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ABSTRACT. In this paper we consider an extremum problem for even periodic functions having small intervals as their supports and subject to restrictions on the sum of the absolute values of their Fourier coefficients.KEY WORDS: extremum problem, periodic functions with small support.With regard to applications in number theory, P. Turan stated the following problem: Problem 1. Let 0 < h _< 1/2. In the class of functions f 6 K1 (h) with the properties

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It was found out that under turning wheel pairs, complete tool failure occurs as a result of ductile fracture of cutting wedge. It was also found out that the process of ductile fracture of a tool is governed by creep of local volumes of tool material adjacent to tool contact areas caused by its severe thermo-mechanical load in machining of hard-drawn surfaces of wheel pairs. The method was proposed to effectively reduce thermo-mechanical load on tool contact areas under machining of wheel pairs, while improving heat transfer from the cutting zone, which provides reduction of thermal stress of cutting wedge and improvement of the efficiency of cutting tool.