On a System of Rational Chebyshev–Markov Fractions

Rouba, Yauheni A.; Potseiko, P. G.; Smatrytski, K.

doi:10.1007/s10476-018-0110-7

Cited by 11 publications

(3 citation statements)

References 8 publications

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“…For any fixed q and even n the relation holds limsup It is known[26, p. 96] that the best uniform polynomial approximation of the considered function possesses the following property:Using similar reasoning, after the necessary transformations from (49) we find that limsup This result is contained in [27] in case of approximation by partial sums of Fourier series with respect to the system of Chebyshev -Markov of algebraic fractions. In particular, when s = 1 we obtain known equality, proved in[28], Now we consider one more result, that follows from the formula (50). Substitution x = sin u leads to the asymptotic estimateIn this relation we put s = 1.…”

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confidence: 94%

On one rational integral operator of Fourier – Chebyshev type and approximation of Markov functions

Potseiko

Rouba²,

Smatrytski³

2020

Journal of the Belarusian State University. Mathematics and Inf

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The purpose of this paper is to construct an integral rational Fourier operator based on the system of Chebyshev – Markov rational functions and to study its approximation properties on classes of Markov functions. In the introduction the main results of well-known works on approximations of Markov functions are present. Rational approximation of such functions is a well-known classical problem. It was studied by A. A. Gonchar, T. Ganelius, J.-E. Andersson, A. A. Pekarskii, G. Stahl and other authors. In the main part an integral operator of the Fourier – Chebyshev type with respect to the rational Chebyshev – Markov functions, which is a rational function of order no higher than n is introduced, and approximation of Markov functions is studied. If the measure satisfies the following conditions: suppμ = [1, a], a > 1, dμ(t) = ϕ(t)dt and ϕ(t) ἆ (t − 1)α on [1, a] the estimates of pointwise and uniform approximation and the asymptotic expression of the majorant of uniform approximation are established. In the case of a fixed number of geometrically distinct poles in the extended complex plane, values of optimal parameters that provide the highest rate of decreasing of this majorant are found, as well as asymptotically accurate estimates of the best uniform approximation by this method in the case of an even number of geometrically distinct poles of the approximating function. In the final part we present asymptotic estimates of approximation of some elementary functions, which can be presented by Markov functions.

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confidence: 94%

On one rational integral operator of Fourier – Chebyshev type and approximation of Markov functions

Potseiko

Rouba²,

Smatrytski³

2020

Journal of the Belarusian State University. Mathematics and Inf

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“…In mathematics, the Fourier series is a way to represent a function as the sum of simple sine waves (Gogoladze and Tsagareishvili, 2016; Raj and Sharma, 2016; Rergis et al, 2018). It decomposes any periodic function or periodic signal into the sum of harmonically related sinusoidal functions (Rouba et al, 2018; Sun and Zhang, 2017; Telyakovskii, 2018). Similarly, a non-sinusoidal vibration waveform can be represented as the sum of sinusoidal vibration waveforms.…”

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confidence: 99%

A novel method for high-frequency non-sinusoidal vibration waveforms with uniaxial electro-hydraulic shaking table based on Fourier series

Wang

Chen

Huang

2020

Journal of Vibration and Control

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In this article, a rotary valve is developed to obtain accurate high-frequency sinusoidal vibration waveforms. Then, a uniaxial electro-hydraulic shaking table controlled by a set of parallel rotary valves is constructed, which can superpose the sinusoidal vibration waveforms. The non-sinusoidal vibration waveforms including triangular vibration waveform, rectangular vibration waveform, sawtooth vibration waveform and trapezoidal vibration waveform are generated by adjusting the spool rotation speed based on the Fourier series. The results show that with one rotary valve, the uniaxial electro-hydraulic shaking table can output accurate high-frequency sinusoidal vibration waveforms and the total harmonic distortion is less than 1% at high vibration frequency. Compared with the standard vibration waveform, the error of the generated vibration waveform is very small. For the rectangular vibration waveform and sawtooth vibration waveform, the error is less than 6%, and for the triangular vibration waveform and trapezoidal vibration waveform, the error is less than 3%. The impacts of the working conditions on the error of the generated vibration waveform are very small. The proposed method for the accurate high-frequency non-sinusoidal vibration waveform is very effective and can be applied in high vibration frequency and different load masses. With the increase of the supply pressure, the amplitude of the generated vibration waveforms increases, while the error changes in a rather narrow range. The amplitude can be adjusted by changing the supply pressure with almost no effect on the accuracy of the vibration waveform.

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“…Преобразуем его. Известно [27], что для алгебраических дробей Чебышева -Маркова первого и второго рода соответственно имеет место представление…”

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The Abel – Poisson means of conjugate Fourier – Chebyshev series and their approximation properties

Potseiko

Rouba

2021

Vescì Akademìì navuk Belarusì. Seryâ fizika-matematyčnyh navuk

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Herein, the approximation properties of the Abel – Poisson means of rational conjugate Fourier series on the system of the Chebyshev–Markov algebraic fractions are studied, and the approximations of conjugate functions with density | x |s , s ∈(1, 2), on the segment [–1,1] by this method are investigated. In the introduction, the results related to the study of the polynomial and rational approximations of conjugate functions are presented. The conjugate Fourier series on one system of the Chebyshev – Markov algebraic fractions is constructed. In the main part of the article, the integral representation of the approximations of conjugate functions on the segment [–1,1] by the method under study is established, the asymptotically exact upper bounds of deviations of conjugate Abel – Poisson means on classes of conjugate functions when the function satisfies the Lipschitz condition on the segment [–1,1] are found, and the approximations of the conjugate Abel – Poisson means of conjugate functions with density | x |s , s ∈(1, 2), on the segment [–1,1] are studied. Estimates of the approximations are obtained, and the asymptotic expression of the majorant of the approximations in the final part is found. The optimal value of the parameter at which the greatest rate of decreasing the majorant is provided is found. As a consequence of the obtained results, the problem of approximating the conjugate function with density | x |s , s ∈(1, 2), by the Abel – Poisson means of conjugate polynomial series on the system of Chebyshev polynomials of the first kind is studied in detail. Estimates of the approximations are established, as well as the asymptotic expression of the majorants of the approximations. This work is of both theoretical and applied nature. It can be used when reading special courses at mathematical faculties and for solving specific problems of computational mathematics.

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On a System of Rational Chebyshev–Markov Fractions

Cited by 11 publications

References 8 publications

On one rational integral operator of Fourier – Chebyshev type and approximation of Markov functions

On one rational integral operator of Fourier – Chebyshev type and approximation of Markov functions

A novel method for high-frequency non-sinusoidal vibration waveforms with uniaxial electro-hydraulic shaking table based on Fourier series

The Abel – Poisson means of conjugate Fourier – Chebyshev series and their approximation properties

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