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Wiener's Tauberian Theorem
In mathematical analysis, Wiener's Tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in L^1 or L^2 can be approximated by linear combinations of translations of a given function. Informally, if the Fourier transform of a function f vanishes on a certain set Z, the Fourier transform of any linear combination of translations of f also vanishes on Z. Therefore, the linear combinations of translations of f cannot approximate a function whose Fourier transform does not vanish on Z. Wiener's theorems make this precise, stating that linear combinations of translations of f are dense if and only if the zero set of the Fourier transform of f is empty (in the case of L^1) or of Lebesgue measure zero (in the case of L^2). Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the L^1 group ring L^1(\mathbb) of the grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...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 produ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals contained between ''I'' and ''R''. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. The set of maximal ideals of a unital commutative ring ''R'', typically equipped with the Zariski topology, is known as the maximal spectrum of ''R'' and is variously denoted m-Spec ''R'', Specm ''R'', MaxSpec ''R'', or Spm ''R''. In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one-sided maximal ideal ''A'' is not necessarily ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy \, x \, y\, \ \leq \, x\, \, \, y\, \quad \text x, y \in A. This ensures that the multiplication operation is continuous with respect to the metric topology. A Banach algebra is called ''unital'' if it has an identity element for the multiplication whose norm is 1, and ''commutative'' if its multiplication is commutative. Any Banach algebra A (whether it is unital or not) can be embedded isometrically into a unital Banach algebra A_e so as to form a closed ideal of A_e. Often one assumes ''a priori'' that the algebra under consideration is unital because one can develop much of the theory by considering A_e and then a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener Algebra
In mathematics, the Wiener algebra, named after Norbert Wiener and usually denoted by , is the space of absolutely convergent Fourier series. Here denotes the circle group. Banach algebra structure The norm of a function is given by :\, f\, =\sum_^\infty , \hat(n), ,\, where :\hat(n)= \frac\int_^\pi f(t)e^ \, dt is the th Fourier coefficient of . The Wiener algebra is closed under pointwise multiplication of functions. Indeed, : \begin f(t)g(t) & = \sum_ \hat(m)e^\,\cdot\,\sum_ \hat(n)e^ \\ & = \sum_ \hat(m)\hat(n)e^ \\ & = \sum_ \left\e^ ,\qquad f,g\in A(\mathbb); \end therefore : \, f g\, = \sum_ \left, \sum_ \hat(n-m)\hat(m) \ \leq \sum_ , \hat(m), \sum_n , \hat(n), = \, f\, \, \, g\, .\, Thus the Wiener algebra is a commutative unitary Banach algebra. Also, is isomorphic to the Banach algebra , with the isomorphism given by the Fourier transform. Properties The sum of an absolutely convergent Fourier series is continuous, so :A(\mathbb)\subset C(\mathbb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always Convergent series, converge. Well-behaved functions, for example Smoothness, smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Convergence
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said to converge absolutely if \textstyle\sum_^\infty \left, a_n\ = L for some real number \textstyle L. Similarly, an improper integral of a function, \textstyle\int_0^\infty f(x)\,dx, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if \textstyle\int_0^\infty , f(x), dx = L. A convergent series that is not absolutely convergent is called conditionally convergent. Absolute convergence is important for the study of infinite series, because its definition guarantees that a series will have some "nice" behaviors of finite sums that not all convergent series possess. For instance, rearrangements do not change the value of the sum, which is not necessarily true for conditionally converge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as because it is a one-dimensional unit -sphere. If is a point on the unit circle's circumference, then and are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, and satisfy the equation x^2 + y^2 = 1. Since for all , and since the reflection of any point on the unit circle about the - or -axis is also on the unit circle, the above equation holds for all points on the unit circle, not only those in the first quadrant. The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk. One may also use other notions of "dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The term ''convolution'' refers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see #Properties, commutativity). Graphically, it expresses how the 'shape' of one function is modified by the other. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution f*g differs from cross-correlation f \star g only in that either f(x) or g(x) is reflected about the y-axis in convolution; thus i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tauberian Theorem
In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series convergent series, converges to some limit then its Abel sum is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/''n'')) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods. There is not yet a clear distinction between Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean. Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series summable by some method that allows it to be summable in the usu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Of A Function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is a solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6=(x-2)(x-3) has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Span
In mathematics, the linear span (also called the linear hull or just span) of a set S of elements of a vector space V is the smallest linear subspace of V that contains S. It is the set of all finite linear combinations of the elements of , and the intersection of all linear subspaces that contain S. It is often denoted pp. 29-30, §§ 2.5, 2.8 or \langle S \rangle. For example, in geometry, two linearly independent vectors span a plane. To express that a vector space is a linear span of a subset , one commonly uses one of the following phrases: spans ; is a spanning set of ; is spanned or generated by ; is a generator set or a generating set of . Spans can be generalized to many mathematical structures, in which case, the smallest substructure containing S is generally called the substructure ''generated'' by S. Definition Given a vector space over a field , the span of a set of vectors (not necessarily finite) is defined to be the intersection of all subsp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
