The identity is related to the Pythagorean theorem in the more general setting of a separable Parseal space as follows. Zygmund, AntoniTrigonometric series 2nd ed. The assumption that B is total is necessary for the validity of the identity. For fornule time signalsthe theorem becomes:. Fformule similar result is the Plancherel theoremwhich asserts that the integral of the square of the Fourier transform of a function is equal to the integral of the square of the function itself.

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Gazuru A similar result is the Plancherel theoremwhich asserts that the integral of the square of the Fourier transform of a function is equal to the integral of the square of the function itself. The interpretation of this form of the theorem is that the total energy of a signal can be calculated by summing power-per-sample across time or spectral power across frequency. Thus suppose that H is an inner-product space.

Alternatively, for the discrete Fourier transform DFTthe relation becomes:. Views Read Edit View history. Geometrically, it is the Pythagorean theorem for inner-product spaces. The identity is related to the Pythagorean theorem in the more general setting of a separable Hilbert space as follows. From Wikipedia, the free encyclopedia.

When G is the cyclic group Z nagain it is self-dual and the Pontryagin—Fourier transform is what is called discrete Fourier transform in applied contexts. Titchmarsh, EThe Theory of Functions 2nd ed. By using this site, you agree to the Ee of Use and Privacy Policy. For discrete time signalsthe theorem becomes:. DeanNumerical Analysis 2nd ed. By using this site, you agree to the Terms of Use and Forule Policy. Let B be an orthonormal basis of H ; i. This is directly analogous to the Pythagorean theorem, which asserts that the sum of the squares of the components of a vector in an orthonormal basis is equal to the squared length of the vector.

Allyn and Bacon, Inc. Translated by Silverman, Richard. Then [4] [5] [6]. Zygmund, AntoniTrigonometric series 2nd ed. Informally, the identity asserts that the sum of the squares of the Fourier coefficients of a function is equal to the integral of the square of the function. Advanced Calculus 4th ed.

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## Dictionnaire de mathématiques

Gazuru A similar result is the Plancherel theoremwhich asserts that the integral of the square of the Fourier transform of a function is equal to the integral of the square of the function itself. The interpretation of this form of the theorem is that the total energy of a signal can be calculated by summing power-per-sample across time or spectral power across frequency. Thus suppose that H is an inner-product space. Alternatively, for the discrete Fourier transform DFTthe relation becomes:. Views Read Edit View history. Geometrically, it is the Pythagorean theorem for inner-product spaces.

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## Théorème de Plancherel

Views Read Edit View history. Alternatively, for the discrete Fourier transform DFTthe relation becomes:. Allyn and Bacon, Inc. When G is the cyclic foormule Z nagain it is self-dual and the Pontryagin—Fourier transform is what is called discrete Fourier transform in applied contexts. It originates from a theorem about series by Marc-Antoine Parsevalwhich was later applied to the Fourier series.