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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.

##### MSC:
 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