mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910....

where the star denotes complex conjugation. The Plancherel theorem is a special case of Parseval's theorem and states: ∑ n = 0 N − 1 | x n | 2 = 1 N ∑ k...

space operators. The Peter–Weyl theorem holds, and a version of the Fourier inversion formula (Plancherel's theorem) follows: if f ∈ L2(G), then f (...

analysis, mathematical physics and algebra, and is known for the Plancherel theorem in harmonic analysis. He was an Invited Speaker of the ICM in 1924...

differentiable. ATS theorem Carleson's theorem Dirichlet kernel Discrete Fourier transform Fast Fourier transform Fejér's theorem Fourier analysis Fourier...

}|F_{\nu }(k)|^{2}\,k\,\mathrm {d} k,} is a special case of the Plancherel theorem. These theorems can be proven using the orthogonality property. The Hankel...

In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its...

the most general form of this property is more properly called the Plancherel theorem. Suppose that A ( x ) {\displaystyle A(x)} and B ( x ) {\displaystyle...

{R} } with α < c < β {\displaystyle \alpha <c<\beta } . Then the Plancherel theorem holds: ∫ − ∞ ∞ e − 2 c t | f ( t ) | 2 d t = 1 2 π ∫ − ∞ ∞ | F ( c...

square-integrable over the interval ( 0 , ∞ ) {\displaystyle (0,\infty )} , then Plancherel's theorem holds: ∫ 0 ∞ | f ( x ) | 2 x 2 c − 1 d x = 1 2 π ∫ − ∞ ∞ | f ~ (...

π ] . {\displaystyle L^{2}[-\pi ,\pi ].} A similar result is the Plancherel theorem, which asserts that the integral of the square of the Fourier transform...

Lauricella's theorem (functional analysis) Paley–Wiener theorem (Fourier transforms) Parseval's theorem (Fourier analysis) Plancherel theorem (Fourier analysis)...

of the Plancherel theorem for unimodular locally compact groups due to Irving Segal and Forrest Stinespring and an abstract Plancherel theorem for spherical...

goal is to provide a general form of the Fourier transform and the Plancherel theorem. This is done by constructing a measure on the unitary dual and an...

is currently known ("satisfactory" means at least as strong as the Plancherel theorem). However, many specific cases have been analyzed, for example, SLn...

as a rather direct corollary of the Plancherel theorem, found in 1910, in combination with the Riesz-Thorin theorem, originally discovered by Marcel Riesz...

d\omega .} It then follows by the Cauchy–Schwarz inequality and Plancherel's theorem that, for all x {\displaystyle x} , | f ( x ) | ≤ 1 2 π 2 a ∫ − a...

Fourier transform and a Plancherel theorem for unimodular separable locally compact groups of type I and a decomposition theorem for arbitrary representations...

(p-b)}\psi (y)\ dy.} This theorem has the immediate implication that the Fourier transform is unitary, also known as the Plancherel theorem. Moreover, ( α h )...

Laplacian operator on 2-dimensional hyperbolic space. More generally, the Plancherel theorem for SL(2,R) of Harish Chandra and Gelfand–Naimark can be deduced from...

first developed by Satake and Ian G. Macdonald. The analogues of the Plancherel theorem and Fourier inversion formula in this setting generalise the eigenfunction...

\mathbb {C} )} principal series and the complementary series. Finally, the Plancherel formula for SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} is...

of ϵ {\displaystyle \epsilon } , can be derived using Plancherel theorem or Parseval's theorem for the Fourier transform. If we substitute in the expression...

in the upper half-plane is a holomorphic function. Moreover, by Plancherel's theorem, one has ∫ − ∞ ∞ | f ( ξ + i η ) | 2 d ξ ≤ ∫ 0 ∞ | F ( x ) | 2 d...

who showed that they are exactly the representations needed for the Plancherel theorem. They were classified by Knapp and Zuckerman, and used by Langlands...

transformation is also geometrical, in a sense made precise by the Plancherel theorem, that asserts that it is an isometry of one Hilbert space (the "time...

{\displaystyle \operatorname {E} [g(X)]} also follows directly from the Plancherel theorem. The expectation of a random variable plays an important role in a...

map gives a geometric interpretation of the Fourier transform, the Plancherel theorem and the Poisson summation formula. The image of Gaussian functions...

that the Fourier transform maps L1(Rd) boundedly into L∞(Rd), and Plancherel's theorem shows that the Fourier transform maps L2(Rd) boundedly into itself...

proved the principal theorems for this transform, the inversion formula, the Plancherel theorem and the analog of the Paley–Wiener theorem. Sigurdur Helgason...