Yudin, V. A. An extremum problem for distribution functions. (English. Russian original) Zbl 0961.42003 Math. Notes 63, No. 2, 316-320 (1998); translation from Mat. Zametki 63, No. 2, 279-282 (1998). Let us consider the functions of the form \[ f( t)= \sum_{\nu \in Z} \widehat {f}_\nu e^{i\nu t},\quad \sum_{\nu \in Z} |\widehat {f}_\nu|< + \infty . \tag{1} \] The following result is proved. Theorem 1. For any \(n \in N\) and any nonnegative function \(f \not\equiv 0\) of the form \((1)\) whose Fourier coefficients satisfy the conditions \[ \widehat f_\nu = {1\over 2\pi} \int_{0}^{2\pi} f(t)e^{i\nu t} dt \leq 0,\quad |\nu|\geq n,\quad \nu \in Z, \] the following inequality is valid: \[ \text{meas}\{ t \in [0,2\pi):f(t) > 0\} \geq {2\pi\over n}. \] The assertion of this theorem is sharp for any \(n \in N\). This is shown by the example of the \(2\pi - \)periodic function: \[ f(t) \equiv f_n (t) = \begin{cases} |\sin nt |,& |t|\leq {\pi\over n},\\ 0,& {\pi\over n} \leq |t|\leq \pi. \end{cases} . \] Reviewer: Boris I.Golubov (Dolgoprudny) Cited in 2 Documents MSC: 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series 42A20 Convergence and absolute convergence of Fourier and trigonometric series 42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) Keywords:extremum problem; distribution function; Fourier coefficients; absolute convergence PDFBibTeX XMLCite \textit{V. A. Yudin}, Math. Notes 63, No. 2, 316--320 (1998; Zbl 0961.42003); translation from Mat. Zametki 63, No. 2, 279--282 (1998) Full Text: DOI References: [1] N. I. Chernykh,Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],88, 71–74 (1967). [2] N. I. Chernykh,Mat. Zametki [Math. Notes],2, No. 5, 513–522 (1967). [3] V. V. Arestov and N. I. Chernykh, in:Approximation and Function Spaces, Proc. Conf. Gdansk, 1979, North-Holland, Amsterdam (1981), pp. 25–43. [4] A. G. Babenko,Mat. Zametki [Math. Notes],35, No. 3, 349–356 (1984). [5] B. F. Logan,SIAM J. Math. Anal.,14, No. 2, 253–257 (1983). · Zbl 0513.42013 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.