Plancherel theorem

In mathematics, the Plancherel theorem is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum.

A more precise formulation is that if a function is in both L(R) and L(R), then its Fourier transform is in L(R), and the Fourier transform map is an isometry with respect to the L norm. This implies that the Fourier transform map restricted to L(R) ∩ L(R) has a unique extension to a linear isometric map L(R) → L(R). This isometry is actually a unitary map. In effect, this makes it possible to speak of Fourier transforms of quadratically integrable functions.

单词	Plancherel theorem
释义	Plancherel theorem 英语百科 Plancherel theorem In mathematics, the Plancherel theorem is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum. A more precise formulation is that if a function is in both L(R) and L(R), then its Fourier transform is in L(R), and the Fourier transform map is an isometry with respect to the L norm. This implies that the Fourier transform map restricted to L(R) ∩ L(R) has a unique extension to a linear isometric map L(R) → L(R). This isometry is actually a unitary map. In effect, this makes it possible to speak of Fourier transforms of quadratically integrable functions.
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