2 Oct
2010
2 Oct
'10
3:40 p.m.
On Fri, 1 Oct 2010 03:25:50 +0200
Robin Gareus <robin(a)gareus.org> wrote:
On Oct 1, 2010, at 2:32 AM, Camilo Polymeris wrote:
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
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?