Three Geometric Forms And Free Control Theory Of Fractal Interpolation

This paper has discussed FIP, analyzed three geometrical forms of FIP and established the theory of free control of three fractal interpolation forms. Firstly, the concepts of similarity and self-similarity of fractal set were generalized, a series of concepts was posed and the dimension theorem between global and local of metric self-similar and quasi-metric self-similar sets were obtained. Secondly, the concepts of fractal curve (defined as non-piecewise smooth) and fractal curve (defined as piecewise smooth curve)were distinguished using smoothness such that the definition of fractal curve becomes strict and easy to judge, also three geometric forms of fractal interpolation were introduced, that's to say, y - value form, 6 -value form and p - value form, distinguished the concepts of interpolation function and interpolation curve, gave the general definition of FIP, disposed unifily of three kinds of FIP and proved three kinds of fractal interpolation curves were continuous parameter curves under affine IFS, holographic IFS and intergraphic IFS. Thirdly, band decomposition of orthogonal coordinate space and X - type decomposition of polar coordinate space were defined and proved that there exists local double Lipschitz mapping between them. The dimension relationship of three kinds interpolation curve of y - value, 9 - value and p- value are also discussed under affine IFS, holographic IFS and intergraphic IFS. Lastly, the free control theory of fractal interpolation was established the theory which included Sheng Zhongping control theory and range control theory, discussed the shape control theory and range control theory of fractal interpolation curve of y - value, 0 - value and p - value, obtained a series of interpolation results and gave general shape control theory. At the same time, several examples of drawing of interpolation curve of y -value, 0- value and p-value under affine IFS, holographicIFS and intergraphic IFS respectively are given. The main results were as follows:1. Generalizing and classifying the concept of similarity and self-similarity of fractal set, the following dimension theory of metric self-similarity was obtained.Theorem A Let D is a bounded closed set on Hn, 9[] s acompact space on D. Let that {wl}=l be an IFS on D, and w is lifting mapping on D Let G C 3[-D] be a fixed set, thenG = w(G) = \J Wi(G)1=1denote G, = Wi(G).(l)If G is a (Holographic) measure self-similarity set or (Intergraphic) measure self-similarity set, then for any arbitrary i(l < i < N),there will be :dim// Gi = dim// G, dim Gl = dim G.(2)If G is a (Holographic) quasi-measure self-similarity set or (Intergraphic) quasi-measure self-similarity set, also let Gn,(t = 1,2, ...,) for nonsingular subset of D, GJt, (t ?1, 2, ...,k) for degeneration subset. Then for any arbitrary nonsingular subset G, (t = 1,2,..., /c), there will be :dim// GJ( = dim// G, dimB GJt = dims G.2. (j/ - value, 9 - value, p - value) three forms interpolation curve of afrme IFS, holographic IFS. and intergraphic IFS (macro function of Lip-schitz continuous), all are hyperbolic have been proved. Also intergraphic I.FS(macro function general continuous) and intergraphic IFS topological conjugate have been proved, correspondence theorem was deduced. Using general intergraphic IFS method, local structure of fractal interpolation curve and free control for whole form were realized, mostly the result was as follows.Theorem B Let \w,}n v ?, ,. Tt0 f um T .1 li be a intergraphjc IFS of FIP. LetUfi,)} interpolation data group,{(, }n . . ...g. lo d llllv l U 111~terpolation data group, [a, b] is the interpolation interval, and D {(, 77) G /?2 : a f 6} is the interpolation domain.Let{oi}=i are indeependent variable compression ratio, {d are factor compression ratio.Let g.t is a macro function in interpolation interval of a Lipschitz continuous, Hl is a Lipschitz constant. Let Ml is the Lipschitz constant of the micro function t, L is the Lips...