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Explicitly, with $B_n$ the centered cardinal B-spline of order $n$, Sommerfeld
shows that, for large $n$ and all $x$,
$$%
B_n(x/2h) \approx (k2h/\sqrt{\pi})\exp(-k^2 x^2),
$$%
with
$$%
kh\sqrt{n} = \sqrt{3/2}.
$$%
(Sommerfeld studies the error distribution of the sum of $n$ random
errors with uniform distribution $[-h .. h]$. Thus the quantity $y$ near the
end of his paper equals $B_n(x/(2h))/(2h)$, since $\int B_n(x/a)dx = a$.)
%
The convergence is quite good. E.g., for $n=10, 100, 400$, $B_n(0)$ is
$$%
0.43041776895944,\ \ 0.13799020407550,\ \ 0.06907291282945,
$$%
while the formula gives the approximations
$$%
0.43701937223683,\ \ 0.13819765978853,\ \ 0.06909882989427 .
$$%
%
Sommerfeld's argument is deliberately cavalier near the end, with [Maurer, L.;
\"Uber die Mittelwerthe der Functionen einer reellen Variabeln;
Math. Ann.; 47; 1896; 267;] cited as the place where the rigorous argument can
be found. A rigorous argument has also beeen supplied by
%
Lei (qv)
%.
Lei reports that Maurer's paper also handles the delicate estimate for the
error in approximating $B_n$ by a Gaussian. Roughly speaking, the
approximation rate is, uniformly,
$$%
1/(n \sqrt n).
$$%
%
Sivakumar mentions that his colleague, Joel Zinn, pointed out to him
that Sommerfeld's result is a special case of
Theorem 1 on page
533 of Feller's book (I believe it is volume II); it appears in the
section on `Expansions for densities' in Chapter XVI (Expansions
related to the Central Limit Theorem). He asked Professor Zinn to
explain it to him in the special case of interest to us. If he has
understood him correctly, this special case is:
Let $A_n:=[-1/2,1/2]/(\sqrt{n/12})$, $f_n(x):=\sqrt{n/12}g_n(x)$,
where $g_n$ = $n$-fold convolution of the function $\chi_{A_n}$ with
itself. Then
%
$f_n(x)-{1\over\sqrt{2\pi}}\exp(-x^2/2)=o(1/\sqrt n)$ uniformly in
$x$.
%
Maurer's estimate would give $O(1/n)$ instead.
\bye