Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. The identity asserts...

^{2}t^{2}+12\pi t+6}}\\[6pt]&={\frac {\pi ^{2}}{6}}.\end{aligned}}} Use Parseval's identity (applied to the function f(x) = x) to obtain ∑ n = − ∞ ∞ | c n |...

Marc-Antoine Parseval des Chênes (27 April 1755 – 16 August 1836) was a French mathematician, most famous for what is now known as Parseval's theorem, which...

complex conjugate. Parseval's theorem is closely related to other mathematical results involving unitary transformations: Parseval's identity Plancherel's theorem...

h\rangle =\langle {\hat {f}},{\hat {h}}\rangle .} This gives us Parseval's identity: ∑ i = 1 N | f ( i ) | 2 = ‖ f ‖ 2 2 = ⟨ f , f ⟩ = ⟨ f ^ , f ^ ⟩...

Bessel's inequality is a stepping stone to the stronger result called Parseval's identity, which governs the case when Bessel's inequality is actually an equality...

logarithmic identities MacWilliams identity Matrix determinant lemma Newton's identity Parseval's identity Pfister's sixteen-square identity Sherman–Morrison...

four-square identity, related to quaternions in the same way Lagrange's identity Other Parseval's identity Pythagorean trigonometric identity acceleration...

arises from the geometric interpretation in Euclidean geometry. Parseval's identity An induction on the Pythagorean theorem yields: if x 1 , … , x n...

(that is, for an orthonormal sequence that is a basis), we have Parseval's identity, which replaces the inequality with an equality (and consequently...

triangle topics Lp space Nonhypotenuse number Parallelogram law Parseval's identity – The energy of a periodic function is the same in the time and frequency...

_{j}(v_{j}^{T}x)^{2}\\[4pt]={}&\lambda _{n}x^{T}x,\end{aligned}}} where we used Parseval's identity in the last line. Finally we obtain that x T M x x T x ≤ λ n {\displaystyle...

less common ways of calculating definite integrals; for instance, Parseval's identity can be used to transform an integral over a rectangular region into...

Exponential sum Dirichlet kernel Fejér kernel Gibbs phenomenon Parseval's identity Parseval's theorem Weyl differintegral Generalized Fourier series Orthogonal...

expansion of x , {\displaystyle x,} and the formula is usually known as Parseval's identity. If B {\displaystyle B} is an orthonormal basis of H , {\displaystyle...

{\displaystyle c=1} . A unit-norm Parseval frame is an orthonormal basis; such a frame satisfies Parseval's identity. A frame is an equiangular frame if...

the fact that the average value of y is zero means that a0 = 0. By Parseval's identity, ∫ 0 2 π y ( x ) 2 d x = ∑ n = 1 ∞ ( a n 2 + b n 2 ) {\displaystyle...

Hermitian adjoint Hilbert matrix Shift operator Symmetric matrix Parseval's identity Rayleigh quotient Reproducing kernel Hilbert space Riesz representation...

a stronger form of Bessel's inequality, and can be used to prove Parseval's identity for Fourier series. Other results are often called the Riesz–Fischer...

can be replaced by summations for discrete representation. Using Parseval's identity it is clear that the three real numbers are the second order moments...

called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It is a generalization of Parseval's theorem;...

\operatorname {im} (P)=M,\operatorname {ker} (P)=M^{\bot }.} Parseval Parseval's identity states: given an orthonormal basis S in a Hilbert space, ‖ x...

infinite-dimensional  real inner product spaces is known as Parseval's identity or Parseval's equation. Particular examples of such a representation of...

uniform distribution over { − 1 , 1 } n {\displaystyle \{-1,1\}^{n}} . Parseval's identity states that ‖ f ‖ 2 = E ⁡ [ f 2 ] = ∑ S f ^ ( S ) 2 . {\displaystyle...

f(t)=\sum _{n=-\infty }^{\infty }L_{n}(f)e^{jnt}} . and in fact we have Parseval's identity | | f | | 2 = ∑ n = − ∞ ∞ | L n ( f ) | 2 {\displaystyle ||f||^{2}=\sum...

Fourier's work. Another application is to solve the Basel problem by using Parseval's theorem. The example generalizes and one may compute ζ(2n), for any positive...

Plancherel, it is still often referred to as Parseval's formula, or Parseval's relation, or even Parseval's theorem. See Pontryagin duality for a general...

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

2 ( t ) {\displaystyle x^{2}(t)} over the time domain, as dictated by Parseval's theorem. The spectrum of a physical process x ( t ) {\displaystyle x(t)}...

{\displaystyle x_{n}} and y n {\displaystyle y_{n}} respectively then Parseval's theorem states: ∑ n = 0 N − 1 x n y n ∗ = 1 N ∑ k = 0 N − 1 X k Y k ∗...