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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: \[ f(t) \equiv f_n (t) = \begin{cases} |\sin nt |,& |t|\leq {\pi\over n},\\ 0,& {\pi\over n} \leq |t|\leq \pi. \end{cases} . \]

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.)
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[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.
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[5] B. F. Logan,SIAM J. Math. Anal.,14, No. 2, 253–257 (1983). · Zbl 0513.42013
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