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Estimates for moduli of coefficients of positive trigonometric polynomials. (English) Zbl 1030.42005
The authors reprove the following theorem of G. Szegö [Math. Ann. 96, 601-632 (1927; JFM 53.0465.04)] and E. Egerváry and O. Szász [Math. Z. 27, 641-652 (1928; JFM 54.0314.01)]: If \(T(\theta)= 1+\sum_{k=1}^{n}(a_k\cos k\theta+b_k\sin k\theta)\) is a nonnegative trigonometric polynomial, then \(\sqrt{a^2_k+b^2_k}\leq 2\cos{{\pi}\over{[n/k]+2}}\), where \(1\leq k\leq n\). Moreover, the equality is attained if and only if \(T(\theta)\) is of the form \[ \tau(\theta)\left\{ 1+{{2}\over{2+p}}\sum_{\nu=1}^p\left((p-\nu+1)\cos\nu\alpha+ {{\sin(\nu+1)\alpha}\over{\sin\alpha}}\right) \cos(\nu k(\theta-\psi))\right\}, \] where \(\tau\) is an arbitrary nonnegative trigonometric polynomial of order \(q\), \(\alpha=\pi/(p+2),\) \(p=[n/k],\) \(n=pk+q\), \(0\leq q<k,\) and \(\psi\) is an arbitrary constant.

42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
47A12 Numerical range, numerical radius
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory