Recursive Holographic Method And Recursive Intergraphic Method For Fractal Interpolation Problem

This paper discusses the fractal interpolation problem (FIP),and puts forwards the recursive holographic fractal interpolation method and the recursive intergraphic fractal interpolation method, by combining The recursive iterated function system(RIFS) with the holographic and the intergraphic fractal interpolation methods. It gives a discrete data set with recursive form,and defines the holographic fractal interpolation recursive iterated function system and the intergraphic fractal interpolation recursive iterated function system, proves that the holographic and the intergraphic attractive set-tuples are all existent and unique. It provides the concepts:the fractal interpolation function space, the holographic fractal interpolation transform and the intergraphic fractal interpolation transform. And it proves that the holography and the intergraphic fractal interpolation functions are all existent and unique. It analyses the relation between the recursive intergraphic fractal interpolation method and the recursive holographic fractal interpolation method, obtains the conjugate relation between the intergraphic and the adjoint holographic recursive iterate function system, and obtains the conjugate relation between the intergraphic and the adjoint holographic fractal interpolation transform. It gives the dimension relation between the recursive intergraphic fractal interpolation function and the adjoint recursive holographic fractal interpolation function, proves that their Hausdorff dimensions are same, and that their box dimensions are same. It put forwards the concept and calculation method of the fractal interpolation dimension, and gives the examples of drawing graph and calculating dimension for fractal interpolation function. It provides a normal form system for the general fractal interpolation method with recursive form by giving the recursive holography and the intergraphic fractal interpolation ones, and it improves and expands the theory of fractal interpolation. The recursive holography and the intergraphic fractal interpolation methods, are a kind of normal form system, and they improve and expand the theory of fractal interpolation. These methods can flexibly deal with the similarity relation between part and part, or between part and all. And they can construct the new fractal interpolation function possessing the mutual embedding similarity for multiple sections. Overall,This paper provides a new structure model and a new theoretical support for the research and development of nonlinear mathematics.