Geodesic
The Inequalities for Polynomials and Integration over Fractal Arcs
Canadian mathematical bulletin, Tome 44 (2001) no. 1, pp. 61-69

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-2001-007-x
Mots-clés : 26B15, 28A80 
Kats, B. A. The Inequalities for Polynomials and Integration over Fractal Arcs. Canadian mathematical bulletin, Tome 44 (2001) no. 1, pp. 61-69. doi: 10.4153/CMB-2001-007-x
@article{10_4153_CMB_2001_007_x,
     author = {Kats, B. A.},
     title = {The {Inequalities} for {Polynomials} and {Integration} over {Fractal} {Arcs}},
     journal = {Canadian mathematical bulletin},
     pages = {61--69},
     year = {2001},
     volume = {44},
     number = {1},
     doi = {10.4153/CMB-2001-007-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2001-007-x/}
}
TY  - JOUR
AU  - Kats, B. A.
TI  - The Inequalities for Polynomials and Integration over Fractal Arcs
JO  - Canadian mathematical bulletin
PY  - 2001
SP  - 61
EP  - 69
VL  - 44
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2001-007-x/
DO  - 10.4153/CMB-2001-007-x
ID  - 10_4153_CMB_2001_007_x
ER  - 
%0 Journal Article
%A Kats, B. A.
%T The Inequalities for Polynomials and Integration over Fractal Arcs
%J Canadian mathematical bulletin
%D 2001
%P 61-69
%V 44
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2001-007-x/
%R 10.4153/CMB-2001-007-x
%F 10_4153_CMB_2001_007_x

[1] [1] Dyachkov, A.M., On the existence of the Stieltjes integral. Dokl. Acad. Nauk. (2) 350 (1996), 158–161. Google Scholar

[2] [2] Feder, I., Fractals. Moscow, 1991. Google Scholar

[3] [3] Harrison, J., Stokes’ theorem for nonsmooth chains. Bull. Amer.Math. Soc. (2) 29 (1993), 235–242. Google Scholar

[4] [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] [5] Harrison, J. and Norton, A., The Gauss-Green theorem for fractal boundaries. Duke Math. J. (3) 67 (1992), 575–586. Google Scholar

[6] [6] Kats, B. A., The gap problem and integral along non-rectifiable curve. Izv. VUZov. Mathematics. 5 (1987), 49–57. Google Scholar

[7] [7] Kats, B. A., The Stieltjes Integral Along Fractal Curve. To appear. Google Scholar

[8] [8] Kats, B. A., The Riemann boundary value problem on open Jordan arc. Izv. VUZov.Mathematics. 12 (1983), 30–38. Google Scholar

[9] [9] Kats, B. A., The Riemann boundary value problem on closed Jordan curve. Izv. VUZov, Mathematics. 4 (1983), 68–80. Google Scholar

[10] [10] Mandelbrot, B. B., The fractal geometry of nature. W. H. Freeman and Co., San Francisco, 1982. Google Scholar

[11] [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] [12] Seifullaev, R. K., The Riemann boundary value problem on non-smooth open arc. Mat. Sb. (2) 112 (1980), 147–161. Google Scholar

[13] [13] Stein, E. M., Singular integrals and differential properties of functions. Moscow, Mir Publishers, 1973. Google Scholar

[14] [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] [15] Young, L. C., General inequalities for Stieltjes integrals and the convergence of Fourier series. Math. Ann. (4) 115 (1938), 581–612. Google Scholar

Cité par Sources :