linuxaudio.org

On Fri, 1 Oct 2010 03:25:50 +0200

Robin Gareus  wrote:
>

quoted text
> On Oct 1, 2010, at 2:32 AM, Camilo Polymeris wrote:

>

> >> The original Fourier Transform as invented by the smart French

> >> guy of the same name does operate on continuous (as opposed to

> >> sampled) data from -inf to +inf.

> >

> > I understand Fourier invented the Fourier Series "only", anyone

> > knows who generalized it to FT?

>

> not me,  but google did:

>

> "Fourier’s initial series lacked the precision of a function, and

> Dirichlet and Riemann would later express the series as a formal

> integral." [1]

> 

mmm, that sounds ambigous to me... what exactly does it mean?

"Integral" here is used as synonimous of "solution of the heat-PDE"? 
What is sure is that Dirichlet was the first to prove a convergence

theorem for Fourier series, and for this he had to first give a new

(both more precise and more general) definition of function. He gave

the famous Dirichlet-function (charateristic of rationals) as an

example of a function - with his definition - not satisfying the

hypotheses of his theorem. 
Then Riemann, Cantor and others proved other convergence theorems with

other hypotheses. Cantor actually developed his set theory for the need

to describe the set of some points regarding Fourier series (something

like the admissable discontinuities, can't remember ATM).

Also I seem to remember that Riemann's integral sprouted from

some problems he had in Fourier analysis.
Unfortunately I still know nothing about the history of the

FT...
renato
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