Parseval's theorem

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The Parseval's theorem is a statement from the functional analysis in the field of Fourier analysis . It says that the -norm of a Fourier series agrees with the -norm of its Fourier coefficients . The statement arose in 1799 from a sentence about mathematical series by Marc-Antoine Parseval , which was later extended to the Fourier series. Parseval, who actually only concentrated on real-valued functions, published his theorem without proof, since he considered its correctness to be obvious. Plancherel's theorem makes a similar statement for the Fourier transform . Often these two sentences are not kept apart, but Plancherel's sentence is named after Parseval.

Statements of Parseval's theorem

Let and two Riemann integrable complex-valued functions over with period and the Fourier series decomposition

and .

Then applies

where is the imaginary unit and denotes the complex conjugation .

There are many different special cases of the theorem. Is z. B. , one obtains

from which the unitarity of the Fourier series follows.

Furthermore, often only the Fourier series for real-valued functions and mean what corresponds to the following special case:

real, ,
real, .

In this case it is

where denotes the real part.

Applications

In physics and engineering , Parseval's theorem is used to express that the energy of a signal in the time domain is equal to its energy in the frequency domain. This is expressed in the following equation:

where is the Fourier transform of with the prefactor omitted and denotes the frequency of the signal.

For discrete-time signals the equation becomes

where is the Discrete Fourier Transform (DFT) of , both of interval length .

See also

credentials

  • Parseval , MacTutor History of Mathematics archive .
  • George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (Harcourt: San Diego, 2001).
  • Hubert Kennedy, Eight Mathematical Biographies (Peremptory Publications: San Francisco, 2002).
  • Alan V. Oppenheim and Ronald W. Schafer, Discrete-Time Signal Processing 2nd Edition (Prentice Hall: Upper Saddle River, NJ, 1999) p 60.
  • William McC. Siebert, Circuits, Signals, and Systems (MIT Press: Cambridge, MA, 1986), pp. 410-411.
  • David W. Kammler, A First Course in Fourier Analysis (Prentice-Hall, Inc., Upper Saddle River, NJ, 2000) p. 74.