Fourier integral operators have become an important tool in the theory of partial differential equations. The class of Fourier integral operators contains...

The Fourier operator is the kernel of the Fredholm integral of the first kind that defines the continuous Fourier transform, and is a two-dimensional...

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent...

equation is called the Fourier integral theorem. Another way to state the theorem is that if R {\displaystyle R} is the flip operator i.e. ( R f ) ( x ) :=...

{\displaystyle Tf} . An integral transform is a particular kind of mathematical operator. There are numerous useful integral transforms. Each is specified...

family of commuting operators). They are also special cases of pseudo-differential operators, and more generally Fourier integral operators. There are natural...

differential algebras and differential rings. Fourier transform Fourier integral operator Oscillatory integral operator Sato's fundamental theorem Operational...

Plancherel theorem Peter–Weyl theorem Fourier integral operator Oscillatory integral operator Laplace operator Laplace equation Dirichlet problem Unit...

partial differential equations. Broadly speaking a singular integral is an integral operator T ( f ) ( x ) = ∫ K ( x , y ) f ( y ) d y , {\displaystyle...

as the inverse Fourier transform of the pointwise product of two Fourier transforms. One of the earliest uses of the convolution integral appeared in D'Alembert's...

particularly desirable properties under the Fourier transform that make it ideal for the study of pseudodifferential operators. Using these methods on a compact...

built from them are called differential operators, integral operators or integro-differential operators. Operator is also used for denoting the symbol of...

represent approximate solution operators for many differential equations as oscillatory integrals. An oscillatory integral f ( x ) {\displaystyle f(x)}...

easier to achieve than in the operator formalism of canonical quantization. Unlike previous methods, the path integral allows one to easily change coordinates...

mathematics, in the field of harmonic analysis, an oscillatory integral operator is an integral operator of the form T λ u ( x ) = ∫ R n e i λ S ( x , y ) a (...

analysis, in particular, the application of pseudo differential and Fourier integral operators to linear partial differential equations".[3] In 2012 he was selected...

functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function...

the context of Fourier analysis—to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in measure...

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of...

pseudo-differential operators, wave front sets, Fourier integral operators, oscillatory integral operators, and paradifferential operators. The term microlocal...

transform Mellin transform, another closely related integral transform Laplace transform: the Fourier transform may be considered a special case of the...

fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform...

Phase Integrals (Russian: Асимптотика многомерных фазовых интегралов). Her other works on mathematical physics include: On Fourier Integral Operators, Mathematics...

{\displaystyle L^{2}} . The Fourier transform "diagonalizes" the momentum operator; that is, it converts it into the operator of multiplication by p {\displaystyle...

vector-valued functions, and operator spaces. Examples of such extensions include vector-valued Laplace transforms and abstract Fourier transforms. Let ( X ,...

simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing...

Titchmarsh wrote the influential Introduction to the theory of the Fourier integral (1937). The current widespread use of the transform (mainly in engineering)...

Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The...

function in any neighbourhood of any point. An example of a nonlocal operator is the Fourier transform. Let X {\displaystyle X} be a topological space, Y {\displaystyle...

transform Wavelet transform (integral) Weierstrass transform Binomial transform Discrete Fourier transform, DFT Fast Fourier transform, a popular implementation...