Microlocal analysis
Encyclopedia
In mathematical analysis
, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transform
s related to the study of variable-coefficients-linear and nonlinear partial differential equation
s. This includes generalized function
s, pseudo-differential operator
s, wave front set
s, Fourier integral operator
s, oscillatory integral operators, and paradifferential operators.
The term microlocal implies localisation not only with respect to location in the space, but also with respect to cotangent space
directions at a given point. This gains in importance on manifold
s of dimension
greater than one.
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
s related to the study of variable-coefficients-linear and nonlinear partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s. This includes generalized function
Generalized function
In mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognized theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing physical phenomena such as point charges...
s, pseudo-differential operator
Pseudo-differential operator
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory....
s, wave front set
Wave front set
In mathematical analysis, more precisely in microlocal analysis, the wave front WF characterizes the singularities of a generalized function f, not only in space, but also with respect to its Fourier transform at each point...
s, Fourier integral operator
Fourier integral operator
In mathematical analysis, Fourier integral operators have become an important tool in the theory of partial differential equations. The class of Fourier integral operators contains differential operators as well as classical integral operators as special cases....
s, oscillatory integral operators, and paradifferential operators.
The term microlocal implies localisation not only with respect to location in the space, but also with respect to cotangent space
Cotangent space
In differential geometry, one can attach to every point x of a smooth manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions...
directions at a given point. This gains in importance on manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s of dimension
Dimension
In physics and mathematics, the dimension of a 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...
greater than one.