0.1 Statement of theorem
Plancherel’s Theorem states that the unitary Fourier transform of functions (the Lebesgue-integrable functions (http://planetmath.org/Integral3)) on extends to a unitary isomorphism on (the square-integrable functions).
Thus, the following two fundamental properties hold for the Fourier transform on functions :
The equalities are as elements of ; in terms of pointwise functions, the equalities hold almost everywhere on .
The Fourier transform preserves norms:
0.2 Extension of the Fourier transform to
are defined as usual, and can be obtained as the limit of .
In the one-dimensional case, a common sequence of approximating sequences to take is ; in that case we have
The inverse Fourier transform can be obtained in a similar way to , using approximating functions :
0.3 Note on different conventions
Here, we have used the convention for the Fourier transform that denotes “ordinary frequency”, i.e. the exponential contains factors of . Another common convention has replaced by denoting the “angular frequency”, with factors occurring not in the exponent, but as multiplicative constants. In this case property (i) above still holds, but property (ii) will not hold unless the multiplicative constants in front of the forward and inverse Fourier transform are chosen properly.
- Folland Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications, second ed. Wiley-Interscience, 1999.
- Katznelson Yitzhak Katznelson. An Introduction to Harmonic Analysis, second ed. Dover Publications, 1976.
- Wiki “ http://en.wikipedia.org/wiki/Continuous_Fourier_transformFourier transform ”, Wikipedia, The Free Encyclopedia. Accessed 22 December, 2006.
|Date of creation||2013-03-22 16:29:00|
|Last modified on||2013-03-22 16:29:00|
|Last modified by||stevecheng (10074)|