean space . It is usually denoted by the symbols ∇·∇, ∇2, or Δ. The Laplacian Δf(p) of a function f at a point p, up to a constant depending on the dimension, is the rate at which the average value of f over spheres centered at p deviates from f(p) as the radius of the sphere grows. In a Cartesian coordinate system , the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable . In other coordinate systems such as cylindrical and spherical coordinates , the Laplacian also has a useful form
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Domain (mathematical Analysis)
In mathematical analysis , a DOMAIN is any connected open subset of a finite-dimensional vector space. This is a different concept than the domain of a function , though it is often used for that purpose, for example in partial differential equations and Sobolev spaces . Various degrees of smoothness of the boundary of the domain are required for various properties of functions defined on the domain to hold, such as integral theorems (Green\'s theorem , Stokes theorem ), properties of Sobolev spaces , and to define measures on the boundary and spaces of traces (generalized functions defined on the boundary). Commonly considered types of domains are domains with continuous boundary, Lipschitz boundary , C1 boundary, and so forth. A BOUNDED DOMAIN is a domain which is a bounded set , while an EXTERIOR or EXTERNAL DOMAIN is the interior of the complement of a bounded domain
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Dimension (mathematics And Physics)
In physics and mathematics , the DIMENSION of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube , a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces. In classical mechanics , space and time are different categories and refer to absolute space and time . That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism
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Special Linear Group
In mathematics , the SPECIAL LINEAR GROUP of degree n over a field F is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion . This is the normal subgroup of the general linear group given by the kernel of the determinant det GL ( n , F ) F . {displaystyle det colon operatorname {GL} (n,F)to F^{times }.} where we write F× for the multiplicative group of F (that is, F excluding 0). These elements are "special" in that they fall on a subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries)
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Lie Group
In mathematics , a LIE GROUP (pronounced /ˈliː/ "Lee") is a group that is also a differentiable manifold , with the property that the group operations are compatible with the smooth structure . Lie groups are named after Norwegian mathematician Sophus Lie , who laid the foundations of the theory of continuous transformation groups . Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures , which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics . They provide a natural framework for analysing the continuous symmetries of differential equations (differential Galois theory
Galois theory
), in much the same way as permutation groups are used in Galois theory
Galois theory
for analysing the discrete symmetries of algebraic equations
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