On the Closure of Certain Assemblages of Trigonometrical Functions

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