Plancherel Theorem
Jump to navigation
Jump to search
Theorem
Let $n \in \Z_{>0}$.
Let $\map \BB {\R^n}$ be the Borel $\sigma$-algebra on $\R^n$.
Let $\lambda^n$ be the Lebesgue measure on $\R^n$.
For each $p \in \closedint 1 \infty$, let:
- $\map {L^p} {\R^n} := \map {L^p} {\R^n, \map \BB {\R^n}, \lambda ^n}$
be the $L^p$ space.
For all $f \in \map {L^1} {\R^n} \cap \map {L^2} {\R^n}$, we have:
- $\norm {\map \FF f}_2 = \norm {\map {\FF ^{-1} } f}_2 = \norm f_2$
where:
- $\map \FF f$ is the Fourier transform of $f$
- $\map {\FF ^{-1} } f$ is the inverse Fourier transform of $f$
- $\norm \cdot_2$ denotes the $L^2$ norm
Proof
This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Source of Name
This entry was named for Michel Plancherel.
Sources
- 2010: Lawrence C. Evans: Partial Differential Equations (2 ed.): $4.3.$ Transform Methods