Ciro Santilli $£ Sponsor €¥ 中国独裁统治 China Dictatorship 新疆改造中心、六四事件、法轮功、郝海东、709大抓捕、2015巴拿马文件邓家贵、低端人口、西藏骚乱

Plancherel theorem

Some sources say that this is just the part that says that the norm of a

L^{2}

function is the same as the norm of its Fourier transform.

Others say that this theorem actually says that the Fourier transform is bijective.

The comment at math.stackexchange.com/questions/446870/bijectiveness-injectiveness-and-surjectiveness-of-fourier-transformation-define/1235725#1235725 may be of interest, it says that the bijection statement is an easy consequence from the norm one, thus the confusion.

TODO does it require it to be in

L^{1}

as well? Wikipedia en.wikipedia.org/w/index.php?title=Plancherel_theorem&oldid=987110841 says yes, but courses.maths.ox.ac.uk/node/view_material/53981 does not mention it.

 Table of contents 135
- The Fourier transform is a bijection in $L^{2}$  link nosplit Plancherel theorem 53
- Every Riemann integrable function is Lebesgue integrable link nosplit Plancherel theorem 7

Ancestors

Incoming links

The Fourier transform is a bijection in $L^{2}$