Parseval's theorem

From Academic Kids

In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series.

Although the term "Parseval's theorem" is often used to describe the unitarity of any Fourier transform, especially in physics and engineering, the most general form of this property is more properly called the Plancherel theorem.

Contents

Theorem

The original theorem, cast into a modern form, could be written as follows. Suppose that one has two series:

<math>A(z) = \sum_{n=-\infty}^\infty a_n z^n \,<math>
<math>B(z) = \sum_{n=-\infty}^\infty b_n z^n<math>

in some coefficients <math>a_n<math> and <math>b_n<math> (here taken to be complex, although Parseval apparently only considered real coefficients starting at n=0). (Here, we neglect the question of when the series converge.) The theorem then states:

<math>\sum_{n=-\infty}^\infty a_n b_n^* = \frac{1}{2\pi} \int_{-\pi}^\pi A(e^{i\phi}) B(e^{i\phi})^* d\phi \,<math>

where i is the imaginary unit and * denotes complex conjugation. Parseval actually presented the theorem without proof, considering it to be self-evident.

There are various important special cases of the theorem. First, for A and B the same series, one immediately obtains:

<math>\sum_{n=-\infty}^\infty |a_n|^2 = \frac{1}{2\pi} \int_{-\pi}^\pi |A(e^{i\phi})|^2 d\phi \,<math>

from which the unitarity of the Fourier series follows, where <math>a_n<math> corresponds to the Fourier-series coefficient <math>F_n<math> of the function <math>f(x) = A(e^{ix})<math>.

In particular, one often considers only the Fourier series for real-valued functions (or real A and B for all φ), which corresponds to the special case: <math>a_0<math> real, <math>a_{-n} = a_n^*<math>, <math>b_0<math> real, and <math>b_{-n} = b_n^*<math>. In this case:

<math>a_0 b_0 + 2 \Re \sum_{n=1}^\infty a_n b_n^* = \frac{1}{2\pi} \int_{-\pi}^\pi A(e^{i\phi}) B(e^{i\phi}) d\phi \,<math>

where <math>\Re<math> denotes the real part. (In the notation of the Fourier series article, replace <math>a_n<math> and <math>b_n<math> by <math>a_n / 2 - i b_n / 2<math>.)

See also

References

  • Parseval (http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Parseval.html), 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 (http://home.att.net/~clairnorman/Eight_Mathematical.pdf) (Peremptory Publications: San Francisco, 2002).

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