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Article title PRESENTATION OF THE WEIERSTRASS FUNCTION AND ITS DERIVATIVE CONTINUED FRACTIONS
Authors I.I. Levin, V.V. Selyankin, V.I. Shmoylov
Section SECTION II. CONTROL SYSTEMS, SIMULATION AND ALGORITHMS
Month, Year 04, 2016 @en
Index UDC 517.524
DOI
Abstract It is considered the approach to the study of nondifferentiable functions, based on the methods of thecontinued fractions theory. It was found that the Weierstrass function at rational point x0 accurately represented by terminating continued fractions. Continued fractions for the Weierstrass function are set from the original trigonometric series by means of recursive Rutiskhauzer’s algorithm. The values of divergent series are found byconstruction of the so-called corresponding continued fractions. This method is used in determining the derivative of the Weierstrass function, which can be written as divergent trigonometric series. The values of the derivative of the Weierstrass function at rational points x0 was set by the summation of divergent series. Besides the derivatives are determined by terminating continued fractions containing the same number of units as continued fractions, which determine the values of the Weierstrass function at the same points. The values of divergent in classical sense function of the Weierstrass were found by means of corresponding continued fractions. The results of numerical experiments connected with the study of the properties of the Weierstrass function are listed. In particular, it is shown that the continuity of the Weierstrass function is appeared in extremely small neighborhood of the selected point x0. Also, the dependence of the accuracy of calculation of the Weierstrass function on the number of members of the series which represent this function was set.

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Keywords Weierstrass function; summation of divergent fraction and series; the derivative of Weierstrass function
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