The Variation And Box Dimension Of A Class Of Multi-Parameter Fractal Interpolation Surface

Fractal interpolation function was first proposed by American mathematician M.F.Barnsley in 1986, it was provided a new interpolation method of approximation of experimental date. In recent years, fractal interpolation method has been more widely used in many practical applications. In this paper, we mainly discusses fractal interpolation surface, extending the traditional polynomial function to the trigonometric function, constructs a class of multi-parameter fractal interpolation surface, So, this paper does the research as follows:First of all, this paper describes the emergence and the development of fractal geometry, discusses the basics knowledge of fractal theory, fractal dimension, the basic concepts of iterated function system and basic theory methods of fractal interpolation.Secondly, this paper studies the general form Fij(x,y,z)=ψij(z)+φij(x,y) of iterated function system of fractal interpolation surface, conditioning in the compression of the functionψij (z), proves the existence of attractor, gives existence and uniqueness of this fractal interpolation surface and discusses the continuity condition of the interpolation surface. This paper proves the nature of oscillation and variation of the bivanate continuous function and by the nature of variation estimates the upper bound of box dimension of interpolation surface.Thirdly, extending the polynomial function to the trigonometric function, this paper constructs a class of multi-parameter iterated function system Fij(x,y,z)=sijz+eijz·siny+fij sin x·siny+rij siny+tij sinx+kij,and proves existence and uniqueness of its attractor, and further proofs this attractor is fractal interpolation surface which passes the given interpolation point sets and obtains box dimension formula of interpolation surface. Basing on matlab this paper gives the numerical experiments of interpolation surface and box dimension.