For $x \in [0,\pi/2]$ and $n \geq 1$ consider the cosine polynomial $P(x)=a_0\cos(x)+a_1\cos(3x)+\ldots+a_n\cos((2n+1)x)$, where the $a_i$'s are positive numbers such that $a_0>a_1>\ldots>a_n$ . Can anything be said about the location of the roots of P ?
Roots of a cosine polynomial
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polynomials
1 Answers
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Assume without loss of generality that $a_0=1$. The roots of $P$ must all lie to the right of the smallest positive root of $$ \cos x+\cos(3\,x)+\dots+\cos(2\,n+1)x=\frac{\sin(2\,(n+1)\,x)}{2\sin x}, $$ which is $\dfrac{\pi}{2(n+1)}$.
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0Can you give more details ? Thanks – 2017-02-17
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0You say "The roots of P must all lie to the right of the smallest positive root of ...". Why is it so ? – 2017-07-23