Approximations of Conjugate Functions by Partial Sums of Conjugate Fourier Series with Respect to a Certain System of Ch
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proximations of Conjugate Functions by Partial Sums of Conjugate Fourier Series with Respect to a Certain System of Chebyshev – Markov Algebraic Fractions E. A. Rovba1* and
P. G. Patseika1**
1 Yanka Kupala State University of Grodno, 22 Ozheshko str., Grodno, 230023 Republic of Belarus
Received October 1, 2019; revised December 12, 2019; accepted December 18, 2019
Abstract—In this paper, we study approximative properties of partial sums of a conjugate Fourier series with respect to a certain system of Chebyshev – Markov algebraic fractions. We cite the main results obtained in known works devoted to studying approximations of conjugate functions in polynomial and rational cases. We introduce a system of Chebyshev – Markov algebraic fractions and construct the corresponding conjugate rational Fourier – Chebyshev series. We obtain an integral representation for approximations of the conjugate function by partial sums of the constructed conjugate series. Moreover, we study approximations of the function, which is conjugate to |x|s , 1 < s < 2, on the segment [−1, 1], by partial sums of the conjugate rational Fourier – Chebyshev series. We obtain an integral representation of approximations, establish their estimates, using the considered method, in dependence of the location of the point x on the segment, and find their asymptotic forms with n → ∞. We also calculate the optimal value of the parameter that makes deviations of partial sums of the conjugate rational Fourier – Chebyshev series from the conjugate function |x|s , 1 < s < 2, tend to zero on the segment [−1, 1] at the highest possible rate. The obtained results have allowed us to thoroughly study properties of approximations of the function conjugate to |x|s , s > 1, by partial sums of the conjugate Fourier series with respect to a system of Chebyshev polynomials of the first kind. DOI: 10.3103/S1066369X20090066 Key words: Chebyshev – Markov algebraic fraction, conjugate function, partial sum of the Fourier – Chebyshev series, exact estimate, asymptotic methods.
1. INTRODUCTION V.P. Motornyi (see, for example, [1]–[3]) studied the problem of the best approximations of singular integrals 1 Sf (x) = π
1 −1
dt f (t) √ , t − x 1 − t2
x ∈ (−1, 1),
(1)
by algebraic polynomials. One understands these integrals in the sense of the Cauchy principal value, while f belongs to various classes of functions continuous on some segment. The research techniques used by V.P. Motornyi and his coauthors are based on intermediate approximation methods proposed by N.P. Korneichuk and A.I. Polovina [4]. Note that one can treat the transform Sf (x) as one of ways to determine the function conjugate to the function f defined on the segment [−1, 1]. The superposition Sf (cos x) can be expressed in a certain way in terms of the function that is trigonometrically conjugate to the induced function f (cos x) [3]. The study of approximations of functions defined by series conjugate to the Fourier trigonometric series leads its history since the beginning of the XXth century. It has to
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