PROCEEDINGS OF THE
NATIONAL ACADEMY OF SCIENCES Volume 13
February 15, 1927
Number 2'
ON THE CLOSURE OF CERTAIN ASSEMBLAGES OF TRIGONOMETRICAL FUNCTIONS By NORBmR WIZNZR DIPARTMUNT OF MATHEMATICS, MASSACHUSETTS INSTITUTE or TUCHNOLOGY
Communicated December 2, 1926
THJROREM: Let [O < Xl <X2 < * *,k+l-Xk 2tA] be a closed set of functions over (0,7r). If COS X,,X
..
Is
P(x)
=
ak COS X*X,
let
Eak c [(X)]2dx,(1 [
C being dependent only on the sequence {Xk1. Let I I2A \2 1 , I < D < i3 s+ (A+2-A-X-,uk
[k =1, 2, 3, *.**.]
Then the sequence cos Aunx is closed over (0,7r). Proof. Let so (x) be any summable function of summable square over (0, 7r). Let dx < e. ak cos
JL
Xkx]
The ak's may be so chosen because of the closure of {cos Xkx}. Let us consider 1 (x) = Eak(cos XkX-COS AkX). 1
We have
x/2 - coI F(u) cos ux du,
24t' (x) sin x where
F(u) =
akFk(U)
o
MA THEMA TICS: N. WIENER
28
Fk(U)
= sgn Qt
xk +
P PROC. N. A. S.
2)-sgn(u-Xk-2)-sgn(U-Ak +
)
+ sgn (u -uk2 It follows that
[F(u) du
4[(X) 2 sin/ < 27r(2+ 1
+A D)D
ak2
from which we conclude that
fW[1 (x)]dx<3-(2 +
+)DEak2
Hence
J, [op (x)-- Eak cos IIkx] dx < 3e
+
<3e+- ~2 +
<3+
2+
A
(2 +
A
) DCf
)DEak2 [
ak Cos
)DC[f [(x)]2dx + 2Ve
xJdx
[((x) 2dX +1e
<.n + of [((x)]2dx, where 0 < 1, and n can be made as small as is desired by taking e small enough. We can hence choose our ak's in such a way that
[,p(x) ]2dx, COS ~lAkx dxdX1., D(x) -Eak J 1 where 0, < 1 and is independent of s. We may now choose a set of numbers bk in such a way that 1n 2 nrakcos/x ak cos kX bk cos IkX dx J (Px) L
E
>E
< 0l
f[(x)-Eakcos Akx1
dx <
#12J [V(X) ]2dx.
By repeating this process, we obtain a trigonometric polynomial in the functions cos IkX such that
Pn(x)
VOL. 13, 1927
MA THEMA TICS: G. Y. RAINICH
f [5a(x)-PP(X)]2 dx . tln
29
[( (x)] dx.
This proves the closure of the set cos ukx. As a special case, let Xk = k-1, C = 2/Xr. Then D can be any number less than
+ -2* In particular, if 9
l
<83,X2+4
3 2
the sequence cos vx, cos (1 + v)x,
cos
(2 + v)x,.
is closed over (0, 7r). I The terms at the beginning may have to be modified slightly because Fk(u) is to vanish for negative u. This is allowed for in the following estimate.
ON A T YPE OF LORENTZ TRA NSFORMA TIONS By G. Y. RAINICH DEPARTMgNT OF MATHEMATICS, UNIVERSITY OF MICHIGAN
Communicated December 23, 1926
1. Under a general Lorentz transformation we understand a linear transformation on the variables x, y, z, t which leaves the form X2 + y2 + Z2 t2 (1) invariant. Usually one type of special Lorentz transformations is considered corresponding to the case in which only t and one of the other variables is affected. The importance of such transformations in the Theory of Relativity is very well known. In what follows another special type of Lorentz transformations is considered, which so far as I know has not been studied, and which seems to have important applications in the study of radiation. 2. We arrive at the desired transformations in a simple way if we look for linear transformations which leave (1) invariant and, at the same time, do not change the vector 1, 0, 0, 1 and the plane x = t, y = 0. The last two conditions give four relations on the 16 coefficients of the general linear homogeneous transformation in addition to the 10 relations which express the condition that (1) remains invariant. The transformation -
