MathDB

Interesting thing related to a kind of Fourier sum

Source: Bogdan Stan 2011

October 7, 2019

functionratioinductionalgebra

Problem Statement

Let be a natural number

n,

two

\text{n-tuplets}

of real numbers

a:=\left( a_1,a_2,\ldots, a_n \right) , b:=\left( b_1,b_2,\ldots, b_n \right) ,

and the function

f:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=\sum_{i=1}^na_i\cos \left( b_ix \right)

. Prove that if the numbers of

b

are all positive and pairwise distinct,
a) then,

f\ge 0

implies that the numbers of

a

are all equal. b) if the numbers of

a

are all nonzero and

f

is periodic, then the ratio of any two numbers of

b

is rational.
Marin Tolosi

Back to Problems View on AoPS