Falconer[1] used the relationship between upper convex density and upper spherical density to obtain elementary density bounds for s-sets at H S-almost all points of the sets. In this paper, following Falconer[1], we ...Falconer[1] used the relationship between upper convex density and upper spherical density to obtain elementary density bounds for s-sets at H S-almost all points of the sets. In this paper, following Falconer[1], we first provide a basic method to estimate the lower bounds of these two classes of set densities for the self-similar s-sets satisfying the open set condition （OSC）, and then obtain elementary density bounds for such fractals at all of their points. In addition, we apply the main results to the famous classical fractals and get some new density bounds.展开更多
The theory of integration to mathematical analysis is so important that many mathematicians continue to develop new theory to enlarge the class of integrable functions and simplify the Lebesgue theory integration. In ...The theory of integration to mathematical analysis is so important that many mathematicians continue to develop new theory to enlarge the class of integrable functions and simplify the Lebesgue theory integration. In this paper, by slight modifying the definition of the Henstock integral which was introduced by Jaroslav Kurzweil and Ralph Henstock, we present a new definition of integral on fractal sets. Furthermore, its integrability has been discussed, and the relationship between differentiation and integral is also established. As an example, the integral of Cantor function on Cantor set is calculated.展开更多
Let E be a compact s-sets of R n.The authors define an orthonormal systemΦof functions on E and obtain that,for any f(x)∈L 1(E,H s),the Fourier series of f,with respect toΦ,is equal to f(x)at H s a.e.x∈E.Moreover,...Let E be a compact s-sets of R n.The authors define an orthonormal systemΦof functions on E and obtain that,for any f(x)∈L 1(E,H s),the Fourier series of f,with respect toΦ,is equal to f(x)at H s a.e.x∈E.Moreover,for any f∈L p(E,H s)(p≥1),the partial sums of the Fourier series,with respect toΦ,of f converges to f in L p norm.展开更多
Corresponding to the irreducible 0 - 1 matrix (a<sub>ij</sub>)<sub>n×n</sub>, take similitude contraction mappings <sub>i</sub>j for each a<sub>ij</sub>=1, in R<...Corresponding to the irreducible 0 - 1 matrix (a<sub>ij</sub>)<sub>n×n</sub>, take similitude contraction mappings <sub>i</sub>j for each a<sub>ij</sub>=1, in R<sup>d</sup> with ratio 0【r<sub>ij</sub>【1. There are unique nonempty compact sets F<sub>1</sub>,…, Fn satisfying for each 1in, Fi = ∪<sub>j=1 a<sub>ij</sub>=1</sub><sup>n</sup> <sub>ij</sub>=（F<sub>j</sub>）. We prove that open set condition holds if and only if F<sub>i</sub> is an s-set for some 1in, where s is such that the spectral radius of matrix (r<sub>ij</sub><sup>s</sup>)<sub>n×n</sub> is 1.展开更多
This paper studies the Hausdorff dimensions, the Hausdorff measures of generalized Moranfrontals and the convergence of the Fourier series of functions defined on some generalizedMoran fractals. A general formula is g...This paper studies the Hausdorff dimensions, the Hausdorff measures of generalized Moranfrontals and the convergence of the Fourier series of functions defined on some generalizedMoran fractals. A general formula is given for the calculatinn of the Hausdorff dimensions ofgeneralized Moran fractals and it is proved that their Hausdorff measures are finite positivenumbers under some conditions. In addition, the authors define an orthonormal system offunctions defilled on generalized Moran s-sets (gMs) and discuss the convergence of the Fourierseries, with respect to of each function f(x) E L1(gMs, Hs).展开更多
基金part by the Foundations of the Jiangxi Natural Science Committee(No:0611005),China.
文摘Falconer[1] used the relationship between upper convex density and upper spherical density to obtain elementary density bounds for s-sets at H S-almost all points of the sets. In this paper, following Falconer[1], we first provide a basic method to estimate the lower bounds of these two classes of set densities for the self-similar s-sets satisfying the open set condition （OSC）, and then obtain elementary density bounds for such fractals at all of their points. In addition, we apply the main results to the famous classical fractals and get some new density bounds.
文摘The theory of integration to mathematical analysis is so important that many mathematicians continue to develop new theory to enlarge the class of integrable functions and simplify the Lebesgue theory integration. In this paper, by slight modifying the definition of the Henstock integral which was introduced by Jaroslav Kurzweil and Ralph Henstock, we present a new definition of integral on fractal sets. Furthermore, its integrability has been discussed, and the relationship between differentiation and integral is also established. As an example, the integral of Cantor function on Cantor set is calculated.
文摘Let E be a compact s-sets of R n.The authors define an orthonormal systemΦof functions on E and obtain that,for any f(x)∈L 1(E,H s),the Fourier series of f,with respect toΦ,is equal to f(x)at H s a.e.x∈E.Moreover,for any f∈L p(E,H s)(p≥1),the partial sums of the Fourier series,with respect toΦ,of f converges to f in L p norm.
基金Partly supported by Natural Science Foundation of Chinapartly by Natural Science Foundation of Hubei Province
文摘Corresponding to the irreducible 0 - 1 matrix (a<sub>ij</sub>)<sub>n×n</sub>, take similitude contraction mappings <sub>i</sub>j for each a<sub>ij</sub>=1, in R<sup>d</sup> with ratio 0【r<sub>ij</sub>【1. There are unique nonempty compact sets F<sub>1</sub>,…, Fn satisfying for each 1in, Fi = ∪<sub>j=1 a<sub>ij</sub>=1</sub><sup>n</sup> <sub>ij</sub>=（F<sub>j</sub>）. We prove that open set condition holds if and only if F<sub>i</sub> is an s-set for some 1in, where s is such that the spectral radius of matrix (r<sub>ij</sub><sup>s</sup>)<sub>n×n</sub> is 1.
文摘This paper studies the Hausdorff dimensions, the Hausdorff measures of generalized Moranfrontals and the convergence of the Fourier series of functions defined on some generalizedMoran fractals. A general formula is given for the calculatinn of the Hausdorff dimensions ofgeneralized Moran fractals and it is proved that their Hausdorff measures are finite positivenumbers under some conditions. In addition, the authors define an orthonormal system offunctions defilled on generalized Moran s-sets (gMs) and discuss the convergence of the Fourierseries, with respect to of each function f(x) E L1(gMs, Hs).
