in January we had a brief discussion about the first rigorous proof
of the inversion formula for Fourier transforms. It was referred to
[Pringsheim 1907]. I think, this is a good source, but I would like
to give a few additions and also to ask further questions. (The
following text can be compiled in LATEX, but I hope that the
formulas can be understood also without compilation.)
1.) [Pringsheim 1907] is mainly an account on the history of the
inversion formula
\begin{equation}\label{star} \frac{f(x+0)+f(x-0)}{2} =
\frac{1}{\pi}\int_0^{\infty}
\int_{-\infty}^{\infty}f(t)\cos[u(t-x)]dtdu \end{equation}
in the 19th century. For a first rigorous proof,
if under very special conditions, Pringsheim refers to Carl Neumann
[1881]. Pringsheim gives in his 1907 paper only the announcement of
the proof of (\ref{star}) under very general presuppositions. The
proof himself appeared in 1910 with a supplement in 1912.
2.) It is a nontrivial question, why, on the one hand, the problem of
the representation of functions by Fourier series played an important
role for the development of analysis in the 19th century, and why, on
the other hand, the ``natural'' extension of this problem, a rigorous
proof of (\ref{star}), was more or less neglected.
3.) One must differentiate between different versions of the
inversion formula. The complex version
\begin{equation}\label{twostar} \frac{f(x+0)+f(x-0)}{2} =
\frac{1}{2\pi}\int_{-\infty}^{\infty}\
int_{-\infty}^{\infty}f(t)e^{iu(t-x)}dtdu
\end{equation} is --- according to Burckhardt [1914] --- due to
Cauchy. During the 19th century the integral concerning the
variable $u$ was silently considered as its principal value
$\lim_{c \to \infty} \int_{-c}^c \cdots du$. In this case,
(\ref{twostar}) is eqivalent to (\ref{star}). In the 20th
century the view was taken, that all integrals are in the Lebesgue
sense, see Samuel Bochner's book [1932]. But, as it seems, the
question about general presuppositions for the validity of
(\ref{twostar}) in the case of improper Riemannian integrals was only
discussed very superficially. In this case, (\ref{twostar}) is true
for a single point $x$, if (\ref{star}) holds, and if
\begin{equation}\label{threestar}\lim_{c \to \infty} \int_0^c
\int_{-\infty}^{\infty}f(t)\sin[u(t-x)]dtdu \end{equation}
exists. There is a remark in [Courant 1930, 67 (footnote)], that
the continuity of $f(t)$ in $t=x$ ($f \in L^1$ being
piecewise smooth) is sufficient for (\ref{threestar}). By the aid of
Pringsheim's methods I can show, that (\ref{threestar}) is met, if $f
\in L^1$, and if
$$\lim_{T \to 0}\int_{-T}^T\left|\frac{f(x+t)-f(x)}{t}\right|dt =
0.$$
Can anybody give me more information about this problem?
References:
Bochner, S. 1932. Vorlesungen \"uber Fouriersche Integrale. Leipzig:
Akad. Verl.-Ges.\\
Burckhardt, H. 1914. Trigonometrische Reihen und
Integrale (bis etwa 1850). Encyklop\"adie der Mathematischen
Wissenschaften II, 1, 2, Artikel IIA12, Leipzig: Teubner, 1904--
1916.\\
Courant, R. (\& Hilbert, D.) 1930. Methoden der Mathematischen
Physik, Bd. 1. (2. Aufl.). Berlin: Springer.\\
Neumann, C. 1881. Ueber die nach Kreis-, Kugel- und
Zylinder-Functionen fortschreitenden Entwicklungen. Leipzig:
Teubner.\\
Pringsheim, A. 1907. Ueber das Fouriersche Integraltheorem.
Jahresbericht der DMV, 16, 2--16.\\
--- 1910. Ueber neue G\"ultigkeitsbedingungen f\"ur die Fouriersche
Integralformel. Mathematische Annalen, 68, 367--408.\\
--- 1912. Nachtrag zu der Abhandlung: Ueber neue
G\"ultigkeitsbedingungen \dots. Mathematische Annalen, 71, 289--298.\\
Hans Fischer