The Localization Of Dimension Of Measures And The Minkowski Dimension Of Graphs Of Functions

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 measuresμat the point x is firstly given, by which we candefine dimensions of measures itself. At the same time, which we can show by dimensionsof sets. Then we define measures ofμx, , then the localization of dimμat the point x isgiven, which we call dim?μ(x). In part three we shall consider the relations between thelocalized dimension dim?μ(x) and the original dimension dim?μ.For the study of the Minkowski dimension of graphs of functions, we firstly give thedefinitions of graphs of functions, Let f : I≡[0, 1]→R be a continuous function. Wedenote byΓ(f,I) the graph of f,then theδ? of f at the point x is given, at the same time theδ? of f on [a,b] is given. Forany a continuous function, the upper (under) Minkowski dimension of graphs of functionsis defined. As we all know, ifIt is natural to ask the following questions :four, we introduce the main aim of this paper. Let f,g be continuous functions on [0, 1]. Weare to establish some basic results on the relationships among dimBΓ(f,I), dimBΓ(g,I)and . Also, we shall consider somepolynomial functions and convergent series of continuous functions. The three questionslisted above will be answered.