analysis, a direct integral or Hilbert integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert...

multiplication operator and as a direct integral, since L 2 ( σ ( A ) , μ ) {\displaystyle L^{2}(\sigma (A),\mu )} is just a direct integral in which each Hilbert...

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process...

Unlike the multiplication-operator version of the spectral theorem, the direct-integral version is unique in the sense that the measure equivalence class of...

by spectral multiplicity theory and requires the use of direct integrals. Using direct integral theory, it can be shown that the abelian von Neumann algebras...

methods include: direct integration of a complex-valued function along a curve in the complex plane application of the Cauchy integral formula application...

we provide a general example of projection-valued measure based on direct integrals. Suppose (X, M, μ) is a measure space and let {Hx}x ∈ X be a μ-measurable...

scalar operators. A finite von Neumann algebra is one which is the direct integral of finite factors (meaning the von Neumann algebra has a faithful normal...

calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of...

antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function f is a differentiable function F whose...

mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear...

Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function...

In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied...

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that...

improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context...

derivatives, the equation contains integrals. A direct comparison can be seen with the mathematical form of the general integral equation above with the general...

rays, cannot penetrate Earth's atmosphere, and direct observations must be made from space. INTEGRAL is an observatory, scientists can propose for observing...

the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form...

of G is of type I, then L2(G) as a unitary representation of G is a direct integral of irreducible representations. It is parametrized therefore by the...

proved that every von Neumann algebra on a separable Hilbert space is a direct integral of factors; he did not find time to publish this result until 1949...

calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the...

whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known...

measure and integral, given by the Haar measure. Every unitary representation of a locally compact group can be described as a direct integral of irreducible...

infinitesimals", it has two major branches, differential calculus and integral calculus. The former concerns instantaneous rates of change, and the slopes...

calculus), a volume integral (∭) is an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially...

The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces...

In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration...

vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary...

continuous function f , an antiderivative or indefinite integral F can be obtained as the integral of f over an interval with a variable upper bound. Conversely...

Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was...