In
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 (m ...
, microlocal analysis comprises techniques developed from the 1950s onwards based on
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
s related to the study of variable-coefficients-linear and nonlinear
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s. This includes
generalized function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
s,
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, e.g. in m ...
s,
wave front set
In mathematical analysis, more precisely in microlocal analysis, the wave front (set) WF(''f'') characterizes the singularities of a generalized function ''f'', not only in space, but also with respect to its Fourier transform at each point. The t ...
s,
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 ...
s,
oscillatory integral operator
In mathematics, in the field of harmonic analysis, an oscillatory integral operator is an integral operator of the form
:T_\lambda u(x)=\int_e^ a(x, y) u(y)\,dy, \qquad x\in\R^m, \quad y\in\R^n,
where the function ''S''(''x'',''y'') is called the ...
s, and
paradifferential operators.
The term ''microlocal'' implies localisation not only with respect to location in the space, but also with respect to
cotangent space
In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
directions at a given point. This gains in importance on
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s of
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
greater than one.
See also
*
Algebraic analysis
Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of functions such as hyperfunctions and microfunct ...
*
Microfunction
Algebraic analysis is an area of mathematics that deals with systems of Partial differential equation, linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of Function (mathematic ...
External links
lecture notes by Richard Melrose
Fourier analysis
Generalized functions
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