华中科技大学 硕士学位论文 测度维数的局部化与函数图象盒维数的若干研究 姓名：杜玉坤 申请学位级别：硕士 专业：基础数学 指导教师：朱三国 2010-05-22 ? ? ?. ? ?, ?x?, ? ?.?, ?, ?. ? ?x, ?dim?x?dim(x), ? ?dim(x)?dim?, ?. ?, ?. ?f : I 0,1 R ?, ?(f,I) ?f ?, (f,I) = (x,f(x) : x 0,1. ?,?f?x?, ?f?a,b?, ? ?, ?(?)?. ?, ?: ? ?dimB(f,I) dimB(g,I), ? dimB(f + g,I) = dimB(f,I). ?, ?: 1. ?dimB(f,I) = dimB(g,I)?, ?dimB(f + g,I)? 2. dimB(f + g,I) maxdimB(f,I),dimB(g,I) ? 3.?dimB(f,I) dimB(g,I), ?dimB(f + g,I)? ?, ?f,g ?0,1?.?dimB(f,I),dimB(g,I) , dimB(f + g,I)?dimB(fg,I)?dimB(f/g,I)?. ?, ?. ? ?. ?; ?; ?;?; ? I ? Abstract This paper considers the localization of dimensions of measures and the Minkowski di- mension of graphs of functions. For the study of the localization of dimensions of measures, the localization of dimensions of measures at the point x is fi rstly given, by which we can defi ne dimensions of measures itself. At the same time, which we can show by dimensions of sets. Then we defi ne measures of x, then the localization of dim at the point x is given, which we call dim(x). In part three we shall consider the relations between the localized dimension dim(x) and the original dimension dim. For the study of the Minkowski dimension of graphs of functions, we fi rstly give the defi nitions of graphs of functions, Let f : I 0,1 R be a continuous function. We denote by (f,I) the graph of f, (f,I) = (x,f(x) : x 0,1. then the of f at the point x is given, at the same time the of f on a,b is given. For any a continuous function, the upper (under) Minkowski dimension of graphs of functions is defi ned. As we all know, if dimB(f,I) dimB(g,I), then dimB(f + g,I) = dimB(f,I). It is natural to ask the following questions : 1. what can we say about dimB(f + g,I) when dimB(f,I) = dimB(g,I)? 2. Is it possible that dimB(f + g,I) maxdimB(f,I),dimB(g,I)? 3. what can we say about dimB(f +g,I) when dimB(f,I) dimB(g,I)? In part four, we introduce the main aim of this paper. Let f,g be continuous functions on 0,1. We are to establish some basic results on the relationships among dimB(f,I),dimB(g,I) and dimB(f + g,I),dimB(fg,I),dimB(f/g,I) .Also, we shall consider some polynomial functions and convergent series of continuous functions. The three questions listed above will be answered. Key words:localization of dimensions; measures; Hausdorffdimensions; Minkowski di- mension; graphs of functions II ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1? 1.1? ?, ?, ?, ? ?,?,?, ?, ?, ? ?, ? ?, ?110. ?, ?, ?, ? ?.?1113. ?,? ?1420. ?, ? ?. ?, ?,?2129. ? ?. ?, ? ?, ?. ?, ?, ? ?. 1.2? ?: (1)?Rd?, ?Rd?, ? dim H = supdim H(x) : x K. (2)?Rd?, ?Rd?, ? dim B = supdim B(x) : x K. ?: (1)?f,g ?0,1 ?, ?dimB(f,I) 6= dimB(g,I), ? dimB(f g) = maxdimB(f,I),dimB(g,I). ?, ?, ?dimB(f,I) dimB(g,I), ? dimB(f + g,I) = dimB(f,I). ?. (2)?f,g ?0,1?, ?dimB(f,I) dimB(g,I), ? dimB(f g,I) = dimB(f,I). 1 ? (3)?n 1, ak(0 k n)?, ?f : 0,1 R, ? Pn(f)(x) = anfn(x) + an1fn1(x) + + a1f(x) + a0. ?Pn(f)?, ?: ?f : 0,1 R+?, ?: dimB(Pn(f),I) dimB(f,I). ?0 k n,?ak 0, ?. (4)?f : 0,1 R ?g : 0,1 0,1 ?. ?f ?s1? ?, ? dimB(fg,I) limsup 0 (2 s1 logVg,(0,1) log ). ?, ?dimB(g,I) = 2 s2, ? dimB(fg,I) 2 s1s2. (5)?f,g : 0,1 R ?, ?f,g 0, ? dimB(fg,I) = dimB(logf + logg,I). ?, ?dimB(f,I) dimB(g,I), ? dimB(fg) = dimB(f,I). (6)?f(x),g(x) : I R+?, ? dimB(f/g,I) = dimB(logf logg,I). ?, ?dimB(f,I) dimB(g,I), ? dimB(f/g) = dimB(f,I). 1.3? ?, ?;? ?; ?. ?, ?, ? ?, ?. ?. 2