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Miklós Schweitzer 1961- Problem 4

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November 22, 2015
college contestsMiklos Schweitzerfunctionmatrixlinear algebraanalysis

Problem Statement

4. Let f(x)f(x) be a real- or complex-value integrable function on (0,1)(0,1) with ∣f(x)∣≤1\mid f(x) \mid \leq 1 . Set
ck=∫01f(x)e−2πikxdx c_k = \int_0^1 f(x) e^{-2 \pi i k x} dx
and construct the following matrices of order nn:
T=(tpq)p,q=0n−1,T∗=(tpq∗)p,q=0n−1 T= (t_{pq})_{p,q=0}^{n-1}, T^{*}= (t_{pq}^{*})_{p,q =0}^{n-1}
where tpq=cq−p,t∗=cp−q‾t_{pq}= c_{q-p}, t^{*}= \overline {c_{p-q}} . Further, consider the following hyper-matrix of order mm:
S=[ETT2…Tm−2Tm−1T∗ET…Tm−3Tm−2T∗2T∗E…Tm−3Tm−2………………T∗m−1T∗m−2T∗m−3…T∗E] S= \begin{bmatrix} E & T & T^2 & \dots & T^{m-2} & T^{m-1} \\ T^{*} & E & T & \dots & T^{m-3} & T^{m-2} \\ T^{*2} & T^{*} & E & \dots & T^{m-3} & T^{m-2} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ T^{*m-1} & T^{*m-2} & T^{*m-3} & \dots & T^{*} & E \end{bmatrix}
(SS is a matrix of order mnmn in the ordinary sense; E denotes the unit matrix of order nn). Show that for any pair (m,n)(m , n) of positive integers, SS has only non-negative real eigenvalues. (R. 19)
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