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The Inequalities for Polynomials and Integration over Fractal Arcs

Published online by Cambridge University Press:  20 November 2018

B. A. Kats
B. A. Kats*
Affiliation:
Kazan State Academy of Architecture and Constructions 1 Zelenaya str. Kazan, Republic of Tatarstan 420043 Russia, email: kats@ksaba.kcn.ru
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Abstract

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The paper is dealing with determination of the integral ${{\int }_{\gamma }}\,f$ along the fractal arc $\gamma $ on the complex plane by terms of polynomial approximations of the function $f$. We obtain inequalities for polynomials and conditions of integrability for functions from the Hölder, Besov and Slobodetskii spaces.

Keywords

26B1528A80

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 44 , Issue 1 , 01 March 2001 , pp. 61 - 69
Copyright
Copyright © Canadian Mathematical Society 2001

References

[1] Dyachkov, A.M., On the existence of the Stieltjes integral. Dokl. Acad. Nauk. (2) 350 (1996), 158–161.Google Scholar
[2] Feder, I., Fractals. Moscow, 1991.Google Scholar
[3] Harrison, J., Stokes’ theorem for nonsmooth chains. Bull. Amer.Math. Soc. (2) 29 (1993), 235–242.Google Scholar
[4] Harrison, J. and Norton, A., Geometric integration on fractal curves in the plane. Indiana Math. J. (2) 40 (1991), 567–594.Google Scholar
[5] Harrison, J. and Norton, A., The Gauss-Green theorem for fractal boundaries. Duke Math. J. (3) 67 (1992), 575–586.Google Scholar
[6] Kats, B. A., The gap problem and integral along non-rectifiable curve. Izv. VUZov. Mathematics. 5 (1987), 49–57.Google Scholar
[7] Kats, B. A., The Stieltjes Integral Along Fractal Curve. To appear.Google Scholar
[8] Kats, B. A., The Riemann boundary value problem on open Jordan arc. Izv. VUZov.Mathematics. 12 (1983), 30–38.Google Scholar
[9] Kats, B. A., The Riemann boundary value problem on closed Jordan curve. Izv. VUZov, Mathematics. 4 (1983), 68–80.Google Scholar
[10] Mandelbrot, B. B., The fractal geometry of nature. W. H. Freeman and Co., San Francisco, 1982.Google Scholar
[11] Matsaev, V. I. and Solomyak, M. Z., On condition of existence of the Stieltjes integral. Mat. Sb. (4) 88 (1972), 522–535.Google Scholar
[12] Seifullaev, R. K., The Riemann boundary value problem on non-smooth open arc. Mat. Sb. (2) 112 (1980), 147–161.Google Scholar
[13] Stein, E. M., Singular integrals and differential properties of functions. Moscow, Mir Publishers, 1973.Google Scholar
[14] Young, L. C., An inequality of the Hölder type, connected with Stieltjes integration. ActaMath., Uppsala, (3–4) 36 (1936), 251–282.Google Scholar
[15] Young, L. C., General inequalities for Stieltjes integrals and the convergence of Fourier series. Math. Ann. (4) 115 (1938), 581–612.Google Scholar